Magnetic coupling interaction under different spin multiplets in neutral manganese dimer: CASPT2 theoretical investigation

Chemical Physics Letters 387 (2004) 395–399 www.elsevier.com/locate/cplett

Magnetic coupling interaction under different spin multiplets in neutral manganese dimer: CASPT2 theoretical investigation Bingwu Wang a, Zhida Chen a

a,b,*

State Key Laboratory of Rare Earth Materials Chemistry and Applications, Department of Chemistry, Peking University, Beijing 100871, China b State Key Laboratory of Structural Chemistry, Fujian Institute of Research on the Structure of Matter, Chinese Academic of Sciences, Fuzhou 350002, China Received 30 December 2003; in final form 15 February 2004 Published online: 10 March 2004

Abstract In order to theoretically study on the magnetic coupling interaction in the Mn–Mn dimer, quantum chemical calculations on the equilibrium Mn–Mn distance Re , total energy ET , binding energies Be and exchange interaction energies EðSÞ under different spin multiplets were performed at CASPT2 level of theory. It is found that the calculated Re , ET , and Be values are correlated with the spin multiplet, and with increasing the total spin S, Re increases, but ET and Be exhibit a decrease trend. The exchange interaction energies EðSÞ deviate significantly from the ‘Lande interval rule’. This unusual magnetic phenomenon is primarily attributed to the biquadratic jðS a
S b Þ2 term contribution in spin Hamiltonian for the Mn–Mn dimer. Ó 2004 Elsevier B.V. All rights reserved.

1. Introduction During the last several decades, transition-metal dimers have received a persistent interest both experimentally and theoretically. Metal dimers are the simplest model systems for studying metal clusters and bimetallic catalysts experimentally. However, their chemical bonding schemes and electronic structures are a severe challenge to theoretical chemists. Among these transition-metal dimers, Mn2 is one of the least understood systems from both the experimental and theoretical point of view. On the other hand, Mn2 is one of the simplest magnetic coupling systems without any organic ligands, meanwhile it is a complicated coupling system because there is a magnetic exchange interaction among 5delectron pairs on Mn atoms, thus studies on the magnetic coupling interaction between two Mn atoms is very informative to understand abundant magnetic molecules composed of Mn atoms or ions. Up to present, a considerable effort has been devoted to studies on electronic structure of Mn2 . Experimen*

Corresponding author. Fax: +86-010-62751708. E-mail address: [email protected] (Z. Chen).

0009-2614/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2004.02.057

tally, Kant et al. [1] observed at first Mn2 dimer under the temperature between 1350 and 1500 K by means of a mass spectrometer, and on the basis of the third law of thermodynamics, they had calculated a dissociation energy Do0 (Mn2 , VDW) ¼ 0.33  0.26 eV for the van der  and a Waals model with an interatomic distance of 3.8 A frequency of 89 cm1 . Subsequently, Haslett et al. [2] had corrected Kant’s dissociation energy to be Do0 (Mn2 , VDW) ¼ 0.27  0.26 eV through the same spectroscopic data but with more recent molecular parameters, while on the basis of the LeRoy–Bernstein analysis on resonance Raman data Haslett obtained De (VDW) ¼ 0.15 eV. Gingerich [3] also reported an average estimated value of 0.44  0.30 eV for the dissociation energy of Mn2 based on the third-law of thermodynamics. Cheeseman et al. [4] found that the Mn–Mn distance varies  for the S ¼ 0 to S ¼ 5 states, according from 3.2 to 3.6 A to the axial anisotropic exchange parameter in cyclopropane matrix. Morse [5] has suggested De 6 0:8 eV to be acceptable. It should be pointed out that so far considerable experiments made efforts to the observed Mn–Mn length and dissociation energy. As far as electronic structure of Mn–Mn dimer is concerned, in 1984 the first experimental information of

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the magnetic coupling in the Mn2 dimer has been obtained by the ESR spectra of the Mn2 isolated in the krypton and xenon matrices [6], which indicated the ground state singlet of Mn2 on the basis of the temperature dependence of the ESR spectrum bands. In subsequent ESR spectroscopy, Baumann et al. [7] obtained ESR lines corresponding to populated multiplet states of total spin S ¼ 1, 2, 3 from 4 to 70 K, and further by using the temperature dependence data on S ¼ 2 transition they found out an antiferromagnetic coupling between the 3d electrons on Mn atoms, with a Heisenberg exchange coupling constant J ¼ 9  3 cm1 . Then the extended ESR work [4] had been done in cyclopropane matrix and observed S ¼ 4, 5 states of Mn2 at temperature ranging from 12 to 110 K. It is evident that these reported experiments consistently indicated the multiplets with S ¼ 0, 1, 2, 3, 4, and 5 to be possibly populated. It should be mentioned that Rivoal et al. [8] also analyzed the temperature variation of the absorption spectroscopy and magnetic circular dichroism (MCD) intensity originating from the S ¼ 2 state in the range 13–17 K, and estimated J ¼ 10:3  0:6 cm1 . More work [9] on ultraviolet–visible and Raman spectroscopy demonstrated that J ¼ 10:0  2:0 cm1 was reasonable for S ¼ 0, 1, 2, 3 states. As mentioned above, almost all experimental results predicted that the ground state of manganese dimer is a van der Waals dimer with the weakly binding energy, longer interatomic distance and antiferromagnetic coupling between the two Mn atoms. Unfortunately, so far theoretical explanation on the Mn2 characters are not yet satisfactory. The first theoretical work on manganese dimer was reported by Nesbet [10], who applied ab initio approximate Hartree–Fock calculation and the Heisenberg exchange Hamiltonian, and predicted an antiferromagnetic ground state with J ¼ 4:1 cm1 , Mn–Mn  and dissociation energy of 0.79 eV. After length of 2.88 A that several density functional theory studies [11–18], including LSDA and GGA approximations, have been done on the equilibrium geometry, binding energy, vibrational frequency. But these density functional theory studies all are based on only one-determinant wave function calculation, hence it is difficult to draw an unambiguous conclusion on the specific multiplicity of the ground state. Also UHF computation [19] and single point calculations [20] at CASSCF level using a limited active space have been performed. It is interesting to note  of the interthat Bauschlicher [20] found that at 2.9 A atomic distance, the 1 Rþ state was 0.013 eV more stable  the 11 Rþ state was than 11 Rþ state, however, at 3.4 A 1 þ 0.002 eV more stable than R state. It appears that theoretically the calculated ground state is quite confusing, where the following three species of the ground state have been reported: a broken symmetry antiferromagnetic singlet state [12,19,21], a triplet state [14,18] and an 11-tuplet state [14–17]. On the other hand, the

Mn–Mn distances calculated are in the range from 1.6 to  and the calculated binding energy ranging from 3.5 A, 0.63 [16] to 1.63 eV [18]. The rather confused electronic structure on Mn2 in theoretical calculations arouses us to take again into account its geometry and magnetic coupling interaction with post-HF calculation at a higher level. As a part of our studies on magnetic coupling interaction in molecules, in the present work our calculations on the equilibrium interatomic distance, the binding energy and the magnetic coupling interaction for the neutral Mn2 dimer will be reported, by using multiconfigurational second order perturbation theory with a CASSCF (complete active space self-consistent field) reference function (CASPT2) [22–25]. In this Letter, we pay more attention to quantitatively examine the magnetic exchange interaction under the different spin multiplets for Mn–Mn dimer. The Letter is organized as follows: Section 2 is involved in computational approach; Section 3 presents the calculated results and discussion. The conclusions are given in the final section.

2. Computational details Due to only one variable in the potential energy surface (PES) of the manganese dimer, we explored the  of Mn–Mn distance, r, with PES from 2.6 to 4.4 A total spin S ¼ 0, 1, 2, 3, 4 and 5, respectively, based on the CASPT2 method. In our calculation, a CASSCF wavefunction is generated to treat the static electron correlation in the manganese dimer. Because the CASSCF method cannot availably predict a bound Mn2 dimer, herein the CASSCF wave function is taken as the zeroth-order wave function in the later perturbation treatment of the dynamic electron correlation, where 14 valence electrons were distributed in the active 3d and 4s orbitals, keeping the 3s, 3p orbitals inactive as frozen core orbitals. The remaining correction of dynamical electron correlation energy was considered through second-order perturbation theory (CASPT2), which expands the first-order wave function in the configuration space spanned by single and double excitations operating on the zeroth-order CASSCF wave function. For describing the manganese atom, we used Dolg’s [26] effective core potential basis set (abbreviated as ECP10mdf), it is that the Ne-like 10e core was replaced by the relativistic small core effective potential, while the 3s 3p 3d 4s 4p valence orbitals were treated by (7s 5p 5d)/[5s 3p 2d] Gaussian type basis functions. In addition, diffuse functions (1s 2p 1d) and a polarized f function were also included in the valence basis set functions. All the calculations have been performed by using MO L P R O code, version 2002.6 [27].

B. Wang, Z. Chen / Chemical Physics Letters 387 (2004) 395–399

The magnetic coupling interaction in the manganese dimer was described through Heisenberg exchange Hamiltonian, which can be written as

-207.520

H ¼ J S a
S b ;

ð1Þ

-207.522

where S a and S b are the electron spin operators on the two manganese atoms, respectively, and J is an exchange coupling constant. The eigenenergies of Eq. (1) are expressed as follows:

-207.523

EðSÞ ¼ ðJ =2Þ½SðS þ 1Þ  2sðs þ 1Þ;

DS;S1 ¼ EðSÞ  EðS  1Þ ¼ JS:

ð3Þ

If the contribution from the biquadratic jðS a
S b Þ2 term is considered, we have the isotropic exchange interaction Hamiltonian: 2

H ¼ J S a
S b þ jðS a
S b Þ :

ð4Þ

The eigenenergies of Eq. (4) can be obtained as J EðS; Sa ; Sb Þ ¼  SðS þ 1Þ  Sa ðSa þ 1Þ  Sb ðSb þ 1Þ 2 2  S þj : ðS þ 1Þ  Sa ðSa þ 1Þ  Sb ðSb þ 1Þ 2 ð5Þ Thus, for Mn2 the energy interval between the adjacent spin states, S and S)1 is,   35 DS;S1 ¼ EðSÞ  EðS  1Þ ¼  J þ j S þ jS 3 ; ð6Þ 2 where the factor 35/2 places the energy origin at S ¼ 0 state.

3. Results and discussion

S=5 S=4 S=3 S=2 S=1 S=0

ET / hartree

-207.521

ð2Þ

where S represents the total spin of the system and s is the atomic spin. Assuming that the magnetic coupling in the manganese dimer is solely a spin exchange interaction between 3d electrons in manganese atoms, one can obtain the energy difference corresponding to the different total spin S, that is the known as Lande interval rule,

397

-207.524

-207.525

-207.526

-207.527 2.8

3.0

3.2

3.4

3.6

3.8

4.0

4.2

4.4

4.6

rMn-Mn / angstrom  for each spin state S Fig. 1. Potential energy surface from 3.0 to 4.4 A at CASPT2/ECP10mdf level.

slowly in the range of the large Mn–Mn distance. In each curve there is a minimum of potential energy, and  of the minimum points are located from the 3.64 A  S ¼ 0 to 3.79 A of S ¼ 5 with increasing S. Herein, the variation trend of our calculated equilibrium Mn–Mn distances is in agreement with the evaluation in the previous report [28]. It should be pointed out that the potential energy curve of S ¼ 0 has the lowest energy. This point clearly indicates that the Mn2 system has a  of Mn–Mn distance. On singlet ground state at 3.64 A the other hand, the fact that each potential energy curve has a minimum, indicates all the low-lying spin multiplets with S ¼ 1, 2, 3, 4 and 5 to be possible populated excited states under lower temperature, which is consistent with the previous ESR experiments [4,6,7]. The calculated equilibrium Mn–Mn lengths, total energy ET , binding energies Be are listed in Table 1. The plots of Re and Be values versus the total spin S are depicted in Fig. 2. From Table 1 and Fig. 2, it is obvious that with increasing the total spin S, the equilibrium Mn–Mn distance Re increases linearly, but not to be a fixed value, which shows that the equilibrium Mn–Mn distance Re indeed is correlated with the total spin S. The calculated  Mn–Mn distance in the ground state (S ¼ 0) is 3.64 A,

3.1. Potential energy curve for Mn2 dimer To our knowledge, the ground state of Mn2 has not yet been confirmed well theoretically [10–21]. Here, we calculated the potential energy curves for Mn2 with the total spin S ¼ 0, 1, 2, 3, 4, 5 by using the CASPT2 method. The potential energy curves for the Mn–Mn  are depicted in distance r ranging from 3.0 to 4.4 A Fig. 1, where potential energy curves have the total spin S ¼ 5, 4, 3, 2, 1, 0 from the top to bottom in sequence. It is evident from Fig. 1 that with increasing the Mn–Mn distance all potential energy curves decrease sharply in the range of the small Mn–Mn distance, but increase

Table 1 Multiplicity structure, Mn–Mn distance Re , total energy ET , binding energy Be , and exchange interaction energy interval DS;S1 calculated at CASPT2/ECP10mdf level Spin

Re  (A)

ET (Hartree)

Be (eV)

0 1 2 3 4 5

3.64 3.66 3.69 3.72 3.76 3.79

)207.5270753 )207.5269955 )207.5268734 )207.5267272 )207.5265767 )207.5264685

0.12 0.12 0.11 0.11 0.11 0.10

DS;S1 (cm1 )

DS;S1 (cm1 )/S

)17.5 )26.8 )32.1 )33.2 )24.0

)17.5 )13.4 )10.7 )8.3 )4.8

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B. Wang, Z. Chen / Chemical Physics Letters 387 (2004) 395–399

Be / eV

0.13 0.12 0.11 0.10

Re / angstrom

0.09

3.80 3.75 3.70 3.65 3.60

S Fig. 2. Equilibrium Mn–Mn distance Re , binding energy Be versus total spin S.

comparable with the experimentally measured value of  [4,7] isolated in matrix. On the other hand, as the 3.4 A total spin S increases, the total energy ET and the binding energy Be both have a decreasing trend. It is suggested that the increase of equilibrium Mn–Mn distance Re is responsible for the decrease of total energy ET and the binding energy Be . It is evident that the calculated binding energy Be is a quite small value (from 0.10 to 0.12 eV). Considering very small zero energy correction, these calculated binding energies Be are consistent well with the small experimental observed values [1–7]. 3.2. The magnetic coupling interaction in Mn2 dimer To the best of our knowledge, a lack of detailed theoretical calculation on magnetic coupling energy in the Mn2 dimer was reported. Only previous theoretical estimation for magnetic coupling constant J ¼ 4:1 cm1 was done by Nesbet [10] on the bases of ab initio Hartree–Fock calculation, but the obtained J value is obviously less than the experimental observation of )9 to )10 cm1 . We herein evaluated the intervals of the exchange interaction energies between Mn and Mn, DS;S1 . The calculated results are listed in Table 1. Our calculated DS;S1 are based on the equilibrium geometry at each spin state. In Table 1, it is obvious that the calculated magnetic coupling DS;S1 is all negative, which indicates an antiferromagnetic coupling characteristic, consistent with experimental observation. It is interesting to note that the calculated DS;S1 =S are not a fixed constant shown by Lande interval rule (cf. Table 1 and Eq. (3)). In the case of the Mn–Mn dimer studied, the calculated DS;S1 =S are found to range from D1;0 =1 ¼ 17:5 cm1 , D2;1 =2 ¼ 13:4 cm1 , D3;2 =3 ¼ 10:7 cm1 , D4;3 =4 ¼ 8:3 cm1 to D5;4 =5 ¼4:8 cm1 , in which the calculated D3;2 =3 ¼ 10:7 cm1 is comparable with the experimentally determined value ()10.3  0.6 cm1 ) for the excited S ¼ 2 state of Mn2 based on ultraviolet spectroscopy [8], and

Fig. 3. DS;S1 versus total spin S with calculated values marked by square and Lande interval rule marked by triangle.

also our calculated average DS;S1 =S ¼ 10:94 cm1 for all spin states is comparable to experimental value ()10.0  2.0 cm1 ) for S ¼ 0, 1, 2, 3 states on ultraviolet– visible and Raman spectroscopy [9]. It is interesting to point out that our calculations herein agree well with ESR measurements of Cheeseman et. al. [28], who found a clear deviation of the magnetic coupling from Land interval rule (cf. Eq. (3)). For clearly showing this deviation in Fig. 3, we give a plot of the calculated DS;S1 versus total spin S where the string marked by triangles is obtained by Eq. (3) assuming D1;0 ¼ J and the curve marked by squares comes from the calculated DS;S1 herein for each spin state at their equilibrium geometries. It is obvious that the curve significantly deviates from the linear line of ‘Land interval rule’ and the larger S more departure from the Land line. In order to further examine the reason of the deviation, we consider the biquadratic jðS a
S b Þ2 term contribution in spin Hamiltonian (cf. Eq. (4)), then applied Eq. (6) to simulate the calculated DS;S1 as mentioned above. The J 0 of )22.1 cm1 and j of )0.41 cm1 are obtained. The larger j value shows that the biquadratic jðS a
S b Þ2 term contribution would be responsible for the deviation from the Lande interval rule in Eq. (3).

4. Conclusion CASPT2 method is used in the study on the geometry, binding energy and magnetic coupling character of the Mn–Mn dimer. The calculated equilibrium Mn–Mn  the binding energy distance in the ground state is 3.64 A, is 0.12 eV. It is found that the calculated energy interval of the exchange interaction DS;S1 =S is not a fixed value, ranging from )4.8 (S ¼ 4 ! 5) to )17.5 cm1 (S ¼ 0 ! 1). CASPT2 results indicate that the Mn2 dimer is in fact a van der Waals bound dimer with the

B. Wang, Z. Chen / Chemical Physics Letters 387 (2004) 395–399

weak antiferromagnetic coupling between two Mn atoms. The calculated equilibrium Mn–Mn distance and exchange interaction energy are in good agreement with the experimental observation. Our calculations also found that with the increasing total spin S in the Mn2 dimer, the equilibrium Mn–Mn distance Re increases but the binding energy Be decreases. The significant deviation of the energy interval of the exchange interaction from the Lande interval rule is primarily attributed to the biquadratic jðS a
S b Þ2 term contribution in spin Hamiltonian for the Mn–Mn dimer.

Acknowledgements This project is supported by the National Nature Science Foundation of China (Grants 20273005), Doctoral Program Foundation of Education Minister (20030001066) and State Key Project of Fundamental Research of China (Grant G1998061305).

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