Portrait of diatomic FeN. A theoretical study

6 June 1997

CHEMICAL PHYSICS LETTERS ELSEVIER

Chemical Physics Letters 271 (1997) 143-151

Portrait of diatomic FeN. A theoretical study Andreas Fiedler, Suehiro Iwata Institute for Molecular Science, Myodaiji, Okazaki 444, Japan Received 24 February 1997

Abstract

The ground and low-lying excited states of the diatomic molecule FeN were systematically studied. For this purpose a density functional/Hartree-Fock hybrid method and the internally contracted averaged quadratic coupled cluster approach with large basis sets were used. In agreement with previous reports the calculations revealed the 2A state to be the lowest in energy and only 3 kcal/mol higher lies the 4II state. In addition, we found another very low-lying 4(I) state and an even less energy demanding state of 6]~+ symmetry. The computations showed excitation energies of 5 and 0.5 kcal/mol, respectively. If the uncertainties of the methods are considered, all four states are good candidates for the real ground state.

1. Introduction

Transition metals in general, and iron in particular, play an essential role in nitrogen fixation which is an important process both in biochemistry and in industrial applications. In order to understand the basic principles of such complex reactions the study of small model systems might be helpful. Thus, it is not surprising that bare diatomic FeN has received interest in basic research, both from theory [1-3] and experiment [4,5]. Siegbahn and Blomberg published two papers on FeN. In the earlier paper [1] they examined two doublet states, 2A and 2E+, with the complete active space self consistent field (CASSCF) [6] method and multi reference configuration interaction (MRCI) [7] wave functions. They concluded that the 2A state is the lowest, the dissociation energy D e being only 21 kcal/mol and the energy separation Te of the two states 30 kcal/mol. Later [2], motivated by the effective medium approach [8] of Raeker and DePistro [3], who obtained a value for De as large as 121 kcal/mol, they applied larger basis sets and extended the study to the doublet states, 2A, 21-I and 2~+, as well as the quartet states, 4A, 4II and 4~+. For three of these states, 2A, 2~+, and 4II, they used the newly developed internally contracted [9] average coupled pair function [ 10] (IC-ACPF) for the many-electron wave function. They again found a 2A ground state and applied empirical corrections to their calculated value for De of 38 k c a l / m o l and estimated it to be about 48 kcal/moi. On the experimental side Chertihin, Andrews, Citra, and Neurock identified FeN in the matrix infrared spectra (w -= 938 cm ~) of the reaction products of laser ablated Fe atoms with molecular nitrogen or nitrogen oxide [4,5]. To support their assignment they carried out calculations using density functional theory [11] (DFT), primarily to evaluate the isotopic shifts of the harmonic frequencies. The non-local functionals of Becke [ 12] and Perdew [13] were applied. Surprisingly, they found the ground state to have quartet multiplicity (the authors did not specify spatial symmetry). 0009-2614/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved. PII S 0 0 0 9 - 2 6 1 4 ( 9 7 ) 0 0 4 4 4 - 2

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In the present paper we report a systematic study on the low-lying states of bare FeN approximately up to the dissociation limit. For the lowest states the geometry has been optimized with a D F T / H F hybrid method [14] (B3LYP functional). To evaluate the excitation energies we applied the recent developed multireference internally contracted [9] averaged quadratic coupled cluster [15] (AQCC) approach using state averaged CAS-orbitals (see below) and large one-particle basis sets. The states are illustrated by MO- and VB-descriptions which have often been shown to be very useful tools for the understanding of chemical and physical properties of molecules containing transition metal atoms [16,17].

2. Computational details Unfortunately, the reliable theoretical treatment of transition metal compounds still constitutes a fundamental challenge to state-of-the-art quantum chemistry [18]. In numerous computational studies, DP-T or D F T / H F hybrid approaches have successfully been applied for a qualitative examination of such systems [19,20]• However, such methods have some fundamental limitations. They can only be applied for the lowest-lying state in a given spin multiplicity and spatial symmetry. In addition, it is well known that a single determinant is not always a spin eigenfunction. This leads to mixed spin states in MO-calculations for low-spin coupled open shells. Surely this can be translated to DFT, but it is not as straightforward since the Kohn-Sham determinant represents only the wave function of a noninteracting reference system [21]. The next difficulty may arise from the fact that most available program codes do not allow calculations with complex orbitals and use symmetry of abelian point groups only. Accordingly, the treatment of systems of non-Abelian point groups, particularly linear molecules, in reduced symmetry may lead to the wave function of mixed spatial states. We shall illustrate this by the following example that arises in FeN. The quartet coupled ~3~1o'1 configuration gives rise to configuration state functions with A = +1 and +3, thus a l l and 40 states [22]. The corresponding Slater determinants

4(I)A= +3 -~

la+2~+2~-27r+ 1Ool

4~A=_3=la+2~_2~_27r_10" 0] 41-IA= + 1 = 1a+2~+2~-27r- lorO] 41-IA= -1 = la+2 6- 26- 2rr+ l°'ol

are projected into C2v space: the complex orbitals 6+ 2 and 7r± l are written in terms of their C2v counterparts

(ar(tx2_y2) +_ i. a2(txy)) and (bl(Trxz) + i- bE(Trvz)), respectively, while ~r straightforwardly transforms to al(o-z). The Slater determinants can be rewritten (normalization omitted, order of orbitals remains 16t587ro'l) as

4 % = +3 =

[al a2-dTb l al l - lal a2-~2be al [ - i( ]at a2-d72bla, ] + ]al a2"dTb2 a, l)

4~a= -3 = [a~a271bla~l - [a~a2-~2b2a~l + i(lala2"~2b~al] + lala271b2al]) alia= +l = la~ a271bl a~ ] + ]al a272b2 al l - i( la~ a272bl al l - [al a271b2 al l) 4IIA= _~ = [ata2~bla~l + la~a272b2a~t + i(]a~a272bla~] - la~a2~bEa~l )

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In order to eliminate the real or the imaginary terms we form linear combinations of the complex determinants with same absolute [AI. 4(I)4= +3 + 4 " 1

-3 =

la,a2~bla, I- laLa2-~2b2all = 4 B 2 ( 4 ( I ) )

4(IDA 43 -- 4(I)4 = -3 = ]ala2-~2b,al] + [a,a2~lb2al] = 4 B , ( 4 q b )

4H~= +l + 4 I I , ~ = - I =

]ala2~blal[ + lala2-~2b2al] = 4 B 2 ( a H )

4H,~= + l - 4 I I . l = - l =

lala2-d~zblall - lala2-~lb2all=4Bl(4II)

"

As shown, both the 4qb and 4H states must be described by two real determinants as well in 4B~ as in 4B 2 symmetry. The states differ in the sign in between these functions. Consequently, a single real determinant represents a 1:1 mixture of 4(I) and 4 II. Fortunately, for the estimation of the energy of these states the mixed state is a rather good approximation; since the states only differ in the A coupling they can be expected to lie very close to each other. We note that for doublet states, 2all0 and 2~I, arising from the g37r~r~ configuration more than one complex determinant is necessary to achieve spin adaption. Here, the ansatz of one real determinant leads to a symmetry and spin mixed state (2Fl, 2qb, 411, 4qb). Because the splitting between states having different multiplicity may be high, a single reference approach could lead to larger errors. Finally, the twofold degeneracy of rr and g orbitals is apparently not ensured in the C2v point group, and thus, unphysical symmetry breaking may occur. Obviously only a multiconfiguration treatment will be able to deliver a complete picture of transition-metal containing diatomic molecules. As a first step the CASSCF approach is straightforward. The easiest choice of the active orbital space is to construct it from all valence orbitals of the atomic fragments. Fortunately, through rapid development in computer technology and improved algorithms it is now possible to treat effectively a rather large number of configuration state functions (CSF) that are necessary in the calculations of transition metal compounds [23]. In order to simultaneously treat excited states of the same symmetry, which otherwise often are not accessible due to root flipping problems, a state averaged procedure is desirable. Further, in our experience this can be a useful tool to minimize the problem of artificial symmetry breaking in calculations of atoms and linear molecules. In this intention all configuration state functions of the different C2, irreducible representations belonging to the same state (see above) are treated simultaneously. Besides the non-dynamical electronic correlation, which is taken into account by CASSCF, dynamic correlation has to be considered extensively even for a qualitative description of metallic systems [18]. On basis of the CAS zeroth order wave function this can be achieved by multireference variants of many-body perturbation theory [24], the averaged coupled pair function (ACPF) method [10], or the recently developed multireference averaged quadratic coupled cluster (AQCC) expansion [15]. Classical truncated multireference CI (MRCI) [7] lacks size consistency and is therefore less advisable [25]. For ACPF and AQCC very effective codes have been implemented recently. The internally contracted variants available in the Molpro96 program [26] allow this type of computations using reasonable large one-particle basis sets. For larger sized systems the AQCC method is probably the most complete treatment of correlation energy yet possible. This approach, however, is very time consuming even for diatomic FeN, and therefore reasonable compromises between accuracy and computational times are still necessary. In the present project we are primarily interested in the energies of the lowest states, and thus, some approximations for the geometry optimizations are tolerable. To account for the effects of larger changes in the equilibrium bond length of the lowest states, we applied the economic B3LYP functional [14] and standard 6-311G ~ basis sets as implemented in the Gaussian94 program [27]. This level of theory has been shown previously to result in geometric structures near to those of highly correlated MO-methods [19]. However, the states are approximated by a single determinant and, as discussed above, for some of them spin a n d / o r spatial contaminations are severe in the calculations. Naturally, no geometry optimizations for the higher excited states could be performed at this level of theory, and thus, in the following high level calculations only vertical excitation energies are accessible (see below). We did

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Table 1 Atomic distances (r), harmonic frequencies (~o), and excitation energies (T) for the low-lying states of diatomic FeN calculated at the B3LYP and AQCC levels of theory State

rFe N/A B3LYP

~o/cm-I B 3 L Y P

T/kcal/mol B 3 L Y P

T/kcal/mol AQCC

2A 2E+ 21I 20 211 20 4Fl 4(I) 41-I 4A 42£+ 4A 4£+ 4£4A 6£+ 6A 6[1 60 611 611 8A sE811

1.626 1.461 1.550 ~

594 1235 899 "

0 31 22 ~

1.633 b

768 b

7 b

1.626

712

11

1.654

715

14

1.770

514

46

2.255

259

40

2.240

225

41

0 19 20 21 25 37 3 5 14 22 27 31 33 38 40 0.5 42 26 31 31 36 35 43 40

" Mixed state of 2II, 20, 41I, 4(I) symmetry due to the single determinant approximation, i.e. the l a l ( 1 6 ) a 2 ( l ~ ) ~ b 2 ( 4 ' r r ) a l - - ' ~ l Kohn-Sham determinant was applied; see computational details. b Mixed state of 411, 40 symmetry.

not choose the alternative C A S S C F geometry optimizations since this method is considerable more elaborate and furthermore trends to considerably overestimate bond length in transition metal systems due to the lack of dynamic correlation [ 18]. Flexible atomic natural orbital ( A N O ) basis sets have been utilized for multi reference based single point calculations using the Molpro94 program package [26]. The metal was described by P o u - A m e r i g o ' s ( 2 1 s l 5 p l 0 d 6 f 6 g ) / [ 7 s 6 p 5 d 4 f 2 g ] basis set [28] and nitrogen by W i d m a r k ' s (14s9p4d3f)/[6s5p3d2f] basis set [29]. The ls2s2p3s3p(Fe) and Is(N) electrons have been treated as inactive in the full valence (4s3d(Fe) and 2s2p(N)) C A S S C F calculations and frozen in the following internally contracted A Q C C computations. In the latter all configuration state functions having a coefficient larger than 0.05 in the C A S S C F were included as reference configurations. In order to figure out the low-lying states of FeN, first the four lowest roots in A 1 and A 2 symmetry for the states with even A and in Bj and B 2 symmetry for odd A, respectively, have been 2 averaged at the C A S S C F level of theory. For the case of B~ and 2B 2 we used five roots since we expected more low-lying states. On the basis of in such way generated orbitals these four (five) roots have been treated simultaneously in the following A Q C C calculations including the references of the two symmetries. At this level o f approximation the vertical state splittings relative to the lowest state for each C2v irreducible representation and multiplicity are reported (in Table 1 separated by a thin horizontal line). The lowest states have been refined by applying the same level of theory, but this time the C A S S C F was carried out for the pure state. We note that the last step shifts the total energies mostly neglectable and always below 5 k c a l / m o l . This shows the usability of the state average method. To determine the dissociation energy ( D ) the ground states of the isolated atoms

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have been calculated using the corresponding state averaged procedure for the roots in D~h. In all multi reference calculations special care was taken that no orbital or root flipping falsified the calculations which turned out to be quite painstaking in some cases. Furthermore, no corrections for relativistic effects [30], core-valence correlation, zero point vibration energy, imperfect basis sets [31], and incompleteness of the correlation treatment [32] have been applied. For the calculation of the dissociation energy these corrections could sum up to the order of 10 kcal/mol but are probably less important for excitation energies. Here the largest uncertainty doubtless results from non optimizing the atomic distances for all the states. In addition, in the AQCC treatment of some higherqying states additional excited configurations appeared with coefficients slightly larger than the 0.05 selection threshold for the references. For these states inclusion of the iron 4p orbitals into the active CASSCF- and reference space would be desirable, however, this exceeds limitations of the used program and our computational resources. Nevertheless. theoretical studies on transition metal systems seldom reach spectroscopic accuracy and thus the reported values should be more regarded as semi-quantitative and as hints for the interpretation of hopefully forthcoming experimental data.

3. Results and discussion

First, we analyze possible electronic states of FeN [22]. At the dissociation limit, the ground state of the system is Fe(s2dr: 5D) + N(s2p3; 4S°), to which the molecular states, 2s+ iA, 2s+ iFi ' and :s+ 1~- (S = 1/2, 3/2, 5 / 2 , 7/2), are correlated. The first excited state at the limit is Fe(sld v. 5F) + N(sZp3; 4S°), from which the molecular states, eS+l~, 2s+ JA, 2s+~II, and 2s+1£+, arise. This asymptote lies 20 kcal/mol above the ground state. Nevertheless, this configuration can be expected to play an essential role in bonding since covalent bonding can be realized by the unpaired electron in the 4s orbital which is much larger than a tight 3d orbital [33] and thus allows better overlap with a p,~ orbital of the ligand. Additionally, in the chemical bonds of transition metal atoms the ionic structures are important. The dissociation limit of the lowest ionic structure is Fe +(sld 6"¢'D) + N

(sep4;4p).

which is nearly degenerate (6 kcal/mol) with Fe+(d7; 4F) + N-(s2p4; 4p). Many molecular states result from these two ionic structures, some of which contribute to the low-lying molecular states. The low-lying states of FeN molecules are adiabatically and diabatically correlated with these states at the dissociation limit, and the manifold creates some difficulties for reliable calculations and to guess the ground state of the molecule. In the VB-picture a formal triple bond for the molecule is suggested. The nitrogen atom has three singly occupied 2p orbitals which are prepared for covalent bonding. If hybridization of one singly occupied 3d,~ orbital and the 4s 2 orbital at iron is assumed, one singly occupied sd,, hybrid orbital will form the ~r bond with the 2p,~ orbital of nitrogen. The second sd~, hybrid orbital, which is doubly occupied, polarizes charge out of the bonding axes to reduce electrostatic repulsion. Two 'rr bonds are formed by the singly occupied 3d,,(Fe) and

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I0~ 4J~

-++-4--++-t-{2 Fig. 1. MO scheme for the valence space of FeN A.

2p~(N) orbitals. There remain three electrons in two 3d~ orbitals on iron. For the molecule this doublet 83 configuration consolidates a 2A state. The corresponding MO scheme (valence space only) and occupation pattern for this state is shown in Fig. 1. Low-lying 7o. is largely the pure 2s orbital of nitrogen. The 8o. orbital is the bonding orbital between 2p,~(N) and the sd,~ hybrid orbital, though it is strongly polarized towards the electronegative nitrogen center. Orbitals 3w also have bonging character. An energetically tight group of non- or slightly anti-bonding orbitals comes next. Orbitals 18 are pure iron d 8 orbitals which find no symmetry match on the ligand. They are followed by the second sd,~ hybrid orbital 9o.. However, the d-character dominates in this slightly anti-bonding orbital in order to avoid the stronger electrostatic repulsion expected for the larger 4s orbital. Orbitals 4 w are anti-bonding in nature. However, the bonding and anti-bonding splitting of 3'rr and 4w is rather small because of the unfavorable overlap of 3d~(Fe) and 2p~(N) orbitals. Thus, w-bonding is rather week and consequently the 4'rr-orbitals lie closely in energy to the 18 and 9{r orbitals. Finally, orbital 10o. is the anti-bonding counterpart to the o.-bond. We note that these orbitals compose the active space in our CASSCF calculations. Actually, the 2A state is calculated to be the lowest in energy at the AQCC level of theory and the dissociation energy amounts to 44 kcal/mol. In the following all given excitation energies (T) refer to this point (Table 1). From Fig. 1 we see that the 7o. 28 o. 23 rr 41839{r 2 configuration corresponding to the 2A state does not follow the simple aufbau principle; if the lowest orbitals should be occupied first the resulting electron configuration would be 840"l, which is the 2~+ state. As shown above, however, the 2~+ state is not correlated to the ground state of the dissociation limit (positive parity, two d-orbitals doubly occupied), and the 84o. ~ main configuration belongs to the excited Fe 5D; sld 7 atomic asymptote. In fact, the 2~+ state is calculated to be 19 kcal/mol higher in energy, reflecting the atomic excitation. However, the yet applied one configuration picture is a coarse simplification. Looking at the wave functions of both states we can consider strong contributions of other CSFs which arise mainly from excitations inside the non-bonding block. In addition, some depopulation of the bonding 3'rr orbitals takes place, and even some occupation of the anti-bonding 10or orbital can be seen. Naturally, rather large weights for double excitations into the virtual space shall be mentioned too. Thus, the similarity of atomic and molecular excitation energy is somewhat fortuitous. It, however, shows that these simple models can be useful for an at least qualitative understanding of trends. We found more doublet states in this excitation regime. Their main configuration is 83o.lw 1, a 9o"---, 4w excitation from the 2A configuration. As shown in Table 1, the lowest among them is 217 ( T = 20 kcal/mol) followed by 2qb, 21I, and 2qb states ( T = 21, 25, and 37 kcal/mol). The 18 ~ 4'rr excitation is more energy demanding ( T = 4 3 kcal/mol); this main 82o"2'rrI configuration belongs to the third 211 state below the dissociation energy. Strong mixing with the lower 217 states can be seen. We note that the configuration 82o"2'rr~ can give rise to states with A up to 5, that is a 2H state. However, these high axial angular momentum

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states obviously lie high in energy since they are not correlated to the neutral dissociation limit. The doublet coupled ~2,rl-rr 2 configuration, resulting from 9or ~ 4nr and 1~ ~ 4"rr double excitations, may even lead to a 2I state. This states as well did not appear in the examined lowest five roots in C2v symmetry. As illustrated, the small energy gaps between the non-bonding orbitals result in some relatively low-lying excited doublet states. It is well known that for states having the same electronic configuration the highest multiplicity is preferred because of the higher exchange energy. The exchange energy terms for 3d electrons in iron are roughly 25 k c a l / m o l [34], and such amount of additional stabilization can even overweight loss gained through occupation of seemingly unfavorable orbitals. Considering the occupation of the lowest 2A state the 9or ~ 4~r excitation with spin inversion leads to the quartet coupled ~3~l"rr~ configuration related to 4H and 4(I) states. We find both states actually much more stable than their doublet counterparts lying as close as 3 and 5 kcal/mol, respectively, to the 2A state. Interestingly, the decrease of the formal bond order (BO = (n - n* ) / 2 ; where n and n* are the occupation numbers and * denotes anti-bonding) from 3 in 2A to 2.5 for these quartet states has little effect on the stability, which further supports our arguments for rather week w-bonds in FeN as discussed above. However, the formal bond order by itself depends on the assignment of bonding, non-bonding, and anti-bonding orbitals. If we consider the ~r-space non-bonding the bond order would be 1 for 2A as well as for these quartet states, which seems to be the somewhat better description. After all, especially for transition metal compounds bond order formalisms are rather fuzzy. In our calculations 411 symmetry is slightly lower in energy than 4qb. The figure for the 4H state agrees with a previous reported value at the ACPF level of theory ( 4 0 was not examined). Nevertheless, this finding is a contradiction to Hund's rule, that is maximum A for a given configuration should be favored. However, 4H might enjoy a small additional stabilization through configuration mixing with the quartet ~20"2Ti'1 configuration, which allows only a maximum A of 1. The latter occupation pattern forms the main configuration of the next root in the CI, and this second 4H state is located 14 kcal/mol above the 2A state. Higher roots clearly exceed the dissociation limit. The quartet states of 4E+ and 4A symmetry lie rather high. Somewhat unexpected, the lowest root in our AQCC calculation results from 9~ ~ 10(r excitation and has a main 1~39cr~ 10(r I configuration. Thus, formally the or-bond is partially broken in this 4A state lying 22 kcal/mol above 2A. However, such picture is much too simple in this case because strong mixing with the two electron excitation ~20"1 "/'r2 configuration and 3"rr ~ 47r excitations stabilize this state. The lowest 4£+ state is only slightly more energy demanding (T = 27 kcal/mol). The wave function is built from ~2cr~Tr2 CSFs having the 13 electron in 9or or 4Ir. It follows the second 4A state ( T = 31 kcal/mol) with the identical main 82(r~Tr2 configuration, however, two electrons are occupying the same 4rr orbital in order to maximize A. It is worth mentioning that a second 4E+ state (T = 33 kcal/mol), a first 4E state ( T = 38 kcal/mol), and the third 4A state ( T = 40 kcal/mol) are found still below the dissociation limit. For sextet multiplicity there is exactly one configuration generated by the occupation of low-lying orbitals only. As expected, the lowest sextet state results from this 82crier 2 configuration with all electrons high-spin coupled in the non-bonding block. Somewhat surprising, this 6~+ state is calculated only 0.5 kcal/mol higher than the lowest-lying 2A state at the AQCC level of theory. Thus, it is even lower than the lowest quartet states in spite of a further decrease of the formal bond order. This can be attributed to the huge exchange energy for this configuration and the absence of self-energy of electrons in the same orbitals. Interestingly, this state corresponds to the ground state of the isoelectronic diatomic FeO + ion. Actually, the computed large dipole moment for FeN 6E+ is as large as 3.08 D, compared to 2.13 D, 2.10 D, and 2.18 D for 2A, 4[I, and 40 states, respectively, which is pointing to the additional strong ionic contribution in bonding. Obviously, the Fe + sLd6 + N - s2p 4 asymptote dominates. Other sextet states must be based on configurations including occupations of the 10~ orbital a n d / o r excitations out of the bonding block. Consequently, they can be expected to be notably more energy demanding. The lowest-lying of these states ( T = 2 6 kcal/mol) is found to have 611 symmetry and the main 7cr28cr23w41829~r~47r~10(r ~ configuration is mixed ca. 2:1 with configurations resulting from additional

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3-rr ~ 4 7 r excitations. In the following almost degenerated 6~ and 6l-I states ( T = 31 kcal/mol) 10or occupation is less pronounced. They have main 7crZ8cr23~r31839cr14~r2 and 7cr28crZ3rr31829~24rr 2 configurations, respectively, and thus suffer from holes in 3w. The next 6[[ state ( T - - 3 6 kcal/mol) is the low A coupled counterpart to 6alP. Finally in the group of sextet spin states, we notice a rather high-lying 6A state (T = 42 kcal/mol). In octet multiplicity the electrons from the ground state atomic asymptote remain uncoupled. Such PESs are mostly repulsive for compounds of the first and second period. However, for transition metal systems these PESs might be lowered by configuration interactions with other low-lying exit channels. Geometry optimizations result in long atomic distances (Table 1) which indicates that only a weaker bound complex can be formed. The lowest state 8A is calculated to be 35 kcal/mol higher in energy than the global minimum 2A, and two further states 8I-I ( T = 40 kcal/mol) and 8 ~ - ( T = 43 kcal/mol) are calculated to be below the dissociation limit. In summary, we found the lowest state to have 2A symmetry stabilized by 44 kcal/mol regarding dissociation to ground state atoms. Slightly more energy demanding we found 411 and 4(I) states (T = 3 and 5 kcal/mol, respectively), and the computed excitation energy to a 6£+ state relative to the 2A state amounts to only 0.5 kcal/mol. The dissociation energy and the doublet-quartet splitting in our calculations are in good agreement with reports of Siegbahn and Blomberg who assigned a 2A ground state and found the 4[I state 3 kcal/mol higher in energy. However, these findings are contrary to results of Chertihin et al. who found quartet multiplicity to be 3 kcal/mol more stable than doublet spin. This is somewhat surprising, since the used BP functional is well known to underestimate exchange energy loss [35], and thus, should even favor the low-spin state. However, the lowest quartet states can not be well described by a single determinant approach, as discussed above. In principle, using only one real determinant should result in a 1:1 mixture of these nearly degenerated states and no larger error for the energy is expected. However, since the used Dgauss program uses no symmetry constrains symmetry breaking might be severe, thus, leading to unreliable results. On the other hand, in line with these authors we find a very low-lying sextet state 6£+ ( T = 0 . 5 kcal/mol). Their calculations predict 8 kcal/mol between the states of sextet and doublet spin. Such a slight overcharging follows the expected trend for DFT calculations. Their very high value for De of 82 kcal/mol and some overestimation of vibration frequencies might be attributed to the same weakness of DFT [35]. We note that the excitation energies evaluated using the B3LYP level of approximation show a still reasonable qualitative agreement to the higher level of theory (Table 1). In addition, the calculated dissociation energy for FeN 2A of 49 kcal/mol is very near the values of the AQCC method (44 kcal/mol) and of the recently reported ACPF approach (39 kcal/mol). However, the computed harmonic frequencies for the lowest-lying states, 2A, mixed 4[][/qb, and 6£+ are all much lower than the reported experimental value of 938 cm -~. We do not know if this displays a failing of the B3LYP method for FeN or if the difference can be charged to matrix effects in the experiments. Thus, unfortunately, we can not identify the ground state from this data.

4. Concluding remarks The low-lying states of FeN have been systematically examined at a reasonable high level of theory. Our approach is the most complete applied yet. On the other hand, the small energy differences between the lowest states might be smaller than the accuracy of any applicable methods for the current time. Thus, we refrain from finally assigning the ground state of FeN to 2A, 4[I, 4qb, or 6~+, which are all good candidates, and rather a spectroscopic examination of this interesting system is desirable. If the calculated order of states should be correct one can expect unusual spectroscopic properties. Since direct transitions between 2A and 6 £+ are very unlikely, rather the more energy demanding quartet states have to be passed as some kind of barrier. We hope

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that the present report will help in the interpretation of the probably quite complex forthcoming experimental data. Surely, this system highlights the variety of available bonding mechanisms for transition metal compounds.

Acknowledgements A postdoc fellowship of the Japanese Society for the Promotion of Science and a research grant from the Japanese Ministry of Education, Science, and Culture, which made possible the stay of AF at IMS are deeply appreciated. This work would not have been possible without the provision of computational resources from the Computer Center of IMS. Special thanks from AF to T. Tsurusawa for continuous technical support.

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