Neutral, Anionic, and Cationic Manganese Dimers through Density

Article pubs.acs.org/JPCA

Neutral, Anionic, and Cationic Manganese Dimers through Density Functional Theory Matteo Barborini* S3 Research Center, CNR-NANO, Via Campi 213/a, 41125 Modena, Italy S Supporting Information *

ABSTRACT: The manganese dimer is the only first row transition metal dimer that presents an antiferromagnetic 1Σ+g ground state bonded by a van der Waals interaction. The various density functional theory (DFT) investigations devoted to the study of the ground state of this molecule and of its anionic and cationic states, are usually based on the generalized gradient approximation, giving results which contradict the experimental observations. In this work, we describe the overall spectroscopic properties of the neutral, cationic and anionic manganese dimers with DFT, focusing on understanding the effects of the percentage of Hartree−Fock exchange and of the different exchange-correlation functionals, on the relative stability of the various potential energy curves. For each of the three species we classify the ferromagnetic and antiferromagnetic states, studying the vertical detachment energies, the ionization energies and the electron affinities. In this way, we locate a hybrid exchange-correlation functional able to give for all the three species, results comparable with the experimental measurements and with previous accurate multiconfigurational calculations, defining a more accurate density functional theory approach to study larger charged or neutral manganese clusters.

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INTRODUCTION The 6S ground state of the manganese atom is characterized by a doubly filled 4s orbital and half-filled 3d orbitals that are responsible for its high magnetic moment. In pure manganese Mnn clusters, the unpaired 3d electrons give rise to local magnetic moments and to a variety of molecular ferromagnetic or antiferromagnetic states. In particular, different experimental and density functional theory (DFT) investigations based on the generalized gradient approximation (GGA), have been devoted to the observation and modeling of a possible ferro- to ferrimagnetic transition, which has been seen to depend on both the size of the clusters and on their total charges.1−7 Unfortunately, already for the manganese Mn2 dimer, the GGA results contradict both the experimental measurements8−14 and the multiconfigurational calculations,15−23 showing the necessity to find a better, but still computationally feasible approach, to describe the chemistry of these particular metallic systems. In the manganese dimer,24 the interaction between two 6 S(Mn) atoms gives rise to a series of lowest energy Σ+ states characterized by a van der Waals bond.17,18,21,22,25 The lowest of these states, which corresponds to the ground state, has been recognized by both experimental measurements and multiconfigurational calculations15−23 to be antiferromagnetic, due to the localization of the 3d electrons on each atomic site with opposite spin. In a first attempt to measure the bonding energy of the manganese dimer through a magnetic-sector mass spectrometer, its weak interaction was found to have a bonding energy of about 0.13(13) eV.8 The characterization of its magnetic states arrived later through electron spin resonance (ESR)9 experiments: depositing manganese atoms into a krypton or xenon matrix at © XXXX American Chemical Society

low temperatures, it was measured that the ground state had an antiferromagnetic character, with a bond length of about 3.4 Å11,12 and a magnetic coupling constant J, estimated through Landé’s interval rule, around −8(4) cm−1.9 Later optical spectroscopy experiments predicted a value of J around −10.3(6) cm−1,10 that was seen to vary in temperature between −8 to −11 cm−1 through ultraviolet and visible absorption spectra.13 The vibrational frequency obtained through Raman spectroscopy was found to have different values: from 74.6 cm−1 26 to 68.1 cm−1 13 and (59, 68) cm−1.14 Due to the spin localization and to the characteristic van der Waals bonding, the correct quantum chemistry description of the ground state of this dimer requires the recovery of both the static and the dynamical electronic correlations, and in the literature, only multiconfigurational approaches have been reported to be fully successful in describing its spectroscopic properties.15−23 As anticipated above, the DFT results have been often contradictory, depending on the exchange-correlation functional used, and sometimes affected by convergence issues.27,28 At first it was reported, that both GGA and hybrid functionals predicted the ground state to be ferromagnetic, even though, while for the hybrid functionals it was found to correspond to the 11Σ+u state, for the GGA functionals it was found to be the 11Πu state.27,29−33 Moreover, with the hybrid functionals only a 1Σ+g state with ⟨S2⟩ = 0 (spin uncontaminated) was identified as a candidate for the antiferromagnetic state, but displayed an higher energy with respect to the dissociation limit,27,30,31,34 with a rather short bond Received: December 12, 2015 Revised: February 18, 2016

A

DOI: 10.1021/acs.jpca.5b12169 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A (1.61−1.70 Å) and high ωe (631−806 cm−1).31 This behavior of the hybrid functionals in describing the antiferromagnetic ground state was later revisited in the work by Yamanaka et al.35 In this investigation, the authors showed that with hybrid functionals it was possible to select a broken-symmetry (BS) solution, with ⟨S2⟩ ≈ 5, which approximated the 1Σ+g state, and which lay below the ferromagnetic 11Σ+u state. The authors also tested a projection procedure to decontaminate the BS solution and benchmarked different exchange-correlation energy functionals, but unfortunately still missed out in obtaining an approach that could describe the overall spectroscopic properties compatibly with the experimental measurements. The purpose of this investigation is thus to find a DFT approximation able to treat the interaction in the Mnn clusters in a more accurate, but still feasible, way, obtaining results comparable with the experimental measurements and able to recover the overall chemical properties of these systems, concentrating in particular on the energetic ordering of the states and on their relative energy differences. For this reason, we extend our study from the neutral manganese molecule to its anionic and cationic states, calculating the ionization energy and the electron affinity and classifying the various potential energy curves as a function of the Hartree−Fock exchange and of different exchange-correlation functionals. Because of the van der Waals nature of the chemical bond in the manganese dimer, we also study the effect of the addition of non-local (NL) dispersion contributions based on VV10 van der Waals functional36 and of atom-triplewise dispersion (D3) corrections37 with Becke−Johnson (BJ) damping38,39 in the description of the spectroscopic properties of this molecule. Moreover, we use the extended Borken symmetry (EBS) procedure40−49 to partially decontaminate the approximate BS antiferromagnetic ground state. This investigation is organized as follows: after describing, in the Computational Details section, the EBS procedure and the computational approach used to study our molecular systems, we divide the Results and Discussion section in three parts, dedicated to the study respectively of the ground states of the Mn2 molecule, of its cation and of its anion. Finally, in the Conclusions we summarize our results, the vertical electron detachment processes, the ionization energies and the electron affinities, that are found to be compatible with the experimental predictions1,8−12,14,50−55 and with the multiconfigurational calculations.1,6,7,15−23,56,57

at first for wave function based methods like the HF theory, to project out the high-spin (HS) component from the BS solution. Later on, the approach was also applied in different DFT investigations.43−49 In the EBS method, the magnetic interaction between spin localizations on two atomic sites 1 and 2, can be modeled through the Heisenberg−Dirac−van Vleck(HDV) Hamiltonian,59−61 defined as the scalar product of the two spin operators Ŝ1 and Ŝ2, multiplied by a magnetic coupling constant J, i.e., Ĥ = −2J Ŝ1·Ŝ2. As Ŝ = S1̂ + Ŝ2 is the total spin operator, the HDV Hamiltonian admits eigenstates with a total spin S that ranges between the values of Smin = |S1 − S2|, that is the spin of the LS state and Smax = (S1 + S2), that is that of the HS one, and the HDV energy for a system of spin S is given by ES = −J[S(S + 1) − S1(S1 + 1) − S2(S2 + 1)]

(1)

For fixed values of S1 and S2, eq 1 gives rise to Landé’s interval rule ES − ES−1 = −2JS which allows one to calculate the energies of all the spin states if the coupling constant J and the energy of one of these states is known. In general for an unrestricted Slater determinant that presents a spin contamination, we have ES = −J[⟨S2⟩S − S1(S1 + 1) − S2(S2 + 1)]

(2)

where ⟨S2⟩ is the expectation value of the Spin square operator over that contaminated state with total spin S. In order to give a possible estimation of J one way is to use the HS and BS states40−42 coupling the equations: 2 ⎧ ⎪ EBS = − J[⟨S ⟩BS − S1(S1 + 1) − S2(S2 + 1)] ⎨ ⎪ 2 ⎩ EHS = −J[⟨S ⟩HS − S1(S1 + 1) − S2(S2 + 1)]

(3)

where EBS and EHS are the energies of the HS and BS states. In this way, combining eq 3 with Landé’s interval rule, J can be estimated according to J=

EBS − EHS 2

2

Smax + ΘHS − (Smin + ΘBS)

=

EBS − EHS ⟨SHS 2⟩ − ⟨SBS 2⟩

(4)

where ⟨S ⟩are calculated using Löwdin’s formulation, and ΘBS = ⟨SBS2⟩ − Smin2 and ΘHS = ⟨SHS2⟩ − Smax2 are respectively the spin contaminations of the BS and HS states. Through this estimation of J in combination with eqs 3 the energy of the LS state can be obtained through the expression: 43−45

2

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ELS = (1 + c)EBS − cEHS

COMPUTATIONAL METHODS Extended Broken Symmetry. As discussed above, the ground state of the manganese dimer is antiferromagnetic due to the localization of five electrons with spin up on one atom and of five electrons with opposite spin on the other, and the total wave function is an eigenstate of S2̂ with eigenvalue S(S + 1) = 0.58 Unfortunately in single reference methods, like Hartree−Fock (HF), density functional theory (DFT), and Møller−Plesset perturbation theory (MP), the only way to treat the spin localization is to use unrestricted Slater determinants. The unrestricted representation, that treats the two spin populations with different sets of molecular orbitals, introduces in the Fermionic wave function a symmetry breaking that enables the spin to localize on different atomic sites. This broken-symmetry (BS) solution, still an eigenstate of the Ŝz operator, fails to be an eigenstate of Ŝ2 introducing a spin contamination in the wave function.58 To partially correct this spin contamination the extended broken symmetry (EBS) approach was proposed,40−42

(5)

where c is a mixing constant defined as c=

Smax − Smin + ΘBS Smax 2 + ΘHS − Smin 2 − ΘBS

=

Smax − Smin + ΘBS ⟨SHS 2⟩ − ⟨SBS 2⟩ (6)

Fitting of the Potential Energy Curve and Anharmonic Vibrational Frequencies. In order to estimate the dissociation energy De, the equilibrium bond length Re, the harmonic vibrational frequencies ωe, the first anharmonicity constant xeωe and consequently the zero-point energy (ZPE) corrections to obtain the experimental dissociation energy D0, we applied perturbation theory.62 The potential energy curve, rescaled to zero by two times the atomic ground state energy, is fitted around the minimum through the equation E(r ) = − De + B

k (r − Re)2 − α(r − Re)3 + β(r − Re)4 2

(7)


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The Journal of Physical Chemistry A

Figure 1. 1Σ+g , 11Σ+u , and 11Πu states of the neutral manganese dimer obtained with the GGA functionals. The curves’ spectroscopic parameters are reported in Table 1. The 1Σ+g and 11Σ+u curves dissociate in two manganese atoms in the 6S(Mn) ground state, while the 11Πu state dissociates in 6S(Mn) + 6 D(Mn) (Table S2 of the Supporting Information).

Table 1. Spectroscopic Properties of the 1Σ+g , 11Σ+u and 11Πu States of the Neutral Manganese Dimer Obtained through GGA Functionalsa 1 + Σg

PBE PW91 BPW91 BLYP

−1

11 + Σu

(BS) −1

−1

11 −1

−1

Πu

Re [Å]

ωe [cm ]

xeωe [cm ]

D0 [eV]

Re [Å]

ωe [cm ]

xeωe [cm ]

D0 [eV]

Re [Å]

ωe [cm ]

xeωe [cm−1]

D0 [eV]

2.70 2.65 2.69 2.68

112.9 118.4 113.8 113.8

0.7 0.7 0.7 0.8

0.433 0.481 0.396 0.282

3.05 3.03 3.05 3.13

122.6 127.9 125.1 104.4

0.6 0.9 0.9 0.4

0.424 0.462 0.390 0.227

2.58 2.57 2.58 2.61

214.3 216.6 213.8 200.0

1.0 1.1 1.0 1.1

1.95 1.99 1.84 1.66

The dissociation energies D0 include the Zero Point energy (ZPE) corrections, and are calculating considering that the two Σ states dissociate in S(Mn) + 6S(Mn), while the 11Πu state dissociates in 6S(Mn) + 6D(Mn) (Table S2 of the Supporting Information).

a

6

where Re is the equilibrium distance of the molecule, k = μωe2 (μ = MMn/2 is the reduced mass), and α and β are additional fitting parameters that take into account the anharmonic corrections. The first anharmonicity constant is defined as xeωe =

β⎤ 3 ℏ ⎡ 5α 2 ⎢ 2 − ⎥ 2 μ ⎣ 2k k⎦

and the ZPE corrections will be 1 1 ZPE = ℏωe − xeωe 2 4

corrections. Second, the CP corrections to the BSSE are essentially negligible with the ma + def2 + QZVPP-DKH basis set, that was thus always used in all further calculations.

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RESULTS AND DISCUSSION Ground State of Mn2. The ground state of the neutral manganese dimer has been confirmed to be an antiferromagnetic 1 + Σg state through both experimental measurements8−14,26 and multireference methods.15−23 As anticipated in the Introduction, the 11Σ+u high-spin state has been found to be higher in energy, lying 0.02 eV above the ground state, according to CASPT2 calculations.23 In order to seek the best DFT exchangecorrelation functional to model this interaction, we have focused our investigation in the study of the three electronic states of the neutral manganese molecule, which are clearly identifiable through DFT: two high-spin (HS) states, the 11Σ+u and 11Πu states, and one BS solution, which can be associated with the 1Σ+g ground state. We must also point out that while the two Σ states dissociate in two manganese atoms in their 6S ground state, the potential energy curve of the 11Πu state dissociates in two atoms of which one is found to be in its excited 6D state17,20,21,35 (Table S2 of Supporting Information). If we consider the GGA functionals, usually used to study the Mnn clusters,3−5 we can see from Figure 1 that the 11Πu state is always lower in energy with respect to the two Σ+ states. The minimum of the BS 1Σ+g state lies between the minima of the other two, while its potential energy curve intersects the 11Σu+HS state around 3.0 Å, independently on the exchange-correlation functional used. By comparing the spectroscopic constants calculated for the three curves, reported in Table 1, we can see that the four GGA functionals predict quite similar results. Only, the BLYP functional gives longer bond lengths, larger frequencies, and smaller dissociation energies, thus predicting a weaker bonding of the molecule. These first results, in agreement with those reported in ref 35, clearly establish the fact that GGA functionals, used in previous

(8)

(9)

In Figure S1a of the Supporting Information, we show the fitting procedure for the B3LYP calculations. In order to guarantee the uniformity of the procedure, we have always fitted a region of 2 Bohr around the minimum of each curve. Computational Details. All the DFT calculations presented in this work are done using the ORCA 3.0.3 program system63 with the TightSCF and Grid7 options to guarantee accurate convergences. Moreover to take into account the relativistic effects we have used the Douglas−Kroll−Hess (DKH) second order corrections.64 The basis set convergence was tested with the minimally augmented version of Ahlrichs basis sets65,66 (ma + def2 + xZVPP) recontracted for the relativistic DKH calculations,67 using the B3LYP hybrid functional (Figure S1 of the Supporting Information). These calculations, that include corrections to the basis set superposition error (BSSE) through the “counterpoise method” (CP) introduced by Boys and Bernardi,68 have been compared with nonrelativistic calculations done with the ma+def2+QZVPP basis sets, and with a relativistic calculation with the ma + def2 + QZVPP-DKH basis set that did not include the BSSE corrections. The spectroscopic parameters obtained from the various curves are compared in Table S1 of Supporting Information. First, the relativistic corrections seem essential to open the gap between the high-spin and low-spin curves, thus all the following calculations have been done including the DKH second order C


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Table 2. Spectroscopic Properties of the 1Σ+g , 11Σ+u , and 11Πu States of the Neutral Manganese Dimer Obtained through Hybrid and Hybrid/Local Meta-GGA Functionalsa 1 + Σg

B3PW91 B3LYP PBE1PBE PW1PW91 B1PW91 B1LYP PBEHHPBE PWHHPW91 BHHPW91 BHHLYP M06 M06-L D3-B3LYP NL-B3LYP D3-PBE1PBE NL-PBE1PBE NL-BHHPW91

−1

11 + Σu

(BS) −1

−1

11 −1

−1

Πu

Re [Å]

ωe [cm ]

xeωe [cm ]

D0 [eV]

Re [Å]

ωe [cm ]

xeωe [cm ]

D0 [eV]

Re [Å]

ωe [cm ]

xeωe [cm−1]

D0 [eV]

3.27 3.43 3.29 3.27 3.30 3.56 3.43 3.42 3.45 3.92 3.23 3.11 3.31 3.32 3.26 3.23 3.38

78.1 54.0 79.4 80.4 76.8 44.4 67.2 67.3 64.4 27.9 72.8 89.4 73.3 66.3 83.2 85.7 72.0

0.9 1.2 0.9 0.9 0.9 1.5 0.9 1.0 1.0 1.6 1.2 0.7 0.7 1.0 0.8 0.8 0.9

0.142 0.050 0.176 0.183 0.136 0.026 0.128 0.132 0.102 0.010 0.174 0.218 0.257 0.114 0.270 0.231 0.153

3.35 3.51 3.38 3.36 3.40 3.64 3.55 3.54 3.57 4.00 3.34 3.23 3.38 3.40 3.35 3.32 3.50

75.4 55.0 76.1 76.9 72.4 44.0 62.1 62.0 59.7 26.6 70.3 92.5 76.1 66.3 80.0 82.7 66.5

1.1 1.6 1.0 1.0 1.1 1.8 0.9 0.9 1.0 1.6 1.4 1.0 0.8 1.3 0.9 1.0 0.9

0.124 0.045 0.152 0.159 0.114 0.022 0.106 0.109 0.081 0.008 0.158 0.214 0.248 0.105 0.245 0.204 0.127

2.58 2.59 2.59 2.58 2.59 2.60 2.65 2.64 2.65 2.64 2.57 2.59 2.68 2.58 2.62 2.58 2.64

218.8 212.4 219.4 221.2 218.5 212.6 208.9 210.6 208.7 208.7 218.5 212.5 198.0 219.0 209.2 223.8 212.1

0.9 0.9 0.8 0.8 0.8 0.8 0.5 0.5 0.5 0.6 1.0 0.9 0.3 0.9 0.5 0.8 0.5

1.417 1.268 1.460 1.473 1.362 1.164 1.377 1.364 1.305 1.036 1.657 2.312 1.429 1.385 1.525 1.554 1.392

The dissociation energies D0 include the zero point energy (ZPE) corrections, and are calculating considering that the two Σ states dissociate in S(Mn) + 6S(Mn), while the 11Πu state dissociates in 6S(Mn) + 6D(Mn) (Table S2 of the Supporting Information).

a

6

Figure 2. 1Σ+g obtained through BS and EBS and 11Σ+u and 11Πu high-spin states of the neutral manganese dimer, obtained with the B3PW91 (a), B1PW91 (b), and BHHPW91 (c) functionals with different percentage of HF exchange. The same behavior is displayed for the B3LYP, B1LYP, and BHHLYP functionals in Figure S2 of the Supporting Information. The curves’ spectroscopic values are reported in Tables 2 and 3

DFT investigations,4,5 cannot be used to correctly predict the interaction in manganese clusters,3 not only because they predict a ferromagnetic ground state, but also because they favor the Π states with respect to the expected Σ ones. Differently from the GGA exchange-correlation functionals, the hybrid and local/hybrid meta-GGA functionals give an ordering of the three states which is compatible with the experimental predictions, as can be seen by the results reported in Table 2. Within these results we can first deduce the role of the HF exchange in selecting the relative stability of the Π or Σ states, as already discussed in the work from Yamanaka et al.35 In Figure 2, we can see how the percentage of HF exchange influences the relative stability between the three states, for the B3PW91, B1PW91, and BHHPW91 functionals that include 20%, 25%, and 50% respectively of HF exchange. As the percentage increases the 11Πu state goes from being the ground state, as for the pure GGA (Figure 1), to being more and more higher in energy with respect to the two Σ states. This same behavior has been reported also for the B3LYP, B1LYP, and BHHLYP functionals in ref 35 (Figure S2 of the Supporting Information).

In general, we can see that the PBE1PBE (or PBE0) the PW1PW91 and B1PW91 1 parameter hybrids with 25% of HF exchange give comparable results, as do the half and half hybrids with 50% of HF exchange (PBEHHPBE, PWHHPW91, BPW91). As already seen for the GGA functionals a different behavior can be observed for the functionals that include the LYP correlation term which give a weaker molecular bonding. In order to correct this behavior we have also tested the introduction of non-local (NL) dispersion contributions based on VV10 van der Waals functional36 and the use of atomtriplewise dispersion (D3) corrections37 with Becke−Johnson (BJ) damping38,39 (Table 2). In particular, for the B3LYP functional, the NL corrections tend to strengthen the bonding giving results that are similar to those obtained through the BHHPW91 functional, but with shorter bond lengths: this is also evident when applying the NL corrections to the BHHPW91 itself, for which, together with the shortening of the bond lengths, we have the increasing of the vibrational frequency and of the bonding energy. Finally, before doing a comparison with the experimental data and with the results obtained with the multiconfigurational D


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Table 3. Spectroscopic Properties of the 1Σ+g State of the Neutral Manganese Dimer Obtained through EBSa

methods, we must add that also both the M06 and M06-L, hybrid and local meta-GGA functionals, predict the correct ordering of the states even though the bond seems underestimated. In conclusion we can say that overall the hybrid or meta functionals reported in Table 2 predict two stable, lower in energy, Σ states, of which the antiferromagnetic one is the ground state, and one excited 11Πu state. By analyzing the KS energies and the symmetries of the molecular orbitals obtained for the two Σ states and the BHHPW91 functional, we can see that in both the 11 + Σu (Figure 3) and 1Σ+g (Figure 4) states two doubly occupied

1 + Σg (EBS)

B3PW91 B3LYP PBE1PBE PW1PW91 B1PW91 B1LYP PBEHHPBE PWHHPW91 BHHPW91 BHHLYP M06 M06-L D3-B3LYP NL-B3LYP D3-PBE1PBE NL-PBE1PBE NL-BHHPW91

Re [Å]

ωe [cm−1]

xeωe [cm−1]

D0 [eV]

JHS/BSb [cm−1]

3.23 3.39 3.25 3.23 3.26 3.52 3.38 3.37 3.39 3.88 3.17 3.02 3.26 3.27 3.22 3.19 3.32

78.6 53.5 80.7 81.8 78.9 45.2 69.4 69.7 66.9 28.3 73.5 87.2 70.4 65.4 83.3 86.9 74.7

0.7 1.0 0.8 0.8 0.8 1.3 0.9 1.0 1.0 1.5 1.0 0.6 0.7 0.8 0.8 0.7 0.9

0.151 0.053 0.187 0.195 0.147 0.028 0.139 0.143 0.113 0.011 0.183 0.226 0.261 0.119 0.281 0.244 0.165

−7.6 −2.4 −9.3 −10.1 −9.3 −1.9 −9.3 −9.6 −9.2 −1.1 −8.0 −8.9 −4.4 −4.4 −10.0 −11.1 −11.1

a

The dissociation energies D0 include the zero point energy (ZPE) corrections and are calculated considering the dissociation limit of 6 S(Mn) + 6S(Mn). bThis coupling constant is calculated between the HS and BS states on the Re equilibrium length of the EBS curves through eq 4.

Figure 3. Orbital KS energies in eV of the 11Σ+u state of the neutral manganese dimer in its minima, obtained with the BHHPW91 exchange-correlation functional.

Different from what was reported in ref 35, the data shown in Table 3 are nearly all in a reasonable agreement, predicting a 1Σ+g (EBS) ground state and a negative J coupling constant of a few cm−1 in magnitude. In order to understand the overall best exchange-correlation functional to describe these three states in Table 4 we have summarized both the available experimental data and the accurate multiconfigurational calculations. From this comparison, we can exclude the functionals including the LYP correlation that give not only a weaker bond, but also tend to underestimate the J coupling constant. The meta-GGA and metahybrid functionals again seem to behave rather well, even if they predict rather short bonds and too high vibrational frequencies and dissociation energies. Finally the half and half hybrids seem to be the best, especially if compared to the experimental measurements, and within the three of them (we are excluding the BHHLYP one) the BHHPW91 seems the most compatible; even though discrepancies with the accurate multiconfigurational calculations appear. In particular our J coupling constant, around −9.6 cm−1 is comparable to the experimental predictions but disagrees of nearly 40% with the multireference calculations.15,17,19,21,23 This is due to the fact that DFT is predicting a larger energy gap between the HS 11Σ+u and BS 1Σ+g states with respect to the multiconfigurational methods. On the other hand, the discrepancy between the experimental measurements and the multiconfigurational calculations has been ascribed essentially to the interaction between the manganese atoms and the noble-gas matrix that cages them in the experiments.19 Another hypothesis was formulated through a CASPT2 investigation23 in which it was seen that the biquadratic j(S1̂ · Ŝ2) 2 contribution to the Hamiltonian had a large effect in deviating from Landé’s interval rule.19,23 This, on the contrary, was not observed with more recent NEVPT2 and MRCI+Q calculations.19

Figure 4. Orbital KS energies in eV of the 1Σ+g state of the neutral manganese dimer in its minima, obtained with the BHHPW91 exchange-correlation functional. In green are given the MOs that appear to be localized atomic orbitals (LAO) in the broken-symmetry solution and that give rise to the strong spin-contamination.

σ and σ* molecular orbitals that combine the 4s atomic orbitals of the Mn 6S ground state are present. In the 11Σ+u state, the atomic 3d orbitals combine together to form a set of nearly degenerate σ, σ*, π, π*, δ, and δ* molecular orbitals, that on the contrary remain localized in the BS solution which approximates the 1Σ+g ground state. This spin localization gives a value of ⟨S2⟩≈ 5 and thus a spin contaminated state that can be partially corrected through the Extended Broken symmetry (EBS) procedure. By applying the EBS between the 11Σ+u and 1Σ+g (BS) states in Table 2, we obtain the 1Σ+g (EBS) potential energy curves whose spectroscopic parameters are shown in Table 3. For the B3PW91, B1PW91, and BHHPW91 functionals the EBS curves are shown in Figure 2, while for the B3LYP, B1LYP, and BHHLYP, the curves are shown in Figure S2 of the Supporting Information. On the EBS minima, for each potential energy curve, we have also estimated the J coupling constant between the HS and BS states through Yamaguchi’s formulation42 (eq 4). E


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Table 4. Comparison between the Spectroscopic Constants of the 1Σ+g , 11Σ+u and 11Πu States of the Neutral Manganese Dimer Obtained through Our DFT Calculations and Those Obtained through Other ab Initio Methods and Experimental Measurements ωe [cm−1]

Re [Å]

xeωe [cm−1]

D0 [eV]

J [cm−1]

Πu

11

MCQDPT2j MRCI+Ql BHHPW91

2.53 2.578 2.65

240.47 226 208.7

1.30 0.5

1.21 1.08 1.305

11 + Σu

MCQDPT2j QD-NEVPT2k MCAS/MRCI+Q/LCg MRCI+Ql s-ACPFl RCCSD(T)h BHHPW91

3.42 3.683 3.853 3.836 3.70 3.69 3.57

62.64 44.7 32.6 37

1.36 0.93 0.84

40.6 59.7

0.84 1.0

MCQDPT2j QD-NEVPT2k MCAS/MRCI+Q/LCg MRCI+Ql s-ACPFl CASSCFf MRMPf BHHPW91 (BS) BHHPW91 (EBS) exptl

3.29 3.698 3.823 3.795 3.60 3.55 3.50 3.45 3.39 3.4a

53.46 43.1 33.7 36

0.126 0.079 0.044 0.046 0.061 0.06 0.081

1 + Σg

0.87 0.79 0.86

68 44 64.4 66.9 68.1,b 76.4,c (59, 68)d

1.0 1.0 0.53,c 1.05b

0.137 0.079 0.047 0.054 0.068 0.11 0.05 0.102 0.113 0.13(13)e

−5.87 −3.6(1) −6.7

−9.2 −8(4),i −9(3)a, (−8 to −11)b

a

Electron spin resonance in rare-gas matrices from ref 11 bUltraviolet−visible and Raman scattering in xenon matrix from ref 13. The J constant was found to vary in temperature between those two values. cRaman scattering in krypton matrix from ref 14. dRaman scattering on two different sites in argon matrix from ref 14. eMagnetic-sector, mass spectroscopy from ref 8. fCalculations with a (18s15p8d4f2g)/[7s6p4d4f2g] Gaussian basis set with active space [14,12] form ref 22, also corrected with the basis set superposition error. gCalculations from ref 19. The J constant is calculated at a bond length of 3.6 Å. hRCCSD(T)+3s calculations with a AV5Z/DK+bf basis set from ref 25. These are similar to the various complete basis set extrapolations reported. iElectron spin resonance in rare-gas matrices from ref 9 jSecond-order quasidegenerate perturbation theory from ref 17. In the article the De values are reported to be 0.14 and 0.13 eV for the 1Σ+g and the 11Σ+u states, respectively. kQuasidegenerate NEVPT2 calculations from ref 18. De is reported to be of 0.082 eV for both the 1Σ+g and the 11Σ+u states. lMultireference configuration interaction and scaled averaged coupled pair functional including second order Douglas−Kroll−Hess relativistic corrections and BSSE corrections: from ref 21.

of the states was also confirmed by later CASSCF and CASPT2 calculations56 that found an energy difference between the two of 0.247 eV (CASPT2/ANO-s), and classified all the intermediate Σ+ states of the cation. This high-spin ground state was also confirmed by X-ray absorption spectroscopy,53,54 and later by X-ray magnetic circular dichroism (XMCD) spectroscopy in an ion trap,55 which also established a limit for the probable equilibrium bond length ≥2.6 Å. To our knowledge two DFT investigations present results for the Mn+2 cation,27,69 both assuming the ground state of the neutral molecule to be ferromagnetic. In the work by Gutsev et al.27 the DFT calculations where done through GGA functionals, for which the ground state of neutral Mn2 was found to be the 11Πu state, as reported above, while the cationic ground state to be a 10Πu state. On the contrary the older work by Nayak et al.69 missed to identify the antiferromagnetic ground state with B3LYP, probably due to convergence problems, and assumed it to be the ferromagnetic 11Σ+u state, while the correct ferromagnetic 12Σ+g ground state for the cation was identified. In order to study the ionization energy of the Mn2 molecule, here we have decided to considered only six of the previous hybrid functionals, namely the B3PW91, B1PW91, BHHPW91, B3LYP, B1LYP, and BHHLYP functionals. In this way we could test the description of the ionization energy as a function of the percentage of HF exchange, and of two different correlation functionals that have given the less (BHHLYP) and the most

Overall, the BHHPW91 functional could be a good candidate in describing the antiferromagnetic or ferromagnetic orderings in the Mnn clusters, giving us the possibility to improve the comprehension of possible transitions. Despite this first success, another important aspect that must be characterized is the description of the interconversion energies between the neutral molecule and its cationic and anionic species, in order to study the change in the relative stability of the ferromagnetic and antiferromagnetic states also for these species. Ionization Energy of the Manganese Dimer: The Ground State of the Mn+2 Cation. The first experimental measurements on the cationic Mn+2 molecule have been done through collision-induced dissociation experiments50 which established a dissociation energy of about 0.85 ± 0.2 eV and an adiabatic ionization potential of 6.9 ± 0.4 eV. A later photodissociation experiment51 gave for the dissociation energy a lower limit of 1.39 eV, while for the Ionization energy of the neutral manganese molecule an upper limit of 6.47 eV. The dissociation energy was then redetermined through a Fourier transform ion cyclotron resonance (FTICR) spectrometer,52 which predicted a value around 2.1 ± 0.3 eV. The cationic ground state was determined to be a 12Σ state through electron spin resonance (ESR) experiments,12 which were in accordance with the theoretical deductions of Bauschlicher57 that through CASSCF and CISD calculations identified it to be a ferromagnetic 12Σ+g state, while a 2Σ+u state with antiferromagnetic coupling was found to lie 0.44 eV higher in energy. This ordering F


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The Journal of Physical Chemistry A (BHHPW91) compatible results with respect to the experimental data. For each of these functionals we have considered the ionization of both the HS 11Σ+u and BS 1Σ+g states, studying the possible transitions with MS ± 1 (being MS the multiplicity of the neutral state). The potential energy curves of the five cationic states that we have identified, namely the 10Σ+g , 12Σ+g , 2Σ+u , 10Πg and 2Πu, have been displayed in Figures S3 and S4 of the Supporting Information, while their spectroscopic parameters have been listed in Table S3. Within this set of states, the two with the lowest energies are the HS 12Σ+g state and the 2Σ+u BS state, the former of which is the cationic ground state, compatible with the experimental measurements and with the multiconfigurational calculations summarized in Table 5. Naturally, as for the neutral manganese

Figure 5. Orbital KS energies in eV for the 12Σ+g state of the manganese cation, obtained with the BHHPW91 exchange-correlation functional.

the HS cationic ground state dissociates in a neutral manganese atom and in its cation, i.e., 6S(Mn)+7S(Mn+), thus presenting one doubly occupied σg Molecular orbital determined primarily by the combination of the 4s atomic orbitals (Table S9 of the Supporting Information). If we compare the 12Σ+g ground state obtained from DFT with the experimental findings, we can see that the half and half functionals that give the lower dissociation energy, still appear to predict a value of about 0.2 eV above the photodissociation limit of ≥1.39 eV.51 Once cataloged the single cationic curves, we have studied the possible ionization energies of the three states of the neutral manganese dimer, the 11Σ+u , 11Πu states, and 1Σ+g (Table S4 of Supporting Information). Of all the possible ionization pathways, those between the Σ+ states of the neutral manganese molecule, and the Σ+ states of its cation are summarized in Table 6. We must point out that, because of the inversion of the stability between the HS and lowspin states of the neutral Mn2 and of its cation, measured from ESR experiments,9,12 we should assume a non trivial ionization

Table 5. Spectroscopic Constants of the 2 Σ+u and 12 Σ+g States of the Mn+2 Cation, Obtained through Our DFT Calculations, with Other ab Initio Methods and from Various Experimental Measurements Re [Å]

ωe [cm−1]

xeωe [cm−1]

D0 [eV]

0.5 0.5 0.6 0.5 0.5 0.5

1.026 1.066 1.104 1.120 1.156 1.176 0.615 1.150

0.5 0.6 0.6 0.6 0.6 0.7

1.511 1.597 1.627 1.448 1.524 1.567 1.053 1.397 0.8(2) ≥1.39 2.1(3)

2 + Σu

BHHPW91 B1PW91 B3PW91 BHHLYP B1LYP B3LYP CASSCFd CASPT2d

3.00 2.95 2.94 3.01 2.96 2.94 3.190 2.863

143.6 142.9 147.4 138.2 136.7 136.4 204.1 12 + Σg

BHHPW91 B1PW91 B3PW91 BHHLYP B1LYP B3LYP CASSCFd CASPT2d CIDa PDb FTICRc

3.05 2.99 2.98 3.08 3.04 3.01 3.178 2.940

147.0 151.2 150.6 137.5 138.2 141.1 214.4

Table 6. High-Spin and Low-Spin Vertical (IEv) and Adiabatic (IEAd) Ionization Energies between the Neutral and Cationic Manganese Dimersa high spin BHHPW91 B1PW91 B3PW91 BHHLYP B1LYP B3LYP

a

From collision-induced dissociation experiments in ref 50. Photodissociation experiments from ref 51. cFourier transform ion cyclotron resonance (FTICR) spectrometer results from ref 52. dAb initio calculations from ref 56, in which the dissociation energies do not include ZPE corrections. b

BHHPW91 B1PW91 B3PW91 BHHLYP B1LYP B3LYP PDb CIDc

molecule, even for the cation the set of intermediate Σ states with different spin-multiplicity are not describable through a single-determinant theory. For these two curves the DFT results seem nearly independent from the XC functional used, and only slightly differ from the other ab initio predictions. In particular the vibrational frequencies are smaller than those given by CASSCF/s-ANO calculations,56 probably due to the fact that CASSCF does not include properly the dynamical correlation effects that are surely responsible for the molecular bonding of the HS state, which is a single reference state. This may also be the reason why CASSCF displays a lower dissociation energy, and a larger equilibrium bond length Re. These two quantities become more compatible with our results when perturbation theory is applied in CASPT2.56 As shown in Figure 5 +

12 + Σg 12 + Σg 12 + Σg 12 + Σg 12 + Σg 12 + Σg

← 11Σ+u ← 11Σ+u ← 11Σ+u ← 11Σ+u ← 11Σ+u ← 11Σ+u low spin

2 + Σu ← 1Σ+g 2 + Σu ← 1Σ+g 2 + Σu ← 1Σ+g 2 + Σu ← 1Σ+g 2 + Σu ← 1Σ+g 2 + Σu ←1Σ+g 12 + Σg ← 12 + Σg ←

IEv [eV]

IEAd [eV]

5.659 5.634 5.730 6.248 6.122 6.107 IEv [eV]

5.486 5.517 5.631 5.912 5.941 5.971 IEAd [eV]

6.103 6.145 6.241 6.558 6.475 6.484

5.942 6.082 6.182 6.244 6.315 6.369 ≤6.47 6.9(4)

a

The vertical ionization energies are evaluated on the equilibrium lengths of the 11Σ+u or 1Σ+g states of the neutral manganese molecule, including the zero point energy corrections. The adiabatic ionization energies include the zero point energy corrections of both the initial and final states. bDerivation through the photodissociation measurments in ref 51. cDerivation through collision-induced dissociation measurments in ref 50. G


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The Journal of Physical Chemistry A process in the molecule. For this reason we have reported in Table 6 the vertical detachment energies, calculated on the equilibrium bond lengths Re of the neutral molecular states, and the adiabatic ionization energies between the potential energy curves’ minima. The values obtained through DFT vary in a range of 0.5 eV but are all compatible with the upper limit predicted through photodissociation of ≤6.47 eV.51 This said, we can also observe that the effect of the increase in the HF exchange percentage is that of lowering the ionization energy of the dimer. Furthermore, the PW91 correlation functional tends to give lower transition energies if compared to the results obtained with the LYP correlation functional. This is also in accordance with the fact that the PW91 correlation term tends to underestimate the ionization energy in the Mn atom (Table S2 of Supporting Information) of about 0.5 eV, that is the same discrepancy observed in the ionization potentials of the molecule. Overall the spectroscopic parameters available for the cationic ferromagnetic and antiferromagnetic states shown in Table 5 display a reasonable agreement with both accurate multiconfigurational ab initio calculations and the available experimental data, thus confirming that the BHHW91 functional may be efficiently used to calculate the possible transitions. Electron Affinity of the Manganese Dimer: The Ground State of the Mn2− Anion. There are few experimental and computational studies of the anionic states of Mnn− clusters1,6,7 which couple photoelectron spectroscopy measurements with DFT calculations. Of these, the only one reporting a detailed analysis on the electron affinity of the Mn2 dimer is the recent work by Gutsev et al.,1 which studied the Mnn− (n = 1−16) clusters through both anion photoelectron spectroscopy and DFT. Within their investigation the authors compared the measured experimental vertical detachment energy (VDE) from Mn−2 to Mn2, of 0.38(8) eV, to the energy differences obtained by various DFT functionals. Differently from the neutral and cationic Mn2 species for which ESR experiments are present, for the anionic species of this molecule the spin multiplicity is unknown. Also, to the best of our knowledge no multiconfigurational investigations have been produced. In the recent DFT analysis,1 the authors have studied all the permitted ionization paths between the Mn2 anion and the neutral manganese molecule with spin multiplicity MS ± 1 (being MS the multiplicity of the anionic state). For the anion two low energy states with high-spin, namely a 10Σ−g state, and low-spin, a spin contaminated 2S + 1 = 2 state were identified. For the 10Σ−g the authors considered the transition to the 11Πu state, which was assumed to be the Mn2 ground state, and to a 9Σ−u state. For the spin contaminated 2S + 1 = 2 solution the authors considered the transition to the spin contaminated 2S+1 = 1 and 2S + 1 = 3 states of the neutral manganese, being the former identifiable as our approximate 1Σ+g ground state. The 11Σ+u state was not considered by the authors, probably due to the fact that the GGA functionals were taken as references, and as discussed above they predict the 11Πu state to be more stable (Figure 1). Unfortunately large discrepancies were found with the experimental measurements, depending on both the multiplicity involved and the exchange-correlation functional used. In order to understand the effect of both the percentage of HF exchange, and of the type of correlation functional used on the description of the electron affinity of Mn2 and on the Vertical detachment energy of Mn−2 , in this section we followed the same procedure previously applied for the cationic manganese species. For each of the six exchange-correlation functionals, the B3LYP, B1LYP, BHHLYP, B3PW91, B1PW91, and BHHPW91 we have

Figure 6. KS energies in eV of the 12Σ+u ground state of the manganese anion in its structural minima, obtained with the BHHPW91 exchangecorrelation functional.

Table 7. Vertical Detachment Energies (VDE) of the Mn2− Anion and Adiabatic Electronic Affinities (EAAd) of the Neutral Manganese Dimera high spin BHHPW91 B1PW91 B3PW91 BHHLYP B1LYP B3LYP

11 + Σu 11 + Σu 11 + Σu 11 + Σu 11 + Σu 11 + Σu

← 12Σ+u ← 10Πg ← 10Πg ← 12Σ+u ← 10Σ+u ← 10Σ+u low spin

BHHPW91 B1PW91 B3PW91 BHHLYP B1LYP B3LYP PEb

1 + Σg 1 + Σg 1 + Σg 1 + Σg 1 + Σg 1 + Σg

← 2Πu ← 2Πu ← 2Πu ← 2Πu ← 2Πu ← 2Πu

VDE [eV]

EAAd [eV]

0.393 0.602 0.779 0.135 0.358 0.600 VDE [eV]

0.393 0.434 0.578 0.126 0.255 0.468 EAAd [eV]

0.395 0.488 0.567 0.141 0.256 0.396 0.38(8)

0.351 0.434 0.511 0.086 0.203 0.339

a

The Energies are all calculated in eV, and the vertical detachment energies are evaluated on the equilibrium lengths obtained for the anionic states and include the zero point energy corrections of the anionic state. The adiabatic electronic affinities all include the zero point energy corrections of both the initial and final states. b Photoelectron measurments from ref 1.

obtained six lower energy states, namely the 2Πu and 2Σ+g states (both spin contaminated as the 1Σ+g ), and the 10Πg, 12Πg, 10Σ−g , and 12Σ+u states (The analysis of the electronic states are reported in Tables S7 and S8 of the Supporting Information). The potential energy curves are displayed in Figures S5 and S6 of the Supporting Information, while their spectroscopic properties have been summarized in Table S5. Here, differently from the case of the cationic molecule, the ordering of the states greatly depends on the percentage of the HF exchange, and is also different when using the PW91 or the LYP correlation functionals. In particular with low percentages of HF exchange the Π states appear always energetically favored, while this changes for the half and half hybrids for which the ferromagnetic 12 + Σu state is always favored. Moreover the functionals with LYP correlation always predict the 10Σ−g state to be lower in energy with respect to the corresponding 10Πg one. As already discussed for the neutral manganese and for the cationic molecule, all the intermediate Σ+ states of the anion H


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Figure 7. High-spin and low-spin vertical detachment energies, ionization energies, and electron affinities calculated with the B3PW91, B1PW91, and BHHPW91 exchange-correlation functionals. The energies involved in these processes are reported in Tables 6 and 7

properties of the potential energy curves. In particular, for the EBS 1Σ+g ground state, described through the BHHPW91 functional, we obtain the value of 3.39 Å for the equilibrium bond length, of 66.9 cm−1 for the vibrational frequencies, of 1.0 cm−1 for the first anharmonicity constant, and of 0.113 eV for the dissociation energy, which are compatible with the experimental measurements. Moreover, the J coupling constant of −9.2 cm−1 is in excellent agreement with that measured through optical spectroscopy.9,10,13 In the second part of this work we have investigated the behavior of six different Hybrid functionals in describing the ionization energy and the electronic affinity, to understand the role played by the percentage of HF exchange, and by the different correlation functionals. A schematic representation of the results obtained for the three B3PW91, B1PW91 and BHHPW91 functionals is shown in Figure 7. Through our study we have confirmed the cationic ferromagnetic 12Σ+g ground state predicted by previous experimental and multiconfigurational calculations, and we have studied the possible ionization processes finding again a good agreement. In this case the BHHPW91 functional tends to underestimate the ionization potential of the molecule of the same quantity, 0.50 eV, it underestimates the ionization potential of the Mn atom. The anionic ground state is seen to strongly depend on the percentage of HF exchange and to correspond to the ferromagnetic 12Σ+u state for the half and half energy functionals. Unfortunately due to the absence of multiconfigurational ab initio calculations or ESR experiments, we could not determine which of the functionals better described the overall properties of the anionic states. The only experimental information available is related to the vertical detachment energy VDE of 0.38(8) eV, estimated through photoelectron spectroscopy,1 that has been found to be compatible with both the 11Σ+u + e− ← 12Σ+u and 1Σ+g + e− ← 2Πu ionization processes, respectively of 0.393 and 0.395 eV, obtained with the BHHPW91 exchangecorrelation functional. In conclusion, despite the remaining uncertainties, we have extensively studied the behavior of DFT in describing the Mn2 molecule and its cationic and anionic species, defining a better

cannot be obtained with DFT; thus, in this work, we have considered and classified only the possible transitions between the six potential energy curves discussed above and the two 11 + Σu and 1Σ+g states of the neutral molecule (Table S6 of the Supporting Information). In Table 7 we summarize only those transitions which happen between the lowest energy states of the anion and neutral dimers, and which satisfy the MS ± 1 rule. Of these transitions, it appears that the most compatible with the VDE estimated in ref 1 through photodissociation, of 0.38(8) eV, is the nearly adiabatic 11 + Σu + e− ← 12Σ+u ionization process estimated through the BHHPW91 functional that displays an energy gap of 0.393 eV. For this functional this process is nearly degenerate in energy with the low-spin 1Σ+g + e− ← 2Πu transition that displays an energy gap of 0.395 eV. In conclusion the half and half hybrid functionals, and especially that BHHPW91, seem to behave better in describing the ionization process of the anionic states, but further accurate multiconfigurational investigations are required for a more detailed comparison. In general we can affirm that the order of the Σ+ states in anion resembles that of the cationic Σ+ states: in both charged molecules the ferromagnetic state appears to be lower in energy with respect to the antiferromagnetic one, inverting the order that appears in the neutral Manganese dimer. In the anionic 12Σu+ state the added electron occupies a σg bonding molecular orbital, which is essentially defined by the linear combination of the 4pz atomic orbitals of the two atoms (Figure 6 and Table S8 of the Supporting Information).

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CONCLUSIONS In this work, we have investigated in depth the possibility to describe the ground states of the cationic, anionic and neutral manganese dimers through DFT. The first part of the work has been devoted to the description of the 1Σ+g ground state of the manganese dimer, and of its corresponding 11Σ+u high-spin state. Different from the results previously reported in the literature2,27,30−32,35 we have selected a group of half and half hybrid functionals able to reasonably describe not only the correct energetic ordering, but also the overall spectroscopic I


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approach to treat its van der Waals bonding and opening the way to a more accurate DFT approach to study charged or neutral Mnn clusters.

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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.5b12169. Basis set convergence data, atomic energies, and various potential energy curves with their spectroscopic parameters (PDF)

■

AUTHOR INFORMATION

Corresponding Author

*(M.B.) E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Computational resources have been granted by the Hydra HPC cluster of the University of Modena and Reggio Emilia

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REFERENCES

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Neutral, Anionic, and Cationic Manganese Dimers through Density

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