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Investigation of the Spin Competition in the Al6 Cluster Isaac L Huidobro-Meezs, and Emilio Orgaz J. Phys. Chem. C, Just Accepted Manuscript • Publication Date (Web): 05 May 2017 Downloaded from http://pubs.acs.org on May 14, 2017

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The Journal of Physical Chemistry

Investigation of the Spin Competition in the Al6 Cluster Isaac L. Huidobro-Meezs and Emilio Orgaz∗ Departamento de F´ısica y Qu´ımica Te´ orica, Facultad de Qu´ımica, Universidad Nacional Autónoma de México, Cd. Universitaria, CP 04510, México, D.F. México E-mail: [email protected]

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Abstract Small sized Aluminum clusters exhibit a possible spin competition at finite temperature. For this reason, it is difficult to standard electronic structure investigations to determine the properties of the ground state. In the particular case of Al6 , a nonmagnetic ground state has been predicted at 0K. However a singlet-triplet intercrossing system could occur at laboratory temperatures in agreement with Stern-Gerlach experiments. We have thus investigated, by means of nonadiabatic transition state theory, the possibility of such singlet-triplet spin competition. We have determined the possible crossing points at the potential energy surface and then identified the most favorable minimum-energy crossing points. For these points, we evaluated the spin-orbit matrix elements, transition probabilities and rate constants for the singlet-triplet equilibrium at different temperatures using the Landau-Zener and weak-coupling formulas. The predicted equilibrium at finite temperatures is consistent with previous ab initio molecular dynamics. We point out the importance of such evaluations in determining physical properties of these kind of systems.

Introduction Magnetic properties in metallic clusters has been an active field of research for their potential technological applications. Several theoretical and experimental efforts have been devoted to describe the physical and chemical properties for this kind of clusters in the past decades. 1–9 In recent investigations of the magnetic properties of small sized metallic aggregates we have pointed out the coexistence of different spin states for a particular cluster. 10,11 Observed magnetic moments have been explained considering a Maxwell-Boltzmann distribution of various isomers exhibiting different spin multiplicities. The possibility of a dynamic spin equilibrium have been mentioned as the underlying process through a spin-orbit coupling mechanism. Interaction between spin-adiabatic electronic states is often neglected in most electronic structure calculations since the nonadiabatic coupling coefficients are very small 2


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for most of the studied systems. However for the description of transitions between electronic states having different spin multiplicities these coefficients need to be evaluated. 12 Coupling between spin-adiabatic states is due to relativistic effects as spin-orbit, spin-spin and higher order interactions. These interactions can be calculated from relativistic four-component electronic structure methods, two-component techniques as the Douglas-Kroll-Hess Hamiltonian 13–15 or introducing relativistic effects as a perturbation. 12,16,17 Perturbative methods are suitable for systems where the small component relativistic wave function is expected to have a small contribution to the total energy and to the wave function having the benefit to be less computationally demanding. In addition, nonadiabatic dynamics are often described with surface hopping algorithms 18–21 or with the ab initio multiple spawning method 22 but these approaches tend to have a heavy computational cost. Nonadiabatic transition state theory (NA-TST) is a statistical approach which assumes that the spin transition occurs at a minimum energy crossing point (MECP) in the crossing seam. This scheme has proven to produce results with no significant difference from more sophisticated methods and reproduce in a satisfactory manner the experimental results in various systems. 22–25 Cox et al 26 have shown via Stern-Gerlach experiments that small sized Aln clusters exhibit a non zero magnetic moment. However, many theoretical investigations of the electronic structure point in several cases to a non-magnetic ground state. Recent detailed investigations of such aluminum clusters indicate that a possible thermal competition between different spin states could occur provided that the spin-orbit mechanism be operative. 11 It is important to note that these results didn’t take into account the explicit spin transition probabilities in order to provide a thorough analysis of the involved physical process. In the present report, we show our results for the particular case of Al6 cluster where a singlet-triplet equilibria seems to be present. By means of NA-TST and taking into account the probability of spin transition mediated by a spin-orbit coupling, we have investigated the singlet-triplet equilibrium on the Al6 cluster. In what follows we summarize the theoretical

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framework and methodological details, and afterwards we describe and discuss our findings.

Theoretical and computational methods Spin-orbit coupling The Breit-Pauli Hamiltonian is one of the most used perturbative methods and includes relativistic effects to the order 1/c2 where c is the speed of light. This includes scalar relativistic corrections (Darwin and mass-velocity), spin-orbit, spin-spin and orbit-orbit interactions. Only the spin-orbit and the spin-spin contributions couple states with different spin multiplicities. Spin-spin effects are often neglected since they tend to be at least one order of magnitude smaller than the spin-orbit effect. 27 The spin-orbit coupling term in the Breit-Pauli operator is 17,27,28 BP HSO

# " X e2 h ¯ X Za = (ri,a × pî ) · sî − (rij × pî ) · (sî + 2sˆj ) 3 2m2 c2 i,a ria i,j6=i

(1)

where me and e are the electron mass and charge respectively and Zα is the nuclear charge. The spin and momentum operators are sˆ and pˆ respectively. Greek subscripts are used for nuclei and roman subscripts for electrons. The spin-orbit Hamiltonian in eq. 1 can be divided in one and two electron terms. For heavy elements the one-electron contribution is dominant and the two-electron contribution is often neglected or averaged using effective nuclear charges. 12,28,29 For the lightest elements the two-electron contribution can not be neglected. The spin-orbit coupling matrix element used to calculate the transition probability between states with spins S and S ′ and magnetic quantum number MS and MS′ can be defined as 2 HSO

=

S X

MS =−S

′

S X

MS′ =−S ′

4

ˆ SO |S ′ M ′ i|2 |hSMS |H S


(2)

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The spin-orbit coupling operator is a first order tensor operator and its matrix elements can be obtained with the Wigner-Eckart theorem 30 employing the Clebsch-Gordan (CG) coefficients. This matrix elements follow the same selection rules as the CG coefficients and are zero if both states are singlets, |∆S > 1| or |∆MS > 1|. Using this selection rules in the case of a singlet-triplet crossing we obtain the spin-diabatic matrix HN R + HSO ES h0, 0|HSO |1, −1i h0, 0|HSO |1, 0i h1, −1|HSO |0, 0i ET 0 h1, 0|H |0, 0i 0 ET SO h1, 1|HSO |0, 0i 0 0

langle0, 0|HSO |1, 1i 0 0 ET

(3)

where ES and ET are the energies of the singlet and triplet states respectively. The energy of the spin-diabatic states can be obtained upon diagonalization of the matrix in eq. 3.

Nonadiabatic transition state theory Kinetics of nonadiabatic transitions between different electronic states can be described according to the nonadiabatic transition state theory (NA-TST). 24,25,31 This statistical approach takes into account the transition probability and the energy of the MECP which describes the height of a potential energy barrier. The equation for the canonical rate constant in this approach can be written as 22,23,32 QM ECP (T ) k(T ) = hQR (T )

Z

∞

Ptrans (E)e−βE dE

(4)

0

where h is the Planck’s constant, β is 1/kB T and E is the energy associated to the reaction coordinate that describes a branching-plane on the electronic surface. QM ECP and QR are the MECP and reactants partition functions respectively, the transition probability (Ptrans ) is a function of the internal energy on the reaction coordinate. The canonical rate constant depends on the form of the Laplace transform of Ptrans as can be easily seen in eq. 4, hence 5



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the employed approach for Ptrans will define the form of the rate constant. Probabilities for spin transitions mediated by spin-orbit coupling can be evaluated with the approximated double-passage Landau-Zener (LZ) formula 23,33 q |HSO |2 µ −2π h LZ ¯ |∆G| 2(E−EM ECP ) Ptrans =2 1−e

(5)

where HSO is the spin-orbit matrix element defined in eq. 2 and ∆G is the gradient difference between the potential energy surfaces (PES). EM ECP is the energy of the MECP referred to the reactants energy and µ is the reduced mass of the degree of freedom in the reaction coordinate orthogonal to the crossing seam. Double-passage LZ formula can be approximated q 2 µ SO | . The LZ formula is defined when 1−pLZ is small, where pLZ = exp −2π |H h|∆G| ¯ 2(E−EM ECP )

only at energies higher than EM ECP and does not account for tunneling and interference between reaction pathways. In order to consider these quantum effects, we employed the weak-coupling (WC) formula 23,34–36 which is based on the Wentzel-Kramers-Brillouin theory. The WC approach is given by WC 2 Ptrans = 4π 2 HSO

# " 2µ|∆G|2 31 2µ 32 × Ai2 − (E − EM ECP ) h ¯ 2 G|∆G| h ¯ 2 G4

(6)

where G is the geometric mean of the PES gradients defined as G = (G1 G2 )1/2 and Ai(x) is the Airy function. Canonical rate constants for the LZ double-passage formula (kLZ (T )) and WC (kW C (T )) have been obtained before 37,38

kLZ (T ) =

p QM ECP π 3/2 s √ kB T eEM ECP /kB T hQR 2 epsilon0

p 1 π 3/2 s QM ECP 1 √ kB T eEM ECP /kB T 1 + e 12s2 (kB T epsilon0 )3 (8) hQR 2 epsilon0 2 1/2 3/2 4H µ and s = ¯hSO G∆G It can be noted that the WC formula rate constant

kW C (T ) = with ǫ0 =

∆G 2GHSO

(7)

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is an improvement of the LZ rate constant since; β3 kW C (T ) = kLZ (T ) 1 + 0.5exp Big) 12s2 ǫ30

!

(9)

where the term in parenthesis is the tunneling factor. Finally for the evaluation of the rate constant the partition functions need to be evaluated. Reactants and MECP partition functions can be obtained from the rigid rotor and harmonic oscillator approximations. The partition function for reactants can be obtained with ease with most electronic structure programs through a geometry optimization and vibrational analysis since it is a stationary point in the 3N − 6 dimensional PES. However the MECP partition function can not be easily obtained since the MECP’s are not stationary points on the PES. MECP are stationary points on the 3N − 7 crossing seam surface, 24,39 therefore we need to introduce an effective Hessian which contains information of the two PES’s via their Hessians (H1 and H2 ) and their gradients (G1 and G2 ) . The effective Hessian matrix is defined as ′ Hef f =

|G1 |H2 ± |G2 |H1 ∆G

(10)

where ∆G is the gradient difference defined as |G1 −G2 |. For sloped intersections (G1 G2 > 0) the minus sign is used meanwhile for peaked intersections (G1 G2 < 0) the plus sign is ′ used. The reaction coordinate or branching plane is projected out of Hef f leaving only the

information of the 3N − 7 frequencies of the crossing seam. 39 The projected Hessian (Hef f ) is then, Hef f =

! ! ∆G ∆G ′ Hef 1 − 1− |∆G| |∆G|

(11)

Computational details Using density functional theory (DFT) we carried out symmetry unconstrained geometry optimizations for ground state (GS) geometries, transition states (TS) and MECP’s for Al6 clusters. Optimizations for ground-state geometries and transition states were per7



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formed with Gaussian 09. 40 In order to investigate crossing points between the singlet and triplet PES we performed intrinsic reaction coordinate (IRC) calculations for every transition state found. Every calculation was performed with the meta-GGA TPSS 41 exchange and correlation functional. The employed basis set was cc-PVTZ. 42 This calculation scheme was chosen since the cohesion energies obtained have the smallest deviation compared with CCSD(T)/QZVP and reproduces experimental data for different aluminum clusters. 11,43 For the singlet state our calculations were performed in both the restricted and unrestricted Kohn-Sham scheme (RKS and UKS respectively). In order to break spin degeneracy for the singlet UKS functions we used a mix of HUMO and LUMO molecular orbitals as initial guess. MECP search 44 was performed with the algorithm implemented in GAMESS 45 code. Single point calculations and vibrational analysis with the effective hessian were done to assure that this geometry was a minimum in the crossing seam and not a saddle point. Spin-orbit coupling constants for the MECP were obtained with the Breit-Pauli spin-orbit operator with the MolSOC 28,46,47 code which takes advantage of the Wigner-Eckart theorem for the evaluation of the matrix elements in a DFT framework. 28,46 The GLOWfreq 48 code was employed to perform a vibrational analysis of the MECP using the effective hessian defined in eq. 11. Rate constants for the non-adiabatic transition between singlet and triplet states were obtained using NA-TST combined with the double passage Landau-Zener and weak-coupling transition probabilities.

Results and discussion Total electronic energies for the GS geometries and TS in singlet and triplet states are summarized in Table 1. Fig. 1 shows the relative energies of these geometries and the found MECP’s. On the left of Fig. 1 we can see the singlet TS and GS as the right side shows the triplet TS and GS. We found that the RKS singlet transition state (1 gT S−R ) structure is

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a crossing-point (CP) in the PES. The IRC obtained from the 1 gT S−R (Available in the SI, Fig. S1) shows that the reaction coordinate connecting the singlet ground state with this TS has no CP’s. The same case occurs for the IRC computed from the triplet transition state 3

gT S (Fig. S2). We observe that the 1 gT S−U geometry has an electronic energy higher than the corre-

sponding triplet (unrelaxed geometry). This means that a forced CP exists in the IRC connecting the 1 gT S−U and the 1 ggs structures as can be seen in Fig.1. We found that the IRC of 1 gT S−U has two crossing points, each one corresponding to the UKS and RKS singlet surfaces as seen in Fig. S3. Both crossings exhibit an energy below the 1 gT S−R structure. For any system having an even number of electrons, the crossing seam has N − 2 degrees of freedom. 49,50 This means that there are, in principle, an infinite number of geometries in which the energies of the singlet and triplet states are degenerate. Hence the CP found with the IRC are not necessarily the MECP. We used these geometries as starting points for a MECP search. In general electronic transitions do not occur at an unique point of the PES. The transition between electronic states can happen at any point of the surface because the transition probability is not necessarily zero thus the evaluation of this probabilities is required in order to consider the possible reaction routes and quantum effects for the electronic transition. The statistical approach of NA-TST gives us the possibility to evaluate rate constants only from the information obtained from the GS geometries, the MECP and the transition probability. The structures corresponding to the MECP have been found and are displayed in Fig. 2. The CP gCP −1 and gCP −3 correspond to intersections of the UKS singlet and triplet PES. For the RKS singlet surface the CP’s found were gCP −2 and gCP −4 . The relative energies of each point are reported in Table 1. The local properties of the CP are important to discard a possible saddle point on the crossing seam. The gCP −1 structure is a transition state on the restricted singlet surface and on the triplet surface. IRC calculations for this CP (Fig. S4) show CP at higher energies.

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It is important to note that this CP connects an unrestricted singlet state with the triplet state so the defined trajectory is not the correct reaction coordinate for the nonadiabatic transition. IRC calculations and scans on the surface of gCP −2 (Fig. S5 and S6 respectively) show that this point is not a MECP but merely a saddle point on the crossing seam since there are other CP’s on the surface defined by the IRC. Structures gCP −1 and gCP −2 have very similar geometries showing a displacement of the perpendicular axis of a near octahedral structure. The vibrational analysis of these structures showed that they are not MECP’s but saddle points on the crossing seam. Structures gCP −3 and gCP −4 have no imaginary frequencies in the projected crossing seam indicating that these geometries represent the MECP of the UKS and RKS surfaces respectively. The gCP −3 structure is close to the triplet ground state geometry (3 ggs ) while gCP −4 is close to the singlet ground state geometry (1 ggs ). These MECP’s have a low relative energy and involve small displacements from the ground state geometries. The gCP −4 IRC is shown in Fig. S7. The reaction coordinate that involves the MECP for the RKS surface (gCP −4 ) includes geometries that are close to the UKS crossing seam as we can find a CP near the gCP −3 geometry. This is a clear indication that the spin transition can undergo by either of the proposed routes favoring the gCP −3 mechanism for its lower activation energy. To define the reaction coordinate for gCP −3 we need to establish the minimum energy path from the gCP −3 to the GS geometries of each spin multiplicity. Since gCP −3 is a TS on the unrestricted singlet surface we performed an IRC to connect this geometry with the 1 ggs structure. In order to find the corresponding trajectory from the gCP −3 to 3 ggs structures, we carried out a PES scan through the possible geometries taking as an advantage their similarities. The resulting reaction coordinate is plotted in Fig. 3. This surface shows the path for the nonadiabatic transition between the singlet and triplet state in Al6 . The splitting of the nonadiabatic states presented on the inset of Fig. 3 at the MECP gCP −3 is 0.08kcal/mol. This splitting is less significant at other geometries in the reaction coordinate. The reaction coordinate also includes the path that involves the RKS surface. The most

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important characteristic for the spin transition that arises from this connection between the RKS and UKS surfaces is the possibility of various transitions on the same reaction coordinate. It is important to say that the RKS-singlet and UKS-singlet are two different electronic states that have the same spin multiplicity and therefore the transition from a triplet state to the singlet state can undergo by any of the CP’s contained in these surfaces. We computed the triplet spin density along the reaction coordinate of gCP −3 (Fig. S8). As the geometry changes from 1 ggs to 3 ggs there is an increase in the β spin density on the atoms located on the perpendicular axis of the octahedral-like structure. The structural change favors an almost octahedral geometry with an slightly elongated axis and a distorted plane for the triplet state. The geometry modifications occurring along the reaction path produce an energy decrease for the triplet state on the HOM O − α molecular orbital (MO) when we go from the 1 ggs to the 3 ggs geometry (Fig. S9). We also see an increase on the energy of the unoccupied molecular orbitals. On the 1 ggs geometry the α HOM O − 4 and HOM O − 5 molecular orbitals are almost degenerate (∆E = 0.00028au), the corresponding β orbitals do not show this near degeneracy. As we advance on the reaction coordinate to the 3 ggs we found a switch on the spin degeneracy. For this the gCP −3 geometry has a complete break down of MO spin degeneracy. At 3 ggs the α HOM O − 4 and HOM O − 5 are not degenerate but their corresponding β orbitals do present this near degeneracy (∆E = 0.00034au). The electronic structure of the Al6 cluster explains the increase on the β spin density on the perpendicular axis of the almost octahedral structure 3 ggs since the corresponding MO gets lower in energy. Metallic clusters at their ground state geometries can present Jahn-Teller (JT) and pseudo Jahn-Teller (PJT) effects breaking orbital symmetry. 51–54 It has been shown that DFT methods treat the JT and PJT contributions in geometry optimizations reproducing the breaking of high symmetry configurations in metallic clusters 52,55 however, spin-orbit effects could compensate JT contributions producing high symmetry structures. 56,57 JT distortions are

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present in both GS and MECP geometries in Al6 clusters. As can be seen in Table 2, SO coupling has a low energy contribution in Al6 (14cm−1 = 0.04kcal/mol on the gCP −3 geometry). This contribution is not large enough to compensate JT and PJT effects on the orbital energies proving that SO coupling won’t heavily distort neither GS nor MECP geometries. The transition probabilities and hence the form of the canonic rate constant depends on the spin-orbit matrix elements and the gradient difference between the involved PES. These contributions can be described as a geometric factor of the PES and a non-adiabatic coupling factor. Transition probabilities were obtained with the Landau-Zener (LZ) and the weak-coupling (WC) formulas given in eq. 5 and 6. The transition probabilities for the gCP −3 route have are plotted in Fig. S10. Weak-coupling describes the tunneling probability through the energy barrier of the MECP and provides a better description of the transition near the energy of the MECP. 22,32 Oscillations on the WC transition probability are due to the interference of different reaction paths and are essential for the description of the spin transition in Al6 since we have shown that there are various probable routes. The parameters involved in the computation of the rate constants are summarized in Table 2. The WC rate constant contains a tunneling factor which can be seen as a correction to the Landau-Zener formula as described in eq. 9. At high temperatures it’s easy to show that kW C ≈ 1.5kLZ . The behavior at low temperatures is dependent on the gradients of the PES’s and the magnitude of the LZ rate constant. The tunneling factor in the WC canonical rate constant depends only on the temperature and the shape of the PES’s via the gradients. The low temperature behavior depends mostly on the tunneling correction; when the argument in the exponential factor is bigger than one the tunneling factor increases and tunneling through the barrier can be the dominant contribution to the rate constant. We have found that for Al6 clusters this is the case. Landau-Zener rate constants don’t describe this quantum effects. LZ and WC rate constants for the gCP −3 route are displayed in Fig. 4. This plot shows

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that at low temperature, rate constants for the non-adiabatic transition are determined by the tunneling factor. Electronic transitions occur prior a rearrangement of the nuclear coordinates, the involved physical process during the transition is the rearrangement of the electron density when we go from one electronic state to another. At low temperatures tunneling through the barrier increases the rate constants for Al6 , therefore the transition is limited by electronic motion when the tunneling effect is significant since the transition can not happen faster than the electronic rearrangement. The gCP −3 route rate constant at 275K is 4.92 × 1010 s−1 , this is around seven orders of magnitude larger than the rate constant involving the gCP −4 route being 1.77 × 103 s−1 (see Fig. 5). Rate constants for both proposed routes show the same behavior at low temperatures. The gCP −3 rate constant is mediated by tunneling at temperatures lower than 50 K meanwhile for gCP −4 this behavior appears at temperatures lower than 25 K. As it can be appreciated in Table 2, the HSO matrix element for the gCP −3 path is 1400 times larger than that involving the gCP −4 route. Meanwhile the gradient dependent factors (G and ∆G) are lower in the gCP −3 path. The order of magnitude of the rate constants strongly depends on the activation energy, spin-orbit coupling and hessian factors. In the canonical WC rate constant, the tunneling factor depends only on the local shape of the PES. If we have a smooth surface in the branching plane the gradient difference and the geometric mean gradient becomes small which, in addition to a large spin-orbit coupling, benefits the tunneling through the barrier since the electronic states are closer in energy near the MECP. In order to estimate the tunneling contribution through the barrier we employed the quotient of the WC rate constant and the LZ rate constant kW C /kLZ . This quotient for both proposed routes is summarized in Table 3. The results show that both routes have a very important tunneling factor at low temperatures. The tunneling is systematically larger for the gCP −3 route, since the coupling between states due to spin-orbit effects is significantly larger in the gCP −3 .

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In Table 3 we show the triplet state population at different temperatures. These populations were computed taking into account the equilibrium constant defined as

Keq =

[3 Al6 ] kS−T = [1 Al6 ] kT −S

(12)

We have an increased triplet population at low temperatures since this is the ground state found at T=0K. However at higher temperatures we have a significant mixture of both spin states. Ab initio molecular dynamics (AIMD) studies on Al6 show a thermodynamical mixture of both spin states at finite temperatures 11 indicating a significant triplet contribution (55% at 600K). In the present NA-TST approach, the triplet population is found close to these values found with AIMD. However, AIMD has no mechanism to differentiate if a transition is possible or not and makes use of two independent dynamics for this comparison.

Conclusions We described the transition between different spin multiplicity states in Al6 by obtaining the MECP and the spin-orbit coupling matrix elements for their use in NA-TST. We propose two different paths for the electronic transition at finite temperature. The spin transition can undergo by any of the proposed paths with different relative importance due to the magnitude of their rate constants. The obtained rate constants suggest that, at least at a macroscopic level, the rate constant is going to be closer to that of the gCP −3 mechanism. We have shown that the low activation energy along with the low gradient difference of the PES produces a nonadiabatic transition rate constant controlled by tunneling through the barrier at low temperatures in Al6 clusters. It is evident now the need to consider unrestricted states for the singlet wave-functions since there is no reason to impose spin-degeneracy and the results obtained are dramatically different (seven orders of magnitude in the rate constant). Both proposed routes give arise 14


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to a mixture of spin states with a favorable triplet population at high temperature. The rate constants obtained are limited by the statistical approach of NA-TST given that they are defined by the MECP and do not consider the different paths that could occur on other points of the crossing seam. Even with these limitations and the lack of consideration of metastable configurations at finite temperature, our results are consistent with the evidence of a non zero magnetic moment experimentally observed by Cox et al. 26 using Stern-Gerlach experiments along with ab initio theoretical studies. 11 A mixed spin state could also be found in other metallic clusters given that there exists a significant spin-orbit coupling and an available crossing point on the PES. This report showcases that the evaluation of the spin-orbit interaction is essential for the description of metallic clusters since it allows the interaction of spin adiabatic states and may alter the cluster geometry, changing their properties as experimental data shows. We expect that this investigation will motivate further experimental studies on the temperature dependent magnetic properties of small sized Al clusters.

Supporting Information Available SI includes the IRC calculations for 1 gT S−R , 3 gT S and 1 gT S−U TS (Fig. S1, S2 and S3 respectively). IRC path for gCP −1 (Fig. S4), IRC and PES scan for gCP −2 CP (Fig. S5 and S6). IRC calculation for the gCP −4 MECP is also available (Fig. S7). We also include spin densities and molecular orbitals for the ground state geometries and the gCP −3 MECP (Fig. S8 and S9). Transition probability for the gCP −3 route is presented in Fig. S10. Cartesian coordinates for the described geometries available on SI. This material is available free of charge via the Internet at http://pubs.acs.org/.

Acknowledgement The authors would like to thank DGTIC-UNAM for providing us part of the computing 15



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facilities and the financial support provided by PAIP-FQUNAM and DGAPA-UNAM under grant number IN113116.

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Table 1: Relative energies and vertical excitation energies between singlet and triplet states for ground states, transition states and minimum energy crossing points (MECP). ∆E is the vertical excitation energy. All energies are reported in kcal/mol and are referred to the triplet ground state 3 ggs energy. Bold letters indicate transition states in the corresponding surface. Geometry 1 ggs 3 ggs 1 gT S−R 1 gT S−U 3 gT S gCP −1 gCP −2 gCP −3 gCP −4

M1 0.228 8.856 6.258 8.800 9.019 6.612 6.276 4.394 2.396

M1-U M3 ∆EM 1−M 3 0.226 8.448 8.220 2.473 0.000 -8.856 4.670 6.254 -0.004 1.490 0.004 -8.796 2.108 0.783 -8.236 3.794 3.805 -2.807 4.691 6.314 0.038 1.027 1.022 -3.373 0.669 2.436 0.040

∆EM 1U −M 3 8.222 -2.473 1.584 -1.486 0.779 0.011 1.624 -0.005 1.767

Table 2: Parameters for the calculation of the non-adiabatic transition rate constant for the Al6 cluster. The reduced mass (µ) for both routes is 26.98 au. MECP gCP −3 gCP −4

G (au.) 7.99×10−3 9.23×10−3

|∆G| (au.) 0.0169 0.0234

HSO (cm−1 ) 14.00 0.01

23

EaS→T


kcal mol

0.80 2.17

EaT →S

kcal mol

1.03 2.40


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Table 3: Comparition between the Landau-Zener (LZ) and the weak-coupling (WC) rate constants at different temperatures and triplet populations at different temperatures. T(K) 25 50 75 100 200 300 600 800

cp3 cp3 kwc /kLZ 2.09×1010 11.63 2.24 1.73 1.52 1.51 1.50 1.50

cp4 cp4 kwc /kLZ 3.60×109 9.53 2.16 1.71 1.52 1.51 1.50 1.50

% TCP 3 99.99 99.39 96.75 92.73 79.82 74.38 69.71 68.79

%TCP 4 99.08 91.47 82.95 76.62 66.73 64.86 64.73 65.03

Figure 1: Relative energies of the ground states (GS), transition states (TS) and minimum energy crossing points(MECP) found. The employed notation describes stationary points or crossing points with subscripts and denotes the belonging surface (singlet or triplet) as a left superscript. All energies in kcal/mol are referred to the triplet ground state 3 ggs energy at the TPSS/cc-PVTZ level of theory. The singlet GS geometry 1 ggs is the same for the restricted and unrestricted Kohn-Sham approach. Two colored lines indicate crossing points on the PES with the exception of 1 ggs .

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Figure 2: MECP structures. gCP −1 and gCP −3 are found at crossing points in the UKS singlet surface meanwhile gCP −2 and gCP −4 in the RKS singlet surface. Bond lengths in ˚ A and selected angles in degrees. Full cartesian coordinates available in the SI.

Figure 3: Branching plane for the spin transition through the MECP gCP −3 . All energies are referred to the energy of 3 ggs . Nonadiabatic states are shown with dotted lines on the figure inset. The splitting of the adiabatic states its equal to 0.08 kcal/mol. Step size for the reaction coordinate is 0.05 a.u.

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Figure 4: Rate constant for the gCP −3 MECP. The rate constant obtained with the weakcoupling transition probabilities is shown with solid lines. Landau-Zener rate constant is shown with dotted lines. A logarithmic scale is used.

Figure 5: Weak coupling rate constants for the gCP −3 and gCP −4 minimum energy crossing points. gCP −3 in solid lines and gCP −4 with dotted lines. A logarithmic scale is used.

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