Theoretical investigation of iron carbide, FeC Demeter Tzeli and Aristides Mavridis Citation: J. Chem. Phys. 116, 4901 (2002); doi: 10.1063/1.1450548 View online: http://dx.doi.org/10.1063/1.1450548 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v116/i12 Published by the AIP Publishing LLC. Additional information on J. Chem. Phys. Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors Downloaded 15 Sep 2013 to 141.161.91.14. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions
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Theoretical investigation of iron carbide, FeCDemeter Tzeli and Aristides Mavridis Citation: J. Chem. Phys. 116, 4901 (2002); doi: 10.1063/1.1450548 View online: http://dx.doi.org/10.1063/1.1450548 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v116/i12 Published by the AIP Publishing LLC. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors
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JOURNAL OF CHEMICAL PHYSICS VOLUME 116, NUMBER 12 22 MARCH 2002
Theoretical investigation of iron carbide, FeCDemeter Tzeli and Aristides Mavridisa)
Laboratory of Physical Chemistry, Department of Chemistry, National and KapodistrianUniversity of Athens, P.O. Box 64 004, 157 10 Zografou, Athens, Greece
~Received 28 September 2001; accepted 21 December 2001!
Employing multireference variational methods~MRCI!, we have constructed full potential-energycurves for the ground state (X 3D) and forty excited states of the diatomic carbide, FeC. For allstates we report potential-energy curves, bond lengths, dissociation energies, dipole moments, andcertain spectroscopic constants, trying at the same time to get some insight on the bondingmechanisms with the help of Mulliken populations and valence-bond–Lewis diagrams. For theX 3Dstate at the MRCI level of theory, we obtain a dissociation energyDe586.7 kcal/mol at a bondlength r e51.581 Å. These values compare favorably to the corresponding experimental ones,De
Continuing our effort to rationalize the electronic struture and bonding in the diatomic metal carbide series~neutralor cations! M–C,1,2 M5first row transition metal elementwe report the theoretical investigation of the iron carb~Fe–C! molecule. In particular, we present high level mulreference variational calculations on a total of 41 states, cering an energy range of about 4 eV.
Due to the obvious interest, practical and/or academithe organometallic bond,3 diatomic metal carbides constitutinteresting, nominally simple systems, which can be thoughly studied and used as models for the more comppolyatomic organometallic compounds. However, desptheir relative simplicity M–C (M5Sc,Ti,V,...) diatomicsare not so easily cracked nuts, the reason being theknown complexity inherent in all systems bearing first rotransition metal atoms.4 Perhaps this is the reason whyabinitio calculations on the M–C series are not abundan2,4
Indeed, we are aware of only threeab initio5–7 and one den-sity functional~DFT! study8 on the FeC molecule.
The Nashet al. work5 deals with the FeC, FeC2, andFeC3 carbides at the SCF, MP4, and DFT level of theoThe Shim and Gingerich work6 involves the examination bymultireference variational methods and double zeta quabasis sets of the ground (X 3D) and a number of low-lyingexcited states of FeC. Their results will be contrasted wours in due course. Hirano and co-workers7 examined theground (3D2,3) and two excited states, namely, the1D and5P1,2, employing multireference methods and large basets including~scalar! relativistic corrections. We touch upotheir results as we move along.
On the experimental side, Balfouret al.,9 using laser-induced fluorescence spectroscopy, determined for thetime that the ground state of the FeC in the gas phase i3D i symmetry, with bond distance~s! r 051.596~1.591! Å for
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the X 3D3 (X 3D2) state. In addition, two electronic excitestates ofV(5uL1Su)53 symmetry and their relative postion with respect to the ground state have been identified9
In 1996 Allen et al.,10 observing for the first time thepure rotational spectrum of FeC, confirmed that the groustate is of 3D symmetry and determined accurately tground state rotational constant of56Fe12C, B09520 075.397 6(66) MHz~50.669 643 18~7! cm21!.
Six new electronic states of FeC have been determiand located relative to the ground state by Brugh aMorse11 by resonant two-photon ionization spectroscopThree of these states haveV53, one hasV54 (3F4), andtwo possessV52. Having also obtained the ionization energy ~IE! of FeC to be 7.7460.09 eV, they were able to~indirectly! determine its bond strength (D0) through theknown dissociation energy of FeC1:12
D0~FeC!5IE~FeC!1D0~FeC1!2IE~Fe!
57.7460.0914.160.31227.9013
53.960.3 eV59067 kcal/mol,
or De5D01ve/214
591.267 kcal/mol.
This is an upper limit value due to the uncertainty in theD0
value of FeC1.12
More recently, Angeliet al.,15 using threshold photoelectron spectra obtained aD0 value of FeC1 of 84.264.1 kcal/mol as contrasted to 4.160.3 eV594.567 kcal/mol of Hettich and Freiser.12 Using the newD0
(FeC1) value, one obtains,D0 (FeC)580.564.6 kcal/mol,or De (FeC)5D01ve/2
14581.764.6 kcal/mol.Aiuchi, Tsuji and Shibuya14 by dispersed fluorescenc
spectroscopy observed a new electronic state 3460 c21
above theX 3D2 , tentatively assigned to a5P2 symmetry~but see below!.
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4902 J. Chem. Phys., Vol. 116, No. 12, 22 March 2002 D. Tzeli and A. Mavridis
Finally, Leung and co-workers16 by laser infrared spectroscopy, they found the spin–orbit splitting between tX 3D2 andX 3D3 states to be 329.809 cm21.
We presently report the theoretical examination ofstates of the FeC diatomic correlating adiabaticallyC(3P)1Fe(a5D,a5F,a3F,a3H). In particular, the ground-state atoms C(3P)1Fe(5D) give rise to 27 states i.e.3S1@1#, 3S2@2#, 3P@3#, 3D@2#, 3F@1#, 5S1@1#, 5S2@2#,5P@3#, 5D@2#, 5F@1#, 7S1@1#, 7S2@2#, 7P@3#, 7D@2#,and 7F@1#, all of which but the3S1@1#, have been com-puted. Moreover, three more states were computed,3D@2#and 3G@1#, correlating to C(3P)1Fe(5F). We have alsocomputed eleven out of the twelve singlets, namely,1S1@1#,1S2@1#, 1P@3#, 1D@3#, 1F@2#, and 1G@1# correlating toC(3P)1Fe(3F), plus one1H ~Greek eta! state correlating toC(3P)1Fe(3H) fragments. Numbers in square bracketsdicate the number of states in each particular symmetry.
Following the philosophy of our previous work on simlar systems,1,2,17 for all 41 states studied we report potentiaenergy curves~PEC!, total energies~E!, binding energies(De), bond distances (r e), dipole moments~m!, spectro-scopic parameters and charge distributions. Emphasisbeen given in deciphering the bonding process with the hof Mulliken populations and simple valence-bond–Lew~vbL! diagrams.
In Sec. II we define the technical approach followed;Sec. III we present some related atomic numerical resuour main body results are discussed in Sec. IV, and sofinal conclusion and comments are presented in Sec. V.
II. TECHNICAL CONSIDERATIONS
For the Fe atom the ANO basis set of Bauschliche18
(20s15p10d6 f 4g) has been used but without theg func-tions; for the C atom the correlation consisted basistriple-z quality of Dunning,19 cc-pVTZ5(10s5p2d1 f ) wasemployed. Both sets were generally contracted@(7s6p4d3 f )Fe/(4s3p2d1 f )C#, thus containing 96 spherical Gaussian functions. However, and in order to monitorresults, three out of the 41 examined states, namelyground (X 3D), and the first two excited states (1D, 3S2),were re-examined using the complete Bauschlicher ba(1g) and the cc-pVQZ for the carbon atom contracted@(7s6p4d3 f 2g)Fe/(5s4p3d2 f 1g)C#, numbering 139 func-tions ~‘‘large basis’’!.
We are confronted here with an inherently multirefeence system of 12 ‘‘valence’’~active! electrons; thereforethe only methodology that can cope with such a systeparticularly if one wishes to construct full potential-enercurves~PEC! and a multitude of excited states, is a compleactive space self consistent field (CASSCF)1configurationinteraction ~CI! approach, specifically, CASSCF1single1double replacements5MRCI. Twelve valence electron(3d64s2 on Fe12s22p2 on C! were distributed to 10 orbit-als ~one 4s and five 3d’s on Fe1one 2s and three 2p’s onC!. Depending on the spin multiplicity and the spatial symetry of the state, our reference space~s! ~CAS! ranges from268 (7S2) to 5220 (3P) configuration functions~CF!, withcorresponding MRCI spaces ranging from 10 218 942 (7S2)to 66 339 660 (3S1) CFs. By applying the internal contrac
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tion ~icMRCI! technique,20 the number of CFs is reducedramatically ranging from 310 000 to 1 210 000 CFs, thmaking the calculations feasible and with tolerable lossetotal energies.17 The corresponding numbers using the larbasis set, for theX 3D state for instance, are 160 182 461 Creduced to about 1 600 000 CFs upon enforcing the icMRapproach. To estimate core (3s23p6) correlation effects, ic-MRCI calculations were performed out of the 12e2 CASSCFspaces including the 8 Fe core electrons, with the smallthe large basis sets, for the groundX 3D state and for the first(1D) and second (3S2) excited states and for a few poinaround the equilibrium. These calculations will be referredas C-MRCI. It is of interest to mention that the numberCFs involved in the C-MRCI~icC-MRCI! computations are723 362 352 ~15 500 000!, and 1 757 617 193~24 340 000!with the small and the large basis sets, respectively forX 3D and 23S2 states. These numbers rationalize our choof the basis set, ‘‘small,’’ and valence correlation approafor obtaining potential-energy curves for more than 40 Fstates.
All calculations were done under C2n symmetry con-straints, nevertheless care was taken for our CASSCF wfunctions to posses the correct axial angular momensymmetry, i.e.,uLu50(S6), 1 ~P!, 2 ~D!, 3 ~F!, 4 ~G!, and5 ~H!. This means thatD andG states are linear combinationof A1 andA2 symmetries,P, F, and H states are combinations of B1 and B2 symmetries, whileS1 and S2 statescorrespond to theA1 andA2 symmetry species, respectivelOf course, MRCI wave functions do not display, in generpure ~spatial! angular momentum symmetry, but ratherA1
~or A2! andB1 ~or B2!. With the exception of the ground, thfirst and second excited states, the state average~SA!approach21 was applied for all other states.
Finally, for the X 3D state coupled cluster calculationwere done, with~C-CCSD~T!! and without~CCSD~T!! the3s23p6 Fe core electrons. However, we were able to coverge these~T! calculations using only CASSCF orbitals tconstruct the~single! reference function. Lastly, the basis ssuperposition error~BSSE! was estimated for the grounstate by the usual Boys–Bernardi approach.22
Due to the relatively large number of active electro~12!, we encountered significant size-nonextensivity prolems, 9–10 mh at the MRCI level and for all states; bycluding the Davidson correction23 for unlinked clusters~1Q!, the nonextensivity error was almost vanished, droping to an average of 0.3 mh.
All calculations were performed with theMOLPRO 2000
suite of codes.24
III. THE Fe AND C ATOMS
Total energies of the Fe5D, 5F, 3F, and C3P terms,and energy splittings Fe(3F, 5F←5D) at the spherically av-eraged CASSCF, MRCI, and MRCI1Q levels of theory arepresented in Table I along with experimental values.13 Theagreement between theoretical and experimental valuesnot be considered as particularly good, the differences be0.176 and 0.180 eV or 20% and 12%, for the5F←5D and3F←5D separations, respectively~MRCI!. Although the
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4903J. Chem. Phys., Vol. 116, No. 12, 22 March 2002 Iron carbide
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TABLE I. Total energies~Hartree! of the Fe(5D, 5F, 3F) and C(3P) atoms, and atomic energy separations~eV!of Fe.
Davidson correction improves significantly, the formleaves the latter intact.
IV. RESULTS AND DISCUSSIONS
Table II gives total energies~E!, equilibrium distances(r e), dissociation energies (De) with respect to the adiabatiproducts, dipole moments~m!, Mulliken charges~q!, har-monic frequencies and anharmonic corrections (ve ,vexe),rotational vibrational couplings (ae), centrifugal distortions(D̄e), and energy gaps (Te) at the CASSCF, icMRCI, andicMRCI1Q level of theory, of forty one states with spatiaspin symmetries1,3,5,7S6, 1,3,5,7P, 1,3,5,7D, 1,3,5,7F, 1,3G and1H. SubscriptsG and L refer to global and local minimarespectively. Figure 1 shows relative energies of all stastudied covering an energy range of 3.7 eV, while Figpresents the totality of PECs. Each excited state has blabeled with a serial number in front of the symmetry symbrevealing its absolute energy order with respect to the gro~X! state, and a number in parentheses indicating its absoorder within the same space–spin symmetry manifold. Fures 3–10 present separately the PECs, according tospin multiplicity.
In the ensuing discussion we analyze first the triplfollowed by the quintets, and then the septets; 26 of thstates correlate to the ground-state atoms, Fe(3D)1C(3P),while three states correlate to the first excited state of5F(d7s1). Then we discuss the 11 singlets correlating tosecond excited state of Fe(3F)(1C(3P)), and finally one1H state correlating to the sixth excited stateFe(3H)(1C(3P)).
A. Triplets: X 3D, 2 3SÀ„1…, 5 3P„1…, 7 3D„2…,
8 3P„2…, 16 3SÀ„2…, 18 3F„1…, 20 3P„3…, 29 3D„3…,
33 3D„4…, and 34 3G„1…
X 3D~;0.81u~1/& !~1s22s23s11px21py
2!
3~1d12 1d2
1 11d11 1d2
2 !&5uA1&1uA2&,CASSCF).
Notice that our orbital notation refers only to the 1valence electrons, i.e., we do not number the doubly ocpied sixs and fourp ‘‘core’’ orbitals; uA1&1uA2& gives thespatial symmetry of the wave functions in C2n , and the ac-ronym ‘‘CASSCF’’ ~or ‘‘MRCI’’ ! refers to the origin of the
o 141.161.91.14. This article is copyrighted as indicated in the abstract.
s2enldte-eir
se
e,e
f
u-
leading variational coefficient~s!, 0.81 in the present case ithe CASSCF wave function.
In agreement with the experimental results9–11,14and re-centab initio calculations,6,7 the ground state of FeC is of3Dsymmetry. As we can see from Table II moving from thMRCI, to C-MRCI, to MRCI/large, and to C-MRCI/largethe X 3D De value increases steadily, to a final value of 86kcal/mol. Taking into account the BSSE correction~20.4kcal/mol! and scalar relativistic effects6 ~22 kcal/mol!, ourbest estimate isDe584.3 kcal/mol, within the range of experimental values, 91.267 kcal/mol ~upper limit!,11 and81.764.6 kcal/mol.15 Similarly our best C-MRCI/large basire51.581 Å, is in respectable agreement with the experimtal value~s!. Notice the shortening of re by about 0.01 Åwhen the Fe 3s23p6 core electrons are included in the MRCcalculation.
At infinity the wave function can be described by thproduct u5D;M562&Fe3u3P;M50&C. As the two atomscome together, Fig. 3, and around 4.5 bohr, theX3D statesuffers an avoided crossing with the 73D(2) state which cor-relatesdiabatically to Fe(5F)1C(3P). As a result thein situequilibrium atoms are of Fe(5F;M562)1C(3P;M50)character, and the bonding can be described pictorially byfollowing valence–bond–Lewis~vbL! icons
The above bonding diagram is in agreement wCASSCF Mulliken equilibrium atomic populations~Fe/C!,
4s0.844pz0.223dz2
1.213dxz1.184px
0.033dyz1.184py
0.03~3dx22y23dxy!3.00/
2s1.822pz0.872px
0.782py0.78.
Clearly we have twop bonds (1.1810.0310.7851.99 e2)32 and ones, the latter caused by the transfer of 0.7 e2
from the Fe 3dz2 to the 2pz orbital of the C atom. The extra;0.2 e2 of C 2pz are pumped from the 2s orbital. The finalresult is a genuine triple bond, with a total transfer of 0.
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.08
05
.2
6
4904 J. Chem. Phys., Vol. 116, No. 12, 22 March 2002 D. Tzeli and A. Mavridis
TABLE II. Absolute energiesE ~Hartree!, bond lengthsr e ~Å!, binding energiesDe ~kcal/mol!, harmonic frequencies and anharmonic correctionve , vexe
~cm21!, rotational vibrational couplingsae ~cm21!, centrifugal distortionsD̄e(cm21), Mulliken charges on CqC , dipole momentsm ~Debye! and energydifferencesTe ~kcal/mol!, at CASSCF, MRCI,a and MRCI1Qb level. Experimental and other theoretical results are also included.
Statec Methods 2E re De ve vexe ae(1023) D̄e(1027) 2qC m Te
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4905J. Chem. Phys., Vol. 116, No. 12, 22 March 2002 Iron carbide
TABLE II. ~Continued.!
Statec Methods 2E re De ve vexe ae(1023) D̄e(1027) 2qC m Te
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6.2
8.0
.49.8
.90.9
2.3
2.8
.83.4
.5
.4
4.1
.45.3
.9.7
.8
1.4
1.4
.92.6
.4
.50
.29.4
.40.4
4906 J. Chem. Phys., Vol. 116, No. 12, 22 March 2002 D. Tzeli and A. Mavridis
TABLE II. ~Continued.!
Statec Methods 2E re De ve vexe ae(1023) D̄e(1027) 2qC m Te
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.3.4
.0.81
.44.8
7.8
4907J. Chem. Phys., Vol. 116, No. 12, 22 March 2002 Iron carbide
TABLE II. ~Continued.!
Statec Methods 2E re De ve vexe ae(1023) D̄e(1027) 2qC m Te
aInternally contracted MRCI.b1Q refers to the multireference Davidson correction.cThe numbers in parentheses refer to the ordering of states according to energy within the same symmetry manifold.dCore 3s23p6 of Fe atom included at the MRCI.eLarge basis set,@7s6p4d3f 2g#Fe/cc-pVQZ/C .fReference 5, MP4~or DFT!, no specification of the state.gReference 8; it is only reported that the ground state is a triplet, the spatial angular momentum is not specified.hReference 6, MRCI/@8s6p3d1f /Fe4s3p1d/C#.iScalar relativistic corrections included.jReference 7, MRCI1Q/@8s7p5d3f 2g/Feaug-cc-pVQZ/c#.k 3D3 .l 3D2 .mReference 9.nReference 10.oDe values corrected for BSSE.pReference 11.qThe De value has been extracted using theD0(FeC1) value of Ref. 12.rThe De value has been extracted using theD0(FeC1) value of Ref. 15.sReference 14.tDG1/2 , DG1/2>ve22vexe .uReference 17.v 5P1 .w 5P2 .x 3F4 .yOur results suggest that this experimentally assigned3F(1) state could be the3P(2) state.z 3F3 .
hc- in-
tionr-nd-
e
~0.23! e2 at the CASSCF~MRCI! level from Fe to C. Due tothe avoided crossing previously mentioned,diabatically, theX 3D state correlates to Fe(5F)1C(3P), therefore, with re-spect to these products, i.e., the internal bond strength86.71DE(5F←5D)5109.6 kcal/mol ~Table I! at theC-MRCI/large level. Clearly, and for obvious reasons, t3dx22y2 (d1) and 3dxy (d2) are nonbonding observer eletrons and the same holds for all states studied.
We believe that this is thesecondexcited state 10.3~6.9!, or14.1~10.8! kcal/mol above theX state at the MRCI~1Q! lev-
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is,
e
els of theory, respectively, competing with the1D statewhose corresponding energy distance~s! from the X state is9.8~9.4!, or 9.7~9.5! kcal/mol ~vide infra!, Table II, Figs. 1and 3. With the exception ofm our results are in relativeagreement with those of Shim and Gingerich~SG!.6 At in-finity we start with the product wave functionu5D;M50&Fe
3u3P;M50&C, the character of which seems to be matained along the internuclear distance despite its interacwith the 163S2(2) around 5.7 bohr. The strong multirefeence character of this state does not allow for a simple boing description; we rather have twop and ones bond, thelatter caused by a transfer of 0.8 e2 to the C 2pz orbital froma (4s4pz3dz2)4.0 hybrid on Fe, in accordance to thCASSCF population analysis~Fe/C!
4s1.244pz0.243dz2
1.623dxz1.284px
0.033dyz1.284py
0.033dx22y21.00 3dxy
1.00/
2s1.822pz1.052px
0.682py0.68.
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mit
e
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anderi-
theI/
4908 J. Chem. Phys., Vol. 116, No. 12, 22 March 2002 D. Tzeli and A. Mavridis
53P~1!~;u1s22s23s11d11 @0.42~1px
21py22p̄y
1!
10.41~1px12p̄x
11py22p̄y
1!20.19~2px21py
22p̄y1!#1d2
1 &
1u1s22s13s̄14s11d11 @0.25~1px
21py22p̄y
1!
10.32~1px12p̄x
11py22p̄y
1!#1d21 &5uB1&,CASSCF).
This is the lowest state of the3P manifold 34.5 kcal/molabove theX state, Table II. Our results are at variance frothe results of SG.6 The PEC shown in Fig. 4 shows thatcorrelates to Fe (5D;M561)1C(3P;M50), maintainingthis character up to the equilibrium~3.5 bohr!. However,there is an avoided crossing at 3.05 bohr~repulsive branch!with the 83P(2) state. Practically, the bonding can be dscribed ass2p2, with the s bond originating from a(4s4pz)
1.92 hybrid on Fe and 0.65 e2 migrating from thishybrid to the 2pz C orbital, as indicated by the CASSCF~B1
symmetry! equilibrium atomic distributions,
4s1.034pz0.243dz2
1.073dxz1.084px
0.043dyz2.004py
0.043dx22y21.00 3dxy
1.00/
2s1.822pz0.832px
0.882py0.94.
A total transfer of 0.5~0.37! e2 is observed from Fe to C athe CASSCF~MRCI! level.
FIG. 1. Relative energy levels of the FeC at the MRC@7s6p4d3f /Fecc-pVTZ/C# level.
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Table II reveals excellent agreement between theoryexperiment11 in theTe value at the MRCI level, however, wpredict an equilibrium distance 0.04 Å larger than the expemental one. Figure 3 depicts the 73D(2) PEC with
FIG. 2. Potential-energy curves of the 41 states of FeC system atMRCI/@7s6p4d3f /Fecc-pVTZ/C# level of theory.
FIG. 3. MRCI potential-energy curves of the tripletsX 3D, 2 3S2(1),7 3D(2), 163S2(2), 293D(3), 333D(4), 343G(1), and3S1(1) of theFeC molecule. Dotted lines are referred to those parts of the 163S2(2),293D(3), 343G(1), and3S1(1) states that have not been calculated.
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d
4909J. Chem. Phys., Vol. 116, No. 12, 22 March 2002 Iron carbide
asymptotic fragments Fe(5D;M561)1C(3P;M561). Moving towards equilibrium the3D(2) state suffers an avoidecrossing around 5 bohr with the 293D(3) state; a second avoided crossing occurs at;4.5 bohr with theX 3D. As a result thein situ equilibrium atoms find themselves in Fe(5D;M562)1C(3P;M50). The equilibrium Mulliken MRCI distributionsare
4s1.014pz0.243dz2
1.153dxz1.114px
0.053dyz1.114py
0.053dx22y21.98 3dxy
1.00/2s1.712pz0.852px
0.842py0.84,
suggesting twop bonds and ones bond between a 4s4pz3dz2 hybrid on Fe and the empty 2pz of the carbon atom.Alternatively, we can say that about 0.6 e2 are transferred from the Fe 4s to the C 2pz orbital, while 0.4 e2 (0.2410.15) arepromoted from the 4s to the 4pz and 3dz2 atomic functions.
The PEC of this state, Fig. 4, experiences two avoidcrossings, one at 4.5 bohr with an incoming 203P(3) state,and a second one just after the equilibrium at 3.05 bohr~videsupra! with the 53P(1) state. Therefore the equilibriumcharacter of the 83P(2) state is that of the 203P(3), whichhas already suffered an avoided crossing~vide supra!, i.e.,Fe(5F;M561)1C(3P;M50). Adiabatically, the 83P(2)state correlates to Fe(5D;M562)1C(3P;M571). OurMRCI Mulliken equilibrium populations~in B1 symmetry!
4s0.934pz0.193dz2
1.353dxz1.314px
0.043dyz1.734py
0.103dx22y21.00 3dxy
1.00/
2s1.722pz0.762px
0.642py1.12,
FIG. 4. MRCI potential-energy curves of the triplets 53P(1), 8 3P(2),183F(1), and 203P(3), of the FeC molecule.
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dimply a p bond ~3dxz12px in B1 symmetry!, and as bondcaused by the transfer of 0.5 e2 from the (4s4pz3dz2)3.0
hybrid on Fe to the 2pz on C. A valence–bond–Lewis~vbL!diagram captures the above description
The bonding is similar to that of the 53P(1) state, the dif-ference being that in the present state thes bond is formedby a 3dz
2→2pz transfer, while in the 53P(1) state by a4s→2pz transfer.
The MRCI De value with respect to the adiabatic proucts is 37.8 kcal/mol, while with respect to the diabatic framents Fe(5F)1C(3P) ~internal bond strength!, we obtain37.81DE(5F←5D)562.0 kcal/mol.
Figure 3 shows the MRCI PEC of the 163S2(2) state;notice the missing part of the PEC between 3.6 and 4.2 bdue to severe problems of convergence in this region atMRCI level. With respect to the adiabatic fragmenFe(5D;M561)1C(3P;M571) we predict a De
526.0 kcal/mol,r e51.799 Å at the MRCI level of theory.An avoided crossing that should occur in the unchar
region 3.6–4.2 bohr with another~not calculated! 3S2 state
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4910 J. Chem. Phys., Vol. 116, No. 12, 22 March 2002 D. Tzeli and A. Mavridis
changes the PEC’s character, so at equilibrium thein situatoms are Fe(5F;M561)1C(3P;M571). Our CASSCFMulliken equilibrium distributions are
4s0.974pz0.363dz2
1.813dxz1.204px
0.033dyz1.204py
0.033dx22y21.00 3dxy
1.00/
2s1.772pz1.072px
0.762py0.76.
Clearly, we have twop bonds and a charge transfer of 0.5e2 from the metal to the carbon atom along thep frame; asto thes frame a 0.16 e2 transfer from C to Fe is observedbut we cannot assert the existence of as bond. Pictorially
with the 3dx22y2(d1), 3dxy(d2), 3dz2(3ds) and 2pz elec-trons coupled into a triplet.
Adiabatically, this3F(1) state correlates to Fe (5D;M562)1C(3P;61), Fig. 4. In conjunction with theCASSCF equilibrium population densities
4s1.224pz0.223dz2
1.393dxz0.984px
0.023dyz0.984py
0.02~3dx22y23dxy!2.98/
2s1.832pz1.322px
0.542py0.54,
and the leading configurations, we can claim that the boing is comprised of 3/2p bonds, and ones bond~3s orbital!due to the interaction of the (4s4pz3dz2)3.0 hybrid on Feand the 2pz orbital on C. A vbL icon consistent with theprevious discussion is the following
The existing experimental results for a3F state11 do notagree with our findings~Table II!. As a matter of fact itseems that the experimental assignement is not correct.sidering the overall good to very good agreement withother experimental results on this system, we believe thre-evaluation of the experimental findings is in order.
203P(3). The leading CFs are the same with thosethe previously described 183F(1) state, but with the~0.45,0.45, 0.32, 0.26! vector of coefficients changed to~0.39,20.39, 0.37,20.29!. As the two atoms come together fro
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infinity, Fe(5D;M50)1C(3P;M561), an avoided crossing occurs at 5 bohr with a~not calculated! 3P(4) state,imparting its character of Fe(5F;M561)1C(3P;M50) tothe 3P(3) curve and resulting to a 7.4 kcal/mol barrier~Fig.4!. A second avoided crossing occurs at about 4.5 bohr wthe 83P(2) state~vide supra!. As a result of the secondcrossing thein situ equilibrium atoms are Fe(5D;M562)1C(3P;M571). Populations and bonding are identicwith the 183F(1) state; numbers are presented in Table
The PECs of the3D(3) and3D(4) states are shown inFig. 3; they correlate to Fe(5F;M562)1C(3P;M50) andFe(5F;M561)1C(3P;M561), respectively. Severe technical difficulties did not allow the construction of the complete PECs. In particular, the parts 4.7–4.1 bohr of the3D(3)PEC, is missing. At the equilibrium thein situ atoms seem tobe Fe(5F;M561,73)1C(3P;M561) for the 3D(3) and3D(4) states, respectively. In both states, we think, the atoare held together by twop and ones bond.
This state correlates to Fe(5F;M563)1C(3P;M561). As Fig. 3 shows, we were unable to construct its coplete PEC due to technical difficulties. As a matter of fact tPEC’s part from 5.0 to 3.6 bohr is missing. Even arouequilibrium our PEC is not ‘‘smooth’’ enough. It seems th
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4911J. Chem. Phys., Vol. 116, No. 12, 22 March 2002 Iron carbide
the in situ equilibrium atoms carry the same character wthe asymptotic fragments and the bonding appears to coof a 1/2p and ones bond.
Finally, we would like to mention our unsuccessful eforts to compute a3S1(1) state. Its asymptotic fragmenare Fe(5D;M561)1C(3P;M571), and we were onlyable to construct its PEC from infinity to 4.3 bohr and fro3.4 to 2.5 bohr~Fig. 3!. However, we do not have any poinclose to equilibrium position, which we surmise to beabout 3.6 bohr.
The asymptotic fragments are Fe(5D;M561)1C(3P;M50). It seems that around 4 bohr an interactitakes place with the 175P(2) state, which already had experienced an avoided crossing with the 245P(3) state. Theresult is a rather rough shaped 35P(1) PEC around thisregion and the character transmission of the 245P(3) to theequilibrium region of the 35P(1), i.e., Fe(5F;M561)1C(3P;M50); see Fig. 5. The Mulliken MRCI densitie(B1 symmetry!
4s0.874pz0.263dz2
1.263dxz1.374px
0.043dyz1.704py
0.143dx22y21.00 3dxy
1.00/
2s1.762pz0.802px
0.582py1.13,
and the leading configuration~s! suggest the formation of apbond ~px in B1 symmetry!, a transfer of 0.13 and a promo
FIG. 5. MRCI potential-energy curves of the quintets 35P(1), 175P(2),245P(3), and 265F(1).
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ist
t
tion of 0.14 e2 from the 3dyz to 2py and 4py , respectively,and the formation of as bond by transfer of 0.56 e2 fromthe (3dz2)2.0 of Fe to the empty 2pz orbital of C. We candraw the following bonding diagram
With the exception ofve our results are in agreemenwith the results of Hirano and co-workers,7 but in disagree-ment with the findings of SG6 ~Table II!. Certainly the DFTDe result of 138 kcal/mol8 is flatly wrong, as is clear fromour De554.6 ~59.2! kcal/mol at the MRCI(1Q) level oftheory. Finally, the recent experimental assignment to a5Pstate,14 3460 cm21 ~9.89 kcal/mol! above theX state, is ques-tionable; instead this transition points to the 11D state, thefirst excited state of the FeC species; see Table II~vide infra!.
3u5D;M50&C, with the corresponding PEC shown in Fig.
FIG. 6. MRCI potential-energy curves of the quintets 45D(1), 6 5S2(1),125S2(2), 155D(2), and 235S1(1).
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4912 J. Chem. Phys., Vol. 116, No. 12, 22 March 2002 D. Tzeli and A. Mavridis
and numerical results presented in Table II along withresults of SG.6 The CASSCF Mulliken population analys(A1 symmetry!
4s0.954pz0.343dz2
1.073dxz1.054px
0.043dyz1.054py
0.043dx22y21.00 3dxy
2.00/
2s1.782pz0.862px
0.902py0.90
in conjunction with the CFs shown in the above-mentionsuggests the following vbL icon~A1 symmetry!
or a p bond with;0.2 e2 migrating from C to Fe via thepframe, and as dative bond originating from the F(4s4pz3dz2)3.0 hybrid; 0.64 e2 are transferred from this hybrid to the empty 2pz C orbital.
This state traces its ancestry to Fe(5D;M50)1C(3P;M50) as shown in Fig. 6. In Table II our results acontrasted with the results of SG,6 where we observe significant differences, particularly between them ~1.71 versus 2.12D!, and Te ~35.8 versus 23.5 kcal/mol! values.
The equilibrium CASSCF atomic Mulliken distributionare
4s1.084pz0.323dz2
1.953dxz1.064px
0.043dyz1.064py
0.043dx22y21.00 3dxy
1.00/
2s1.792pz0.852px
0.892py0.89.
We can claim that the atoms are held together by ap and asbond; the latter is due to a transfer of 0.64 e2 from a(4s4pz)
2.0 hybrid on Fe to the empty 2pz orbital of carbon.In pictures
The adiabatic fragments are Fe(5D;M561)1C(3P;M571); as we move toward the equilibrium and around 4
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e
,
8
bohr, an avoided crossing occurs with a~not calculated!5S2(3) state correlating to Fe(5F;M561)1C(3P;M571), Fig. 6, transferring its character at the same time toequilibrium atoms. The CASSCF populations,
4s1.014pz0.383dz2
1.653dxz1.234px
0.033dyz1.234py
0.033dx22y21.00 3dxy
1.00/
2s1.802pz1.132px
0.742py0.74,
in conjunction with the leading CFs reveal twop bonds anda repulsives interaction. The bonding is succinctly represented in the vbL icon that follows
As we approach from infinity @Fe(5D;M561)1C(3P;M561)# toward equilibrium, Fig. 6, the5D(2)suffers an avoided crossing at 4.5 bohr with the~not calcu-lated! 5D(3) state, the latter correlating to Fe(5F;M562)1C(3P;M50). An ‘‘interaction’’ also occurs with the lower4 5D(1) state at about 4 bohr. The CASSCF populations abonding are similar to that of the 45D(1) state~vide supra!,i.e., ap and as bond keep the two atoms together.
The adiabatic fragments are Fe(5D;M562)1C(3P;M571). Moving to the left from infinity an avoided crossintakes place with the 245P(3) state at 4.4 bohr; a seconavoided crossing occurs around 4 bohr with the 35P(1)state; see Fig. 5. Due to the two consecutive avoided cring the in situ equilibrium atoms are Fe(5D;M561)1C(3P;M50). The MRCI Mulliken populations~symme-try B1! are
4s1.084pz0.243dz2
1.143dxz1.184px
0.063dyz1.864py
0.093dx22y21.00 3dxy
1.00/
2s1.682pz0.802px
0.762py1.03.
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4913J. Chem. Phys., Vol. 116, No. 12, 22 March 2002 Iron carbide
The bonding is similar to that of the 35P(1) state, the dif-ference being that in the present state thes bond is due a4s2(Fe)→2pz(C) instead of 3dz2(Fe)→2pz(C) electrontransfer.
At infinity we have Fe(5D;M561)1C(3P;M571).As we approach the PEC’s minimum and around 4.5 bohravoided crossing takes place with the higher~not calculated!5S1(2) state resulting to a barrier of 2 kcal/mol with respeto the asymptotic products~Fig. 6!, and a character change othe in situ equilibrium atoms to Fe(5F;M561)1C(3P;M571). Our MRCI atomic Mulliken populations and leadinCFs
4s0.934pz0.263dz2
1.153dxz1.114px
0.073dyz1.114py
0.073dx22y21.46 3dxy
1.46/
2s1.792pz0.892px
0.812py0.82
imply a s and ap bond as shown in the bonding diagram
The cause of thes bond is a 0.3 e2 transfer from thepz
orbital on C to the (4s4pz3dz2)2.0 hybrid on Fe; to the op-posite direction and via thep frame 0.63 e2 migrate fromthe Fe to C giving rise to thep bond.
MRCI, global ~G! minimum!.This is a complicated state whose PEC, shown in Fig
suffers multiple avoided crossings. Moving from infinitFe(5D;M50)1C(3P;M561), toward equilibrium, thefirst avoided crossing occurs at 4.8 bohr with an incom~not calculated! 5P state correlating to Fe(5F;M561)1C(3P;M50), and giving rise to a barrier of 4.2 kcal/mowith respect to the asymptote. Moving further to the leftsecond avoided crossing takes place at 4.4 bohr with175P(2) state~vide supra! correlating to Fe(5D;M562)1C(3P;M571). At 3.9 bohr a third avoided crossing intervenes with a5P state of unknown lineage; in addition,interacts with the previously mentioned 175P(2) state atabout 3.7 bohr. Therefore, at the~G! minimum the identitiesof the in situ atoms are not clear. Perhaps we can say thatbonding is of s2p2 character. Passing equilibrium,r e
53.677 bohr, the 245P(3) state suffers a fourth avoide
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crossing at 3.25 bohr with another5P state of, also unknownorigin, resulting to a formal~L! minimum atr e53.19 bohr,Fig. 5.
265F~1!~;0.56u1s22s13s11px2~2px
11py21d1
1 1d22
11py22py
11d12 1d2
1 !&
5uB1&, CASSCF, global~G! minimum).
This state correlates to Fe(5D;M562)1C(3P;M561), maintaining this character up to the first local~L! mini-mum, r e53.838 bohr, Fig. 5, andDe513.1 kcal/mol; seeTable II. Due to an avoided crossing at 3.5 bohr with the~notcalculated! 5F(2) state correlating to Fe(5F;M563)1C(3P;M50), these atomic characters are transferredthe global ~G! minimum at r e53.203 bohr. The MRCIatomic populations of the L and G minima are given belo
L: 4s1.034pz0.433dz2
1.083dxz1.044px
0.033dyz1.044py
0.033dx22y21.49 3dxy
1.49/
2s1.812pz1.472px
0.502py0.50
G: 4s0.794pz0.163dz2
0.863dxz1.354px
0.093dyz1.354py
0.093dx22y21.48 3dxy
1.48/
2s1.692pz0.562px
1.002py1.00.
Notice the totally different distributions of the C atom btween the L and G ‘‘isomers’’ characteristic of the M561,and M50 carbon terms, respectively; corresponding diffences in the metal are not so distinct. With the help ofpopulations and leading CFs, the following bonding dgrams can be drawn for the L and G minima, respectivel
The previous diagrams imply 3/2p bonds and a strongsinteraction in the L case, and ap and 1/2s bond in the Gcase.
C. Septets: 9 7D„1…, 10 7P„1…, 13 7SÀ„1…, 22 7D„2…,
25 7P„2…, 27 7 S¿„1…, 28 7SÀ
„2…, 30 7P„3…, and31 7F„1…
9 7D(1), 227D(2). Asymptotically, these two states ardescribed by the product wave functionsu5D;M562&3u3P;M50& and u5D;M561&3u3P;M561&, respec-tively, with their PECs given in Fig. 7. The equilibriumCASSCF leading CFs~symmetryA1! for the 97D(1) stateare
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4914 J. Chem. Phys., Vol. 116, No. 12, 22 March 2002 D. Tzeli and A. Mavridis
For the 227D(2) state the CFs are the same, but the cosponding vector coefficients are 0.58,20.27, 0.39, 0.23,0.19, 0.24,20.17,20.20, and20.26. Therepulsive charac-ter of the 227D(2) state, Fig. 7, is interrupted at 4.7 bowhere an avoided crossing occurs with an incoming7D state,possibly correlating to Fe1 (6D;M562)1C2(4S), ~experi-mentally! 6.2 eV higher than the ground-state neutral atomThe ionic character is imparted to the equilibrium of thstate ~r e52.081 Å, Table II!, and perhaps to the 97D(1)state through an interaction between the two7D statesaround 4.5 bohr. The atomic CASSCF populations of thestates are practically identical, those of the 97D(1) being
4s0.974pz0.383dz2
1.043dxz1.024px
0.043dyz1.024py
0.043dx22y21.00 3dxy
2.00/
2s1.762pz0.852px
0.932py0.93.
The bonding for both states can be described pictoriby the following superposition of vbL icons, indicating iessence the formation of a singles bond and the transfer oabout 0.5 e2 from Fe to C
FIG. 7. MRCI potential-energy curves of the septets 97D(1), 137S2(1),227D(2), 277S1(1), and 287S2(2).
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While the r e , ve , andm values of SG6 are in relativegood agreement with ours~Table II!, there is complete dis-agreement between theTe values, the present one beintwice as large.
Asymptotically, we have Fe(5D;M561)1C(3P;M50) and this character is conserved up to the equilibriuthe PEC is shown in Fig. 8. An avoided crossing exists inrepulsive part of the PEC around 3.1 bohr with the 257P(2)~vide infra!. The CASSCF atomic distributions are the folowing
4s1.004pz0.343dz2
1.053dxz1.034px
0.043dyz2.004py
0.043dx22y21.00 3dxy
1.00/
2s1.782pz0.822px
0.922py0.94.
FIG. 8. MRCI potential-energy curves of the septets 107P(1), 257P(2),307P(3), and 317F(1).
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4915J. Chem. Phys., Vol. 116, No. 12, 22 March 2002 Iron carbide
The examination of the leading CFs in conjunction wthe Mulliken analysis indicates the formation of as bondaccording to the scheme~B1 symmetry!
Again, a strong disagreement is observed between ouTe
The asymptotic description is Fe(5D;M50)1C(3P;M50), and the same character prevails in the equilibrium, F7. The atomic Mulliken equilibrium distributions,
4s1.054pz0.353dz2
1.983dxz1.034px
0.043dyz1.034py
0.043dx22y21.00 3dxy
1.00/
2s1.782pz0.832px
0.932py0.93,
clearly indicate the formation of as bond with a concomi-tant transfer of 0.6 e2 from a (4s4pz)
2.0 hybrid on Fe to theempty 2pz orbital of C, while;0.1 e2 are moving back fromC to Fe through thep frame. Pictorially,
Finally, as in the previously described 97D(1) and 107P(1)states, ourTe value is at variance with the correspondingTe
As we move from infinity, Fe(5D;M562)1C(3P;M571), the 257P(2) suffers its first avoided crossing aabout 4.5 bohr with the 307P(3) state, therefore acquiring aequilibrium the latter’s character, Fe(5F;M561)1C(3P;M50); see Fig. 8. In the PEC’s repulsive partsecond avoided crossing occurs around 3.4 bohr againthe 307P(3) state, imparting its character to the 257P(2)state, Fe(5F;M50)1C(3P;M561), and finally, a thirdavoided crossing takes place at 3.0 bohr with both 107P(1)and 307P(3) states. Our equilibrium CASSCF atomic poplations are practically the same with those of 107P(1) ~see
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.
ith
above-mentioned!, indicating the formation of as bondcaused by the transfer of 0.6 e2 from the hybrid(3dz24pz)
2.0 on Fe to the empty 2pz orbital of carbon. Also,;0.1 e2 are fed back to the Fe atom via thep nonbondingframe.
277S1~1!~;0.84u1s22s13s11px22px
11py22py
11d11 1d2
1 &
5uA1&,MRCI).
The asymptotic products are Fe(5D;M561)1C(3P;M571). As we move to the left, the PEC presena Pauli repulsive interaction with a maximum at 4.5 bohr aa height of 3.4 kcal/mol; see Fig. 7. The following vbL icoexplains what is meant
Between 4.5 and 3.8 bohr an attractive interaction setdue to a transfer of 0.25 e2 ~at 4.2 bohr! to the 2pz C orbitalfrom the 4s2 of Fe, with a synchronous promotion of 0.40 e2
to the 4pz of the Fe atom. At 3.8 bohr an avoided crossioccurs with the~not calculated! 7S1(2) state correlating toFe(5F;M50)1C(3P;M50). Notice the change of the Mvalues from~61, 71! to ~0, 0!, resulting to a favorable forsbonding electronic distribution. The Mulliken MRCI equilibrium atomic populations
4s0.864pz0.213dz2
0.893dxz1.714px
0.103dyz1.714py
0.103dx22y21.01 3dxy
1.01/
2s1.692pz0.472px
1.082py1.08.
and the leading CFs at equilibrium dictate the followinbonding picture
suggesting a formal 1/2s bond. From Table II we read thaat re51.766 Å, De515.5 kcal/mol with respect to the adiabatic products, or, diabatically, the internal bond strength15.51DE(5F←5D)539.7 kcal/mol.
Notice that the leading configurations of the 137S2(1)state described previously and the present one are the sThe asymptotic products are given by the product wave fution u5D;M561&Fe3u3P;M571&C, but the situation atequilibrium is not clear. An avoided crossing takes place4.5 bohr, Fig. 7, but we do not understand where the invening7S2(3) state correlates to, therefore, we do not kn
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4916 J. Chem. Phys., Vol. 116, No. 12, 22 March 2002 D. Tzeli and A. Mavridis
the equilibrium atomic characters. Besides the numerfindings presented in Table II, nothing much can be sabout the bonding of this state.
307P(3). This is a rather complicated state with twminima, a local~L! and a global~G! one ~see Fig. 8!, andfive avoided crossings along its PEC. The L and G leadequilibrium CFs are
As the two atoms Fe(5D;M50)1C(3P;M561) ap-proach from infinity, the first avoided crossing occurs abohr with a ~not calculated! 7P(4) state correlating toFe(5F;M561)1C(3P;M50), giving rise to a barrier of6.8 kcal/mol~Fig. 8!. A second avoided crossing occurs4.5 bohr with the 257P(2) state~vide supra! correlating toFe(5D;M562)1C(3P;M571). At 4.13 bohr~52.186 Å!the L minimum appears with aDe58.74 ~8.8! kcal/mol atthe MRCI~1Q! level. At this point the L Mulliken atomicdensities are
4s1.074pz0.423dz2
1.063dxz1.014px
0.023dyz1.014py
0.023dx22y21.50 3dxy
1.50/
2s1.852pz1.512px
0.512py0.51.
The leading CFs and the above-mentioned distributisuggest the following bonding picture
The bonding is comprised of a 1/2p bond and as interac-tion due to a transfer of 0.35 e2 from the Fe 4s2 orbital tothe 2pz C orbital, and a promotion of 0.42 e2 to the Fe 4pz
orbital.Leaving now the L minimum, a third avoided crossin
takes place at 3.65 bohr with the~not calculated! 7P(4)state, which has suffered already at least a second avocrossing~the first one was described previously!, imparting,finally, a rather ionic character to the G minimumFe1(6D;M561)1C2(4S). We hasten to add, howevethat we are not certain about this conclusion because it iscorroborated by the electron transfer as deduced frompopulation analysis.
Two more avoided crossings are encountered at thepulsive part of the PEC: The fourth one at 3.4 bohr with t
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257P(2) state imparting a character of Fe(5F;M561)1C(3P;M50) to the PEC of this region, and finally a fiftone at 3.0 bohr, again with the 257P(2) state~vide infra!.The final character of the repulsive part of the 307P(3) stateis Fe(5D;M561)1C(3P;M50). ~The L and G minimaare practically degenerate at the MRCI level, but at the1Qlevel the energy of the G minimum drops by 6 mh wirespect to the L.!
317F~1!~;0.67u1s22s23s14s11px1~1py
12py11d1
1 1d22
12px11py
11d12 1d2
1 !&5uB1&,CASSCF).
This state correlates to Fe(5D;M562)1C(3P;M561) and this character is transferred to the equilibriumometry; see Fig. 8. It has the lowest binding energy ofother states presented in Table II,De58.2 ~8.4! kcal/mol atthe MRCI~1Q! level, and the longer internuclear distancre52.184 Å. The similarity of the 317F(1) with the above-described 307PL(3) state is striking: same configurationbond distances, binding energies, and, of course, bonnature. In the repulsive part of its PEC, the 317F(1) statesuffers an avoided crossing at about 3.5 bohr with a~notcalculated! 7F(2) state, which rather correlates tFe(5F;M563)1C(3P;M50).
D. Singlets: 1 1D„1…, 11 1P„1…, 14 1G„1…, 19 1S¿„1…,
As was previously mentioned@state 23S2(1)# our re-sults indicate that the1D(1) is the first excited state bein9.8~9.4!, 9.5~9.1!, 9.7~9.5! kcal/mol above theX state in thesequence of MRCI~1Q!, MRCI/large~1Q!, C-MRCI/large~1Q! calculations, respectively~Table II!. Notice thatthe Te(
1D←X) value remains, practically, method indepedent. The difference from theX state is a spin flip of thed1
orbital, d̄11 instead ofd1
1 ~or d̄21 instead ofd2
1 in A2 sym-metry!. It traces its lineage to Fe(3F;M562)1C(3P;M50), while the asymptotic atomic characters are transferto equilibrium ~Fig. 9!. The CASSCF atomic populations
4s0.914pz0.203dz2
1.273dxz1.164px
0.033dyz1.164py
0.033dx22y21.00 3dxy
1.99/
2s1.792pz0.812px
0.792py0.79,
coupled with the leading CFs indicate the formation of onesand twop bonds. The corresponding vbL icon is clearly~A2
symmetry! the following
Through thep frame about 0.4 e2 migrate from the C to themetal, and through thes skeleton 0.60 e2 are transferredfrom the (3dz2)2.0 of the metal to the empty 2pz C orbitalgiving rise to thes bond. The bonding is identical to that o
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4917J. Chem. Phys., Vol. 116, No. 12, 22 March 2002 Iron carbide
the X 3D state~vide supra!, and this is reflected to, practcally, the same bond length between the two states and rasimilar bond strengths~Table II!. As we can see from TableII, the r e , ve , andvexe values are in fair agreement witrecent experimental results obtained by Aiuchiet al. by dis-persed fluorescence spectroscopy.14 The same workers determined the transition, first excited state←X 3D, T0
53460 cm21(59.893 kcal/mol) or Te5(T01Dve/2)59.786 kcal/mol, suggesting also that the first excited sis of 5P symmetry. Our results clearly indicate that the fiexcited state is of1D symmetry; the same conclusion wareached by SG6 and Hiranoet al.;7 see Table II. Finally, wewould like to mention that the1D state has the largest binding energy,De5107.4 ~109! kcal/mol at the MRCI~1Q!level, and the smallest internuclear distance,r e51.585 Å ofall states studied. The corresponding MRCI/large~1Q! andC-MRCI/large~1Q! values are 110.0~112!, 111.0~114! kcal/mol, and 1.575, 1.566 Å, respectively.
111P~1!~;0.56u1s22s23s̄11px21py
22py11d1
1 1d̄21 &
10.32u1s22s23s11px21py
22p̄y11d1
1 1d̄21 &
10.21u1s22s13s21px21py
22py11d̄1
1 1d̄21 &
5uB1&,MRCI).
Figure 10 presents the PEC of the1P(1) state correlat-ing to Fe(3F;M561)1C(3P;M50); this character ismaintained at the equilibrium,r e51.741 ~1.717! Å at theMRCI~1Q! level, Table II. At this geometry the MRCI Mulliken densities are
FIG. 9. MRCI potential-energy curves of the singlets 11D(1), 141G(1),191S1(1), 211S2(1), 321D(2), and 401D(3) of the FeC molecule. Dot-ted line is referred to that part of the 191S1(1) state that has not beecalculated.
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er
tet
4s0.954pz0.243dz2
1.233dxz1.204px
0.043dyz1.924py
0.083dx22y21.01 3dxy
1.01/
2s1.712pz0.802px
0.742py0.96,
which, in conjunction with the main CFs and the symmeof the in situ atoms, support the following bonding pictur(B1 symmetry!
The two atoms are held together by ap and as bond,the latter formed by a transfer of 0.50 e2 from the Fe(3dz2)2
orbital to the~empty! 2pz C orbital, with the synchronouspromotion of 0.24 e2 to the 4pz ~Fe! orbital; via thep frame,0.25 e2 migrate from the C to the Fe atom. Our numericresults are in qualitative agreement with the correspondresults of SG6 ~Table II!.
141G(1). The PECpresented in Fig. 9 shows that thfragments at infinity are Fe(3F;M563)1C(3P;M561).It is of interest to give the CASSCF CFs in both symmetrA1 andA2 ‘‘compatible’’ with the G symmetry.
141G~1!/A1;0.59u1s22s23s21px21py
2~1d12 11d2
2 !&
141G~1!/A2;0.83u1s22s23s21px21py
21d11 1d̄2
1 )&.
Of course, a trueG state is a linear combination of theA1 andA2 components, 2(0.59)2'0.832.
Moving from infinity towards equilibrium, an avoidedcrossing occurs around 4.2 bohr with the1G(2) state possi-
FIG. 10. MRCI potential-energy curves of the singlets 111P(1), 351H(1),361P(2), 371F(1), 381F(2), and 391P(3). Inset: MRCI potential-energy curves of 371F(1) and 381F(2) states.
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4918 J. Chem. Phys., Vol. 116, No. 12, 22 March 2002 D. Tzeli and A. Mavridis
bly correlating to Fe(3H;M563or75)1C(3P;M561).This atomic character is imparted to equilibrium with tfollowing Mulliken CASSCF distributions,
4s1.204pz0.233dz2
1.623dxz1.314px
0.033dyz1.314py
0.033dx22y21.01 3dxy
1.01/
2s1.812pz1.082px
0.662py0.66.
In A2 symmetry the bonding can be visualized by ticon
Through thep frame 0.3 e2 are transferred from Fe to Cwhile, clearly, two p bonds are formed. Thes bond isformed due to the 2pz~C!1(4s4pz3dz2)3.0 ~Fe! interaction;less than 0.1 e2 seems to be transferred from C to Fe throuthe s frame. ADe566.1 kcal/mol~MRCI! is predicted withrespect to Fe(3F)1C(3P), however, with respect to the dabatic products Fe(3H)1C(3P), the internal bond strengthis 66.11DE(3H←3F)13586.6 kcal/mol.
191S1~1! ~;0.56u1s22s23s21px21py
2~1d12 11d2
2 !&
20.22u1s22s13s̄14s21px21py
21d22 &5uA1&,CASSCF).
At infinity this state is represented by the product wafunction, u3F;M50&Fe3u3P;M50&C; see Fig. 9. Severeconvergence problems, probably caused by an avoided cing, prevented us from calculating the PEC’s part betwe5.5 and 4.0 bohr. The assumed avoided crossing betweeand 4.0 bohr is due to a~not calculated! 1S1(2) state corre-lating rather to Fe(3H or 3P;M561)1C(3P;M571),thusly transferring this character to thein situ equilibriumatoms. The binding mode is very similar to that of the pviously discussed1G(1) state as is also reflected in theDe
The in situ atomic characters at equilibrium are the sawith the asymptotic fragments, Fe(3F;M561)1C(3P;M571); see Fig. 9. The MRCI atomic distributions atr e
51.820 Å are
4s0.984pz0.403dz2
1.883dxz1.174px
0.053dyz1.174py
0.053dx22y21.00 3dxy
1.00/
2s1.702pz0.962px
0.772py0.77,
consistent with the formation of twop bonds and the transfeof 0.54 e2 via thep frame from Fe to the C atom. As usuthes interaction is not so clear; we can only say that 0.242
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ss-n5.0
-
e
migrate back from the C to the Fe atom through thes frame,populating the 4pz orbital of the latter: 0.40 e2'0.2410.12 ~from 3dz2! 10.02 ~from 4s!. In other words, thesinteraction is caused by promoting 0.40 e2 to the 4pz ~emptyat infinity! Fe orbital. However, we would like to remark thathe PEC’s shape~Fig. 9! is not very ‘‘natural,’’ reminiscentof an harmonic oscillator potential. It should be mentionthat we were unable to converge CASSCF energy pointthe region 3.9 to 4.4 bohr for this state.
321D(2), 401D(3). Thestates1D(2) and1D(3) corre-late to Fe(3F;M573,61)1C(3P;M561), respectively~Fig. 9!. Their main equilibrium CASSCF (1D(2)) andMRCI (1D(3)) CFs are
At 3.8 bohr an avoided crossing between these twoDstates causes the exchange of the M values of the mTheir equilibrium CASSCF populations are as follows fthe 1D(2) and1D(3) states~populations of1D(3) in paren-theses!
4s1.03~0.94!4pz0.24~0.29!3dz2
1.02~1.10!3dxz1.19~1.15!4px
0.03~0.03!
3dyz1.19~1.15!4py
0.03~0.03!3dx22y21.94~1.93!3dxy
1.00~0.95!/2s1.74~1.80!
2pz0.98~0.86!2px
0.78~0.87!2py0.78~0.87! .
Without doubt, in both states, the bonding consists of twopand ones bonds as shown in the vbL diagram~A2 compo-nents!,
The net electron transfer from Fe to C is 0.30 and 0.40 e2 forthe 1D(2) and1D(3) states, respectively.
351H~1!~;0.47u1s22s23s11px21py
22p̄y11d1
1 1d̄21 &
10.38u1s22s23s11px22p̄x
11py2~1d1
2 21d22 !&
20.27u1s22s23s̄11px21py
22py11d1
1 1d̄21 &,CASSCF).
This is the highestL(55) state of the present report;
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4919J. Chem. Phys., Vol. 116, No. 12, 22 March 2002 Iron carbide
correlates to Fe(3H;M565)1C(3P;M50), Fig. 10, andthe last state formally bound with respect to the ground-sproducts by just 0.2 kcal/mol, but strongly bound~De
562.4 kcal/mol at the MRCI level! with respect to theasymptotic products. Although we were unable to locateavoided crossings, we cannot exclude an interaction with1H(2) state originating from Fe(b 3H;M565)1C(3P;M50); the Fe(b 3H) state is 19.3 kcal/mol abovthe Fe(a 3H). The equilibrium CASSCF atomic densities
4s0.894pz0.303dz2
1.163dxz1.584px
0.043dyz1.584py
0.043dx22y21.00 3dxy
1.00/
2s1.802pz0.832px
0.872py0.87,
suggest the following bonding icon~B1 component!
Through thep interaction 0.26 e2 move from the C to the Featom. Thes bond seems to be caused, given the reservatexpressed before concerning the interaction with an1H(2)molecular state, by the transfer of 0.63 e2 from the 4s2 Feorbital to the 2pz ~empty at infinity! C orbital, and the pro-motion of 0.3010.16 e2 to the 4pz and 3dz2 Fe orbitals,respectively. Alternatively, from the hybrid (4s4pz3dz2)3.0
on the Fe center, 0.63 e2 diffuse to the 2pz C orbital.
The energy difference between the above1P states is 4.4~3.6! kcal/mol at the MRCI~1Q! level ~Table II!. They cor-relate to Fe(3F;M50,62)1C(3P;M571), states 36 and39, respectively. For both states nothing much can beabout the bonding. An interaction between them occwithin 3.8–4.0 bohr, while the equilibrium character of thin situ atoms of the1P(2) state seems to be the same wthe asymptotic fragments; we are not sure for the equilibriatomic character of the1P(3) state. Also, for the latter, wewere unable to construct the PEC’s region between 4.2bohr due to severe technical problems~Fig. 10!. Given ourreservations, we claim that the bonding for both states csists of a singlep bond.
The two states, 37 and 38, correlate to Fe(3F;M563,62)1C(3P;M50,61), respectively, with their corresponding PECs shown in Fig. 10~see also inset!. As we movetowards the equilibrium and around 5 bohr, the1F(2) statesuffers an avoided crossing with a~not calculated! 1F(3)state correlating to Fe(3H;M564or72)1C(3P;M571),and at 3.65 and 3.25 bohr two additional avoided crossioccur between the1F(1) and1F(2) states. As a result the1F(1) presents a global minimum atr e~G!51.756 Å(53.32 bohr) with atomicin situ characters of Fe(3H;M564or72)1C(3P;M571) and a local minimum atr e~L!51.745 Å(53.30 bohr) within situ atomic charactersof the asymptotic fragments. The minimum of the1F(2)appears at 3.25 bohr, on top of the second avoided crosbetween theF~1! andF~2! states; see Fig. 10.
Of course, for the1F(2) state, spectroscopic constanlack of meaning and only formalDe andr e values are givenin Table II. Also, the type of leading CFs involved in thMRCI expansion of the1F(1) state do not allow any cleadescription of the bonding in the global minimum. As a mater of fact conventional bonding pictures break downsuch states, so there is no meaning in discussing the bonmechanism.
V. SYNOPSIS AND REMARKS
For the FeC diatomic, a 12 active electron system,have performed multireference variational calculatiospecifically, CASSCF11125MRCI, with a@7s6p4d3 f /Fecc-pVTZ/C# basis set. For three out of 4states examined, i.e., the ground FeC(3D), and the first twoexcited states~1D and 3S2!, calculations were also done athe larger basis set@7s6p4d3 f 2g/Fecc-pVQZ/C# ~MRCI/large!. In addition, for these three states we examinedresults of core correlation effects by including the Fe 3s23p6
electrons at the MRCI level~C-MRCI/large!. For reasons ofcomparison theX 3D state was also examined at thCCSD~T!/small level of theory.
For all states we report total energies, equilibrium bodistances, dissociation energies, dipole moments, andmost common spectroscopic constants~ve , vexe , ae , andD̄e! obtained by a Dunham analysis. Full potential-enecurves were also constructed for almost all states withpurpose of following the evolution of the atomic states alothe reaction coordinate. A synopsis of our findings and cclusions follows.
~1! The FeC molecule is a genuine multireference systetherefore it is close to impossible to be tackled by asingle reference-type method. For many of the exained states the largest variational CASSCF~or MRCI!expansion coefficients do not exceed the value of 0Also, the existing DFT data seems to be in stark d
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4920 J. Chem. Phys., Vol. 116, No. 12, 22 March 2002 D. Tzeli and A. Mavridis
agreement with corresponding experimental resultsquality ab initio values. The same general disagrement between DFT and high levelab initio results wasalso recorded for the diatomic carbides ScC2 andTiC.25
~2! For the ground state (X 3D) and at the highest level oour calculations~C-MRCI/large!, we predict a bindingenergy De586.7 kcal/mol. By correcting this valufor BSSE and scalar relativistic effects,6 we obtainDe584.3 kcal/mol. These values should be contraswith two experimental values 91.267 kcal/mol~upperlimit !,11 and 81.764.6 kcal/mol,15 or with their aver-age of 86.565.8 kcal/mol.
~3! The recently assigned by dispersed fluorescence stroscopy5P state of FeC,;10 kcal/mol above theXstate, certainly has a different symmetry. Two stacompete for that energy position in our calculations1D and a3S2. Our final results at the C-MRCI/larglevel suggest, with some confidence, that the1D is thefirst excited state 9.7 kcal/mol above theX state. At thesame level of theory the corresponding splitting for t3S2 state is 14.1 kcal/mol; however, the Davidscorrection brings those numbers to 9.5 and 10.8 kmol, respectively, or that DE (3S221D)51.3kcal/mol.
~4! With the exception of 183F(1) state all our results arin good agreement with existing experimental daFor the3F(1) state all our calculated parameters ain disagreement with experimental values. Our resusuggest that, perhaps, this experimentally assigned3Fstate is of3P symmetry, specifically the 83P(2) state.
~5! All calculated states are bound with respect to thasymptotic products; with respect to ground state frments, 35 states are bound, the last one being1H(1) state. The1D(1) state has the largest bindinenergy~with respect to its asymptotic products!, andthe shortest bond length of all calculated statnamely, De5111.0 kcal/mol at r e51.566 Å at theC-MRCI/large level.
~6! In all states a net charge transfer is observed fromto C ranging from about 0.50 e2 to 0.25 e2 at theCASSCF level. At the MRCI this Fe→C electrontransfer is diminished on the average by 0.1 e2.
~7! All our computed states correlate to the ground-stcarbon atom, C(3P;M50,61). In all cases where thein situ equilibrium C atom has the M50 character, i.e.,two electrons distributed ‘‘perpendicularly’’ to the internuclear axis, a strong transfer of electrons isserved from Fe to C along thes frame, thus, resultingin a very strongs bond and rather highDe values; see,for instance, theX 3D and 11D(1) states. In casewhere thein situ C finds itself in a M561 state, thesinteraction is turned off or it is very weak, and thcharge transfer from Fe to C is forced to occur mainthrough thep frame.
~8! In theX and the first~1! excited state, as well as in th2, 7, 14, 19, 29, 32, 33, and 40 states, the two atoare held by a triple bond.
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.
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-
s
~9! For all states and for obvious reasons, thedx22y2(d1)anddxy(d2) electrons remain strictly localized on thFe atom; therefore, they do not play any significarole in the binding process.
~10! For many states of the FeC system the entanglingstates of the same symmetry creates severe techdifficulties, sometimes unsolved, particularly if onwishes to compute complete potential-energy curvHowever, in most cases complete PECs are necesbecause the bonding depends on the detailed historthe atoms from infinity to equilibrium.
~11! The Te’s of the following pairs or groups differ byabout 1 mhartree and obviously the real orderingpossibly different, $8 3P(2),97D(1)%, $107P(1),111P(1)%, $111P(1),125S2(1)%, $175P(2),183F(1)%, $183F(1),191S1(1)%, $235S1(1),245P(3)%, $245P(3),257P(2)%, $257P(2),265F(1)%, $265F(1),277S1(1)%, $307P(3),317F(1)% and $371F(1),381F(2)%. The ordering isreversed by adding the zero-point energy correctDve/2 for the pairs$8 3P(2),97D(1)%, $265F(1),277S1(1)% and$307P(3),317F(1)% by 0.2, 0.1, and0.1 kcal/mol, respectively; the two states 371F(1) and381F(2) cannot be distinguished, the second stateing lower than the zero-point energy of the first onLast, by applying the Davidson1Q correction, theordering is reversed for the groups$111P(1),125S2(1),107P(1)%, $141G(1),137S2(1)%,$163S2(2),183F(1),191S1(1),175P(2),155D(2)%,$277S1(1),265F(1),235S1(1),245P(3),227D(2),257P(2)%, $321D(2),317F(1)%, and $361P(2),351H(1)%.
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4921J. Chem. Phys., Vol. 116, No. 12, 22 March 2002 Iron carbide
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24MOLPRO 2000is a package ofab initio programs written by H.-J. Werneand P. J. Knowles, with contributions by R. D. Amos, A. BernhardssonBerninget al.
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