Detailed ab initio first-principles study of the magnetic anisotropy in a family of trigonal pyramidal iron(II) pyrrolide complexes.
ABSTRACT A theoretical, computational, and conceptual framework for the interpretation and prediction of the magnetic anisotropy of transition metal complexes with orbitally degenerate or orbitally nearly degenerate ground states is explored. The treatment is based on complete active space self-consistent field (CASSCF) wave functions in conjunction with N-electron valence perturbation theory (NEVPT2) and quasidegenerate perturbation theory (QDPT) for treatment of magnetic field- and spin-dependent relativistic effects. The methodology is applied to a series of Fe(II) complexes in ligand fields of almost trigonal pyramidal symmetry as provided by several variants of the tris-pyrrolylmethyl amine ligand (tpa). These systems have recently attracted much attention as mononuclear single-molecule magnet (SMM) complexes. This study aims to establish how the ligand field can be fine tuned in order to maximize the magnetic anisotropy barrier. In trigonal ligand fields high-spin Fe(II) complexes adopt an orbitally degenerate (5)E ground state with strong in-state spin-orbit coupling (SOC). We study the competing effects of SOC and the (5)E⊗ε multimode Jahn-Teller effect as a function of the peripheral substituents on the tpa ligand. These subtle distortions were found to have a significant effect on the magnetic anisotropy. Using a rigorous treatment of all spin multiplets arising from the triplet and quintet states in the d(6) configuration the parameters of the effective spin-Hamiltonian (SH) approach were predicted from first principles. Being based on a nonperturbative approach we investigate under which conditions the SH approach is valid and what terms need to be retained. It is demonstrated that already tiny geometric distortions observed in the crystal structures of four structurally and magnetically well-documented systems, reported recently, i.e., [Fe(tpa(R))](-) (R = tert-butyl, Tbu (1), mesityl, Mes (2), phenyl, Ph (3), and 2,6-difluorophenyl, Dfp (4), are enough to lead to five lowest and thermally accessible spin sublevels described sufficiently well by S = 2 SH provided that it is extended with one fourth order anisotropy term. Using this most elementary parametrization that is consistent with the actual physics, the reported magnetization data for the target systems were reinterpreted and found to be in good agreement with the ab initio results. The multiplet energies from the ab initio calculations have been fitted with remarkable consistency using a ligand field (angular overlap) model (ab initio ligand field, AILFT). This allows for determination of bonding parameters and quantitatively demonstrates the correlation between increasingly negative D values and changes in the σ-bond strength induced by the peripheral ligands. In fact, the sigma-bonding capacity (and hence the Lewis basicity) of the ligand decreases along the series 1 > 2 > 3 > 4.
- Joseph M Zadrozny, Dianne J Xiao, Mihail Atanasov, Gary J Long, Fernande Grandjean, Frank Neese, Jeffrey R Long[Show abstract] [Hide abstract]
ABSTRACT: Single-molecule magnets that contain one spin centre may represent the smallest possible unit for spin-based computational devices. Such applications, however, require the realization of molecules with a substantial energy barrier for spin inversion, achieved through a large axial magnetic anisotropy. Recently, significant progress has been made in this regard by using lanthanide centres such as terbium(III) and dysprosium(III), whose anisotropy can lead to extremely high relaxation barriers. We contend that similar effects should be achievable with transition metals by maintaining a low coordination number to restrict the magnitude of the d-orbital ligand-field splitting energy (which tends to hinder the development of large anisotropies). Herein we report the first two-coordinate complex of iron(I), [Fe(C(SiMe3)3)2](-), for which alternating current magnetic susceptibility measurements reveal slow magnetic relaxation below 29 K in a zero applied direct-current field. This S = complex exhibits an effective spin-reversal barrier of Ueff = 226(4) cm(-1), the largest yet observed for a single-molecule magnet based on a transition metal, and displays magnetic blocking below 4.5 K.Nature Chemistry 05/2013; 5(7):577-581. · 21.76 Impact Factor - Po-Heng Lin, Nathan C Smythe, Serge I Gorelsky, Steven Maguire, Neil J Henson, Ilia Korobkov, Brian L Scott, John C Gordon, R Tom Baker, Muralee Murugesu[Show abstract] [Hide abstract]
ABSTRACT: Two mononuclear high-spin Fe(II) complexes with trigonal planar ([Fe(II)(N(TMS)(2))(2)(PCy(3))] (1) and distorted tetrahedral ([Fe(II)(N(TMS)(2))(2)(depe)] (2) geometries are reported (TMS = SiMe(3), Cy = cyclohexyl, depe = 1,2-bis(diethylphosphino)ethane). The magnetic properties of 1 and 2 reveal the profound effect of out-of-state spin-orbit coupling (SOC) on slow magnetic relaxation. Complex 1 exhibits slow relaxation of the magnetization under an applied optimal dc field of 600 Oe due to the presence of low-lying electronic excited states that mix with the ground electronic state. This mixing re-introduces orbital angular momentum into the electronic ground state via SOC, and 1 thus behaves as a field-induced single-molecule magnet. In complex 2, the lowest-energy excited states have higher energy due to the ligand field of the distorted tetrahedral geometry. This higher energy gap minimizes out-of-state SOC mixing and zero-field splitting, thus precluding slow relaxation of the magnetization for 2.Journal of the American Chemical Society 09/2011; 133(40):15806-9. · 10.68 Impact Factor - [Show abstract] [Hide abstract]
ABSTRACT: The determination of anisotropic magnetic parameters is a task of both experimental and theoretical interest. The added value of theoretical calculations can be crucial for analyzing experimental data by (i) allowing assessment of the validity of the phenomenological spin Hamiltonians, (ii) allowing discussion of the values of parameters extracted from experiments, and (iii) proposing rationalizations and magneto-structural correlations to better understand the relations between geometry, electronic structure, and properties. In this review, we discuss the model Hamiltonians that are used to describe magnetic properties, the computational approaches that can be used to compute magnetic parameters, and review their applications to transition metal and (to a lesser extent) lanthanide based complexes. Perspectives concerning current methodological challenges will then be presented, and finally the need for further joint experimental/theoretical efforts will be underlined.Physical Chemistry Chemical Physics 10/2013; · 4.20 Impact Factor
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r2011 American Chemical Society
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pubs.acs.org/IC
Detailed Ab Initio First-Principles Study of the Magnetic Anisotropy in
a Family of Trigonal Pyramidal Iron(II) Pyrrolide Complexes
Mihail Atanasov,*,†,§,||Dmitry Ganyushin,†Dimitrios A. Pantazis,†,‡Kantharuban Sivalingam,†and
Frank Neese*,†,‡
†Institute for Physical and Theoretical Chemistry, University of Bonn, Wegelerstrasse 12, D-53115 Bonn, Germany
‡Max-Planck Institute for Bioinorganic Chemistry, Stiftstrasse 32-34, D-45470 M€ ulheim an der Ruhr, Germany
§Institute of General and Inorganic Chemistry, Bulgarian Academy of Sciences, Acad. Georgi Bontchev Street 11, 1113 Sofia, Bulgaria
)
D? epartement de Chimie, Universit? e de Fribourg, Ch. du Mus? ee, 9, CH-1700 Fribourg, Switzerland
b
S Supporting Information
I. INTRODUCTION
In this work we study the origin of the magnetic anisotropy of
a family of trigonal pyramidal iron(II) complexes supported by
derivatives of the tris(pyrrolyl-R-methyl) ligand. It was recently
discovered that these systems possess unusually large magnetic
anisotropies.1,2The computational and theoretical results re-
ported here are used to analyze the effects of the geometrical
distortions due to the first coordination sphere (Jahn?Teller
effect)andtheinfluence oftheremoteligand substituentson the
magnetic anisotropy.
Single-molecule magnets (SMMs) are molecules that exhibit
slowmagneticrelaxation,whichoriginatesfromanenergybarrier
to inversion of the total molecular spin. The magnetic moment
caneventuallybeblockedifthethermalenergyissmallerthanthe
barrierheight. Thisslowrelaxation enablessuch moleculestoact
as molecular magnets, similar to the classical ones. This behavior
manifests itself by the presence of a magnetic hysteresis at low
Received:January 28, 2011
ABSTRACT: A theoretical, computational, and conceptual
frameworkfortheinterpretationandpredictionofthemagnetic
anisotropy of transition metal complexes with orbitally degen-
erate or orbitally nearly degenerate ground states is explored.
The treatment is based on complete active space self-consistent
field (CASSCF) wave functions in conjunction with N-electron
valence perturbation theory (NEVPT2) and quasidegenerate
perturbation theory (QDPT) for treatment of magnetic field-
and spin-dependent relativistic effects. The methodology is
applied to a series of Fe(II) complexes in ligand fields of almost
trigonal pyramidal symmetry as provided by several variants of
the tris-pyrrolylmethyl amine ligand (tpa). These systems have recently attracted much attention as mononuclear single-molecule
magnet (SMM) complexes. This study aims to establish how the ligand field can be fine tuned in order to maximize the magnetic
anisotropybarrier. Intrigonal ligandfieldshigh-spin Fe(II)complexes adoptanorbitallydegenerate5Egroundstatewithstrong in-
statespin?orbitcoupling(SOC).WestudythecompetingeffectsofSOCandthe5EXεmultimodeJahn?Tellereffectasafunction
of the peripheral substituents on the tpa ligand. These subtle distortions were found to have a significant effect on the magnetic
anisotropy. Using a rigorous treatment of all spin multiplets arising from the triplet and quintet states in the d6configuration the
parameters oftheeffective spin-Hamiltonian(SH)approachwerepredictedfromfirstprinciples. Beingbased onanonperturbative
approachweinvestigateunderwhichconditionstheSHapproachisvalidandwhattermsneedtoberetained.Itisdemonstratedthat
already tiny geometric distortions observed inthe crystal structures of four structurally and magnetically well-documentedsystems,
reported recently, i.e., [Fe(tpaR)]?(R = tert-butyl, Tbu (1), mesityl, Mes (2), phenyl, Ph (3), and 2,6-difluorophenyl, Dfp (4), are
enough to lead to five lowest and thermally accessible spin sublevels described sufficiently well by S = 2 SH provided that it is
extended with one fourth order anisotropy term. Using this most elementary parametrization that is consistent with the actual
physics, the reported magnetization data for the target systems were reinterpreted and found to be in good agreement with the ab
initioresults. The multiplet energies fromtheab initio calculationshavebeen fitted withremarkableconsistencyusingaligandfield
(angular overlap) model (ab initio ligand field, AILFT). This allows for determination of bonding parameters and quantitatively
demonstrates the correlation between increasingly negative D values and changes in the σ-bond strength induced by the peripheral
ligands. In fact, the sigma-bonding capacity (and hence the Lewis basicity) of the ligand decreases along the series 1 > 2 > 3 > 4.
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ARTICLE
temperature. The discovery of SMMs nearly 20 years ago3has
induced large-scale research efforts since such species might
ultimately find applications in high-density informationstorage,4
quantum computing,5?7or even magnetic refrigeration.8How-
ever, successful design of SMMs rests on the ability to find
molecules with a sufficiently large magnetic anisotropy such that
the blocking temperature is raised from nearly zero to more
practical values, ideally room temperature. Clearly, this ultimate
and ambitious goal requires a clear understanding of the magne-
to-structuralcorrelationsthatdominatethemagneticanisotropy.
Experimental and theoretical studies on manganese-based,
oxo-bridged SMMs have been used to explore the parameters
that govern the spin-reversal barriers in such complexes.9This
barrier has been expressed as U = S2|D|, where S is the spin
ground state and D is the axial zero-field splitting parameter that
quantifies the magnitude of the magnetic anisotropy. In addition
to S and D, the magnitude of the magnetic exchange coupling
between constituent metal centers in a SMM, J, serves to define
the temperature range over which the barrier is effective, since it
determines how well isolated in energy the spin ground state is
from excited states. Despite the requirement of large S and D,
efforts to increase these parameters simultaneously have been
prohibited by their interrelationship, where it has been shown
already in ref 10 that D is inversely proportional to S2and thus
U = S2|D| does not or only weakly depends on the ground state
total spin. The importance of this result for single-molecule
magnetismhasbeenonlyrecentlyrecognized.11?13Notethatthe
proportionality of D to 1/S2is not specific to the case of
magnetically interacting ions in clusters but is a fundamental
result of rigorous theory as described in detail in ref 10. It has
already been verified experimentally that clustersexhibiting large
values of S tend to show small values of D.9For instance, in the
highestspingroundstateyetobservedforamolecule,S=83/2,14
D is so small that no slow magnetic relaxation could be detected,
despite the comparatively large local anisotropy due to the
Mn(III) ions involved in the cluster.
InMn(III)-basedoxo-bridgedSMMsthemagneticanisotropy
stems from nearly parallel alignments of the Mn(III) local spins
arising from Jahn?Teller splitting of its5E ground state and the
spin?orbit coupling (SOC) of the resulting orbitally nonde-
generategroundstatewithexcitedelectronicstates(5T2)bearing
angular momenta. Magnetic anisotropies described by a total D
of this origin cannot become large. Thus, recently attention has
been turned to transition metal ions with orbitally (nearly)
degenerate ground states. Such states with strong in-state SOC
ariseincertaindNconfigurations,providedthata3-or4-foldaxis
is present. In such systems, the energy gap U can become larger
than 100 cm?1even in complexes with first-row transition
metals.1,2Due to much larger SOC the magnetic anisotropies
are largely enhanced in rare earth (4f) or actinide (5f)
complexes.15?19Orbital moments of transition 3d, 4d, and
5d complexes are usually quenched by off-axial geometric
distortions;20?28however, using geometrically constrained and
sterically bulky macrocyclic ligands it was possible to stabilize
low-coordinate high-spin iron(II) complexes29?34with axial D
values as large as ?50 cm?1in a planar (β-diketiminate)FeCH3
complex.29
Recently, hybrid ligand scaffolds of Fe(II) with trianionic tris-
(pyrrolyl-R-methyl)amines35havebeenreportedtodisplaya3-fold
coordination geometry around the Fe(II) center and an unusually
large value of D = ?40 cm?1as found in K[Fe(tpaMes)].1Slow
relaxation of the magnetization inthe presence of a small dc field
withaneffectiverelaxationbarrier ofUeff=42cm?1providesthe
first example of a mononuclear transition metal complex with a
SMM-like behavior.1
Expanding on this discovery, a series of four (Figures 1 and 2)
structurally and magnetically well-documented compounds,
[Fe(tpaR)]?(R = tert-butyl, Tbu (1), mesityl, Mes (2), phenyl,
Ph (3), and 2,6-difluorophenyl, Dfp (4) have been reported to
display similar properties tuned by a variation of the substituents
R.2Usingthesamecompoundsoxygen-atomtransferproperties,
intramolecular aromatic C?H hydroxylation (for 3, Me3NO f
Me3N + O), activation of nitrous oxide (N2O f N2+ O), and
intermolecular hydrogen-atom abstraction (for 2) have been
demonstrated.35
Figure1. StructuresoftrigonalpyramidalFeN4tpa-basedcomplexeswithSMMproperties.Orange,blue,yellow,andgrayellipsoidsrepresentFe,N,F,
and C atoms, respectively. Hydrogen atoms have been omitted for the sake of clarity (adopted from ref 2).
Figure 2. Geometrical parameters, numbering of ligator atoms of
FeN4?complexes, definition of the geometrical parameters r, R, and β,
and orientation of the Cartesian axes with respect to the
molecular frame.
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Calculations of the electronic structure and magnetic proper-
ties of complexes with orbitally degenerate or nearly degenerate
ground states represent a real challenge for first-principles
methods. DFT-based perturbation theory is not directly applic-
able because of the multireference nature of such ground states.
Note, however, that a general DFT-based ligand field approach
to the parameters of the spin Hamiltonian (SH), LFDFT, has
been proposed.27,36?38At the same time, the systems under
consideration are too large to be treated by the usual implemen-
tations of variational configuration interaction (CI) approaches
that cover dynamic correlation effects. Even the application of
complete active space self-consistent field (CASSCF)39in con-
junction with second-order perturbation corrections of the
CASSCF energies (such as the complete active space perturba-
tion theory, CASPT2,40?44or the N-electron-valence-perturba-
tion theory, NEVPT2) is very challenging.45?48The potential of
using such approaches to calculate SH parameters of
mononuclear13,49and binuclear50?52transition metal complexes
has been recently demonstrated. A general first-principles meth-
od to calculate the spin-dependent part of the energy of ground
and excited multiplet energies in larger polynuclear complexes
has been proposed.53Here, we apply the recent implementation
of the CASSCF and NEVPT2 methods in our computer code
ORCA54to perform large-scale correlated calculations on sys-
temsofanunprecedentedsize.Theresultspresentedinthiswork
provide a theoretical, computational, and conceptual framework
for interpretation, analysis, and prediction of the magnetic
anisotropy in transition metal complexes with orbitally degen-
erate or nearly degenerate ground states.
Using this first-principles method we study the effects of the
small5EXε multimode Jahn?Teller distortions and the influ-
ence of the substituents of the tpa ligand on the magnetic
anisotropy by a full rigorous treatment of all spin multiplets
stemming from the quintet and triplet states within the d6
configuration of Fe(II) and by an approximate effective SH
approach. The comparison between these two independent
treatments allows one to specify in which cases and under which
conditions the SH approach is still valid. Furthermore, it
becomes evident which terms are minimally required in the SH
in order to describe the actual level structure correctly. We show
below that the tiny geometric distortions observed in the crystal
structures of the well-documented systems reported recently
([Fe(tpaR)]?2, 3, and 4 (Figure 1) are enough to lead to only
five low-lying and thermally accessible spin S = 2 sublevels
described sufficiently well by a S = 2 spin Hamiltonian, provided
that it is extended with one fourth-order tensor spin operator.
Using this approach, the reported magnetization data have been
reinterpreted and found to be in good agreement with the ab
initio results. Furthermore, the multiplet energies from the ab
initio calculations have been fitted with remarkable consistency
using a ligand field (angular overlap) model (ab initio ligand
field), thus allowing one to deduce bonding parameters. Using
this approach one is able to explore the dependence of D on the
geometric and electronic structure of the ligand system.
II. THEORY AND COMPUTATIONS
II.1. Computational Details. Since the results of the calcula-
tions are expected to be sensitive to small structural details, we
conducted the calculations on the four compounds (1?4) on the basis
of both crystal and optimized structures. Geometry optimizations
have been performed using the nontruncated systems along with the
Perdew?Becke?Ernzerhof (PBE) functional,55empirical van der
Waals corrections56for the DFT energy, the scalar relativistic zero-
order regular approximation (ZORA),57and the scalar relativistically
recontracted (SARC)58version of the def2-TZVP basis set.59
Ground and excited state energies and wave functions as well as
magnetic properties were calculated on geometries from X-ray diffrac-
tion data1,2and from DFT geometry optimizations using the CASSCF
moduleofORCA(toaccountforstaticcorrelation)togetherwiththeN-
electron valence perturbation theory (NEVPT2)45?48(to account for
dynamic correlation). Unlike the popular CASPT2 method, NEVPT2
doesnotsufferfromintruderstateproblemsbecausetheimportanttwo-
electron interactions inside the active space are already included in the
definition of the zeroth-order Hamiltonian.60This results in a spectrum
of the zeroth-order Hamiltonian that is much closer to the spectrum of
the full Hamiltonian than what one can achieve with a one-body zeroth-
order Hamiltonian. This ensures properly positive and sufficiently large
energy denominatorsthat areimperative forastableperturbation series.
In CASPT2 one needs to introduce level shift parameters in the energy
denominators in order to avoid divergence. The final energies then
depend on the user defined level shift.61,62
The nontruncated systems 1?4 were used in the calculations. It is
worth emphasizing that this is essential to obtain realistic results.
Truncated versions of the systems only capture a fraction of the
differences in magnetic properties observed experimentally. This will
be elaborated below.
For the correlated calculations, basis sets of def2-TZVPP, def2-
TZVP, def2-SVP, and def2-TZVP(-f) quality for Fe, N, H, and C,
respectively, alongside with the corresponding auxiliary sets have been
used.54In this set of calculations only the metal d orbitals were included
in the active space. According to conventional wisdom, this set is too
small as a second d shell is usually required in CASPT2 calculations
along with the corresponding metal?ligand bonding orbitals. Techni-
cally, these extensions do not present essential problems; however, our
preliminarycalculationswithextendedactivespacesdidnotimprovethe
results noticeably. Hence, in keeping with Ockham’s razor, we used the
smallest possible active space that cleanly maps onto ligand field theory.
Thefactthatinclusionofmetal?ligandbondingorbitalsisnotnecessary
probablystemsfromthelimitedcovalencybetweenhigh-spinFe(II)and
the ligand. We do not expect this to be a universal conclusion for
CASSCF/NEVPT2 calculations.
Realistic treatment of SOC is crucial for successful modeling of the
magnetic properties. In a5E orbitally degenerate ground state SOC
occurs in the first order of perturbation theory (PT) but generally at
second order for orbitally nondegenerate states with S > 1/2. This leads
to mixing of states which differ in their spin by ΔS= (1,0. Through this
mixing, SOC reintroduces some orbital angular momentum into the
electronic ground state that is otherwise well known to be quenched
through low symmetry. In QDPT,63one starts by obtaining an approx-
imate solution of the Born?Oppenheimer (BO) Hamiltonian of a
multireference type such as CASSCF in the form given by
jΨSS
Iæ ¼∑
μ
CμIjΦSS
μæ
ð1Þ
where the upper indices SS stand for a many-particle wave function
(configuration state function, CSF) with a spin quantum number S and
spin projection quantum number MS= S. SOC lifts the (2S + 1)
degeneracy ofthe total spin Sof^ HBOeigenfunctions. Thus, thebasis for
the SOC treatment are the |ΨI
calculated in the first step of the procedure and Ms= ?S...S labels all
membersofagiventerm.MatrixelementsofSOCoverthe|ΨI
functions are easily generated making use of the Wigner?Eckart
theorem, since all (2S + 1) term components share the same spatial
part of the wave function.64In this way, both the SOC and the Zeeman
interaction can be accurately accounted for.
SSæ states, in which I extends to all states
SMsæbasis
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ARTICLE
The Zeeman interaction can be accounted for by diagonalization of
the matrix representation^ HBO+^ HSOC+^ HZin the |ΨI
SMsæ basis
ÆΨSMs
I
j^ HBO þ^ HSOC þ^ HZjΨS0M0s
¼ δIJδSS0δMsM0
Thecomplete manifoldof5quintet and 45triplet states wasincluded in
the calculations, andSOCwas accountedforbythe meanfield(SOMF)
Hamiltonian.65,66Evaluation of the matrix elements of the orbital
momentum operators between the |ΨI
terms of one-electron matrix elements within the MO basis. This
procedure carries us beyond the perturbative regime and accounts
for strong SOC effects to all orders. Test calculations additionally
including the 50 singlet states did not change the results. Similar to
the procedures followed in the closely related CASSCF/CASPT2
methodology with inclusion of SOC,44matrix elements were calculated
using the state-averaged CASSCF (SA-CASSCF) wave functions and
NEVPT2 corrections are only included in the diagonal of the QDPT
matrix.
An approximate counterion/solvent modeling using the conductor-
like screening model (COSMO)67,68was also attempted. However,
possibly due to the limited total charge of ?1 of the target systems, the
results arealmostidenticaltotheonesinwhichtheCOSMOmodelwas
not included.
In order to study vibronic coupling effects (the present systems
represent5EXε Jahn?Teller problems) we used a truncated model
obtained by freezing the geometry of each Fe(tpa) complex unit and
replacingthebulkysubstituents atthepyrrolylfragmentswithhydrogen
atoms (C?H bond distances and H?C?C bond angles were reopti-
mized with the Fe(tpa) geometry unchanged).
II.2. Magnetic Properties. II.2.1. Exact Treatment within the
Manifold of the5E Ground Term. Under the combined action of SOC
and vibronic coupling, the5E ground state of trigonal Fe(II) splits into
10 sublevels spaced in a narrow interval of 500 cm?1. This manifold is
wellseparatedfromallotherexcitedstatesbyanenergygapofmorethan
5000 cm?1.
TheleadingtermintheHamiltonianconsistsofthediagonalenergies
resulting from SA-CASSCF eigenvalues, corrected for dynamical corre-
lation by NEVPT2. Within the complete triplet and quintet manifold of
the d6configuration, 160 microstates arise that interact via^ HSOC. This
operatoraswellasthesetofspin^Siandangularmomentumoperators^Li
(i =x, y, z)are represented byoff-diagonal(complex valued) 160 ?160
matrices. Denoting the 10 lowest CI eigenvectors of ^ HSOCby the
rectangular submatrix C(1:160,1:10), the matrices of^Siand^Liare
transformed into the (10 ? 10)^Si0and^Li0
subspace (eqs 3 and 4)
J
æ
sES
Iþ ÆΨSMs
I
j^ HSOC þ^ HZjΨS0M0s
J
æ
ð2Þ
SMsæ basis functions is done in
imatrices of the5E model
^Si
0¼ C†^SiC
ð3Þ
^Li
0¼ C†^LiC
Denoting the diagonal matrix of the 10^ HSOClowest eigenvalues by Λ ^,
we then solve the eigenvalue equation for the operator matrix^ H = Λ ^+
^ HZ(^ HZ= the Zeeman matrix) on a grid of points on a unit sphere
defined by the vectors (nx, ny, nz) (eq 5) using the value of the pro-
bing magnetic field B = Bo(in T) and for the sake of numerical
differentiation (see below) two more incremental values B = Bo+ 0.01
and B = Bo+ 0.02.
ð4Þ
^ H ¼ Λ ^
þ βB½nxð^Lx
þ nzð^Lz
0þ g0^S0
0þ g0^Sz
xÞ þ nyð^Ly
0þ g0^Sy
0Þ
0Þ?ð5Þ
The field-dependent adiabatic magnetization ofacrystalline powder has
been calculated using a numerical integration over all magnetic field
directions (eq 6, NA= the Avogadro number, kB= the Boltzmann
constant,Z=thepartitionfunction,i.e.,thesumofBoltzmannfactorsfor
all states under consideration) defined by the vector (nx, ny, nz) or
alternatively by the polar angles j and θ.
Zπ
Mav¼ NAkBT
0
Z2π
0
d
dBðj,θÞlnðZðBðj,θÞÞÞ
??1
4πsin2θ dθ dj
ð6Þ
Notethatthistreatmentisbydefinitionmoreaccuratethanthedirect
diagonalization of theSOC and magneticfield together in the basis of
the nonrelativistic magnetic sublevels of the5E term. This is because
the SOC of the5E state with all other quintet and triplet ligand field
states is accounted for to all orders. The only thing that is missing is
the magnetic field-induced mixing of the 10 lowest SOC-corrected
relativistic eigenstates with the other states. This must be tiny given
that the orbital Zeeman matrix elements are on the order of 1 cm?1
while the energy differences to the next low-lying SOC-corrected
states are, by construction, higher than 5000 cm?1. Technically, the
simultaneous diagonalization of the SOC and magnetic field in the
entire ligand field manifold would not be a problem of course.
However, the present method has the advantage that the effective
Hamiltonian obtained in the SOC-corrected 10 ? 10 spaces maps
most cleanly onto the spin Hamiltonian to be discussed in the next
section.
II.2.2. Connection to the Spin Hamiltonian. The5E ground state of
Fe(II) in trigonal ligand field is described by a spin S =2 and two singly
occupied orbitals e(dxz) or e(dyz) which carry an extra electron in
addition to the half-filled [(dxzdyz)2dz21(dx2?y2dxy)2] shell. These states
give rise to Ml= (1 eigenfunctions of the angular momentum operator
^Lz(eq 7).
ffiffiffi
Since the states|5E,1æand|5E,?1æcannotmix by^Lxand^Ly(Mlchanges
by 1 under the action of these operators) and the excited |5A1,0æ state is
muchhigherinenergy69(seesectionIII.1,Figure5),the5Egroundstate
SOC operator^ HSOCin this approximation takes the simple form
j5E, ( 1æ ¼ ð1=
2
p
Þðj5E,dxzæ ( ij5E,dyzæÞð7Þ
^ HSOC¼ ? ðζ=4Þ^Lz^Sz
Thus, within the |Ms,Mlæ basis^ HSOCis represented by a diagonal 10 ?
10 matrix with elements: ?(ζ/4)MsMl(Ms= 0, (1, (2 and Ml=
(1).70?74ζ is the effective (covalently reduced) SOC ‘constant’ of
Fe(II)(seeref75foradetaileddiscussion).Onsymmetryloweringfrom
C3vtoCsorevenC1,the5E(dxz)and5E(dyz)sublevelssplitandmixwith
each other, as described by the two energy parameters δ1and δ2,
respectively (eq 9). Thus, within the basis of eq 7 the ligand field opera-
tor^ HLF(eq 10) is off-diagonal; It mixes the terms |Ms,1æ and |Ms,?1æ.
ð8Þ
5EðdxzÞ5EðdyzÞ
?δ1
δ2
δ2
δ1
"#
ð9Þ
^ HLF¼ δ ^L2
The physical origin of δ will be thoroughly analyzed in section II.3.
Analyticalexpressionsfortheeigenvaluesofthematrix^ HSOC+^ HLF(5E)
are given in eq 11 (see Supporting Information for a derivation) along
withsymmetrynotationspertainingtotheD3holohedrizedsymmetry.It
is worth noting that within the many-electron basis of the5E ground
term, the A1and A2states (eq 11) remain accidentally degenerate in the
x?^L2
y
??
;δ ¼ ?δ1 þ iδ2
ð10Þ
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ARTICLE
absence of SOC mixing with excited states (see below).
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
EðEÞ ¼ (1
4
EðEÞ ¼ ( δ
EðA1,A2Þ ¼ (1
2
4δ2þ ζ2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p
16δ2þ ζ2
p
ð11Þ
From eqs 7?10 the origin of the magnetic anisotropy immediately
emerges; without off-axial distortions (δ = 0) there is a 1:1 mixing
between the xz and yz sublevels of the5E term by SOC, and this
leadstoadditionofanorbitalangularmomentumcontributionof1/2
(L = 1 = (1/2)go, go= 2, the electron free spin g factor) to the spin-
only value of S = 2 in the ground state and subtracts the same amount
from the excited state, leading to total spin moments J of these two
states of 5/2 (a ‘sextet’ ground state) and 3/2 (a ‘quartet’ excited
state), respectively. However, these states are split further by SOC
even in zero field. It follows immediately (within the approximations
inherent in eq 11) that a SH of the usual form for J = 5/2 and 3/2
applies to this case. Since^Lx=^Ly= 0, this SH is of the Ising type.
Energies and their corresponding^Jz=^Sz+ (1/2)^Lzeigenvalues (MJ,
Table 1)along with the spin Hamiltonian of eq 12 (E = 0 in this case)
have been applied to obtain the ZFS and g-tensor parameters
included in Table 1; the value of D is correspondingly negative for
J = 5/2 (the ground state, Table 1, second row, first column) and
positive for the J = 3/2 (the excited state, Table 1, third row, first
column), and one additional fourth-order term (B40) in the param-
etrization emerges.
h
þ B40 35^J4
þ3JðJ þ 1ÞðJðJ þ 1Þ ?2Þ?
Turning now to the other extreme, δ . ζ, and applying perturbation
theory, energy expressions for the zero-field split levels of the two
S = 2 states can be derived. The resulting expressions for D are listed
^ Heff
ZFS¼ D^J2
z?JðJ þ 1Þ=3
h
i
þ ðE=2Þ^J2
þþ^J2
?
??
z?ð30JðJ þ 1Þ?25Þ^J2
z
ð12Þ
in Table 1 (third column); a SH of the broadly used form of eq 11
with a negative (positive) D for the ground (excited) state applies in
this case
^ Heff
ZFS¼ D½^S2
z?SðS þ 1Þ=3? þ E^S2
x?^S2
y
??
ð13Þ
In the Fe(II) complexes considered that possess close to trigonal
geometries, ζ .δ, i.e., much closer to the case with δ = 0 (eq 12). It
is interesting to observe that the lowest A1 and A2 states are
nondegenerate and thus (being 1:1 mixtures of functions with
Ml= (1, see Supporting Information) nonmagnetic, i.e., MJ= 0 in the
absence of a magnetic field. However, they become polarized
(mixed) by an applied magnetic field which recovers values of
MJ= (5/2 (see Table 1, first column). This causes a Zeeman splitting
of 5βB. Off-axial distortions lead to a first-order (δ splitting of the
second E (MJ= (1/2)term which tends to recover pure5E(dxz)and
5E(dyz) many-electron wave functions starting from |Ms, (1æ
(eq 7). Thus, it competes with the Zeeman splitting. If δ . βB/2,
the Zeeman splitting is suppressed, thus leading to a dominant MJ=
Ms= 0 situation which remains even with an applied magnetic field.
In this particular case one can redefine the SH of eq 12 in terms of a
formal spin of S = 2 for two noninteracting electronic states with D
negative (positive) in the ground (excited) state and correspond-
ingly modified g-tensor values of gz= 2.5(1.5) (gx= gy= 0). Under
these conditions the SH of eq 13 is still applicable, when extended
with a fourth-order tensor operator term represented by the para-
meter B40, eq 14. This is similar to eq 12 (for energy expressions for
D and B40see Table 1, second column). Both the signs and the
magnitudes of D and B40are dominated by SOC (ζ)but are reduced
by the distortions (δ, Table 1, second column).
h
þB40 35^S4
þ 3SðS þ 1ÞðSðS þ 1Þ? 2Þ?
We should note that eq 11 is not exact but contains approximations,
i.e., the neglect of mixing via SOC of5E with the excited5A1and5E
andthetripletstates.Uponaccountingforsuchamixingthetopmost
pair of states A1and A2, which are accidentally degenerate in eq 11,
split and shift downward in energy (Figure 3). Comparison with
^ Heff
ZFS¼ D^S2
z? SðS þ 1Þ=3
z? ð30SðS þ 1Þ ?25Þ^S2
i
þ ðE=2Þ^S2
þþ^S2
?
??
z
h
ð14Þ
Table 1.
Hamiltonian Parameters in Trigonal FeN4Complexes in
Dependence of Spin?Orbit Coupling (ζ) and off-Axial
Splitting (δ)a
5E Ground State Multiplet Energies and Spin-
δ = 0
ζ . δ . βB/2
δ . ζ
EMJ
EMS
EMS
A1,A2: ζ/2
E: ζ/4
E: 0
E: ?ζ/4
A1,A2: ?ζ/2 (5/2
(3/2
(1/2
(1/2
(3/2
ζ/2
ζ/4
δ, ?δ
?ζ/4
?ζ/2
(2
(1
0,0
(1
(2
δ + ζ2/(8δ)
δ + ζ2/(32δ)
δ, ?δ
?δ ? ζ2/(32δ) (1
?δ ? ζ2/(8δ)
(2
(1
0,0
(2
D: ζ/8
B40: 0
gz: 2
gx,y: 0
D: (3ζ/28) ? (δ/7)
B40: ?(ζ/840) + (δ/140)
gz: 1.5
gx,y: 0
D: (ζ2/32δ)
B40: 0
gz: 2
gx,y: 2
D: ?9ζ/112
B40: ζ/3360
gz: 2
gx,y: 0
aEntries in the second and third rows include SH paramaters for the
lower and upper S = 2 nonrelativistic spin multiplets of the5E state.
D: ?(3ζ/28) + δ/7
B40: (ζ/840) ? (δ/140)
gz: 2.5
gx,y: 0
D: ?(ζ2/32δ)
B40: 0
gz: 2
gx,y: 2
Figure 3.
splitting of5E (2δ) and the effective spin?orbit coupling parameter ς
deduced from NEVPT2 calculation on [Fe(tpatbu)]?including the full
setofS=2(5states)andS=1(45states).Theplothasbeenconstructed
using the AOMX program package,85along with ligand field parameters
obtained from a fit to CASSCF results for (1, Table 6b) allowing for a
variation of δ.
5E split sublevels originating from the interplay between the
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Inorganic Chemistry
ARTICLE
exact calculations (see section II.2.1 and results below) shows that
the lowest five thermally accessible spin sublevels are well described
by the spin Hamiltonian of eq 14 in a quite wide range of δ values;
already small distortions described by δ lead to a splitting pattern
typical for a S = 2 spin with a negative D as illustrated in Figure 3.
Within the basis of the five spin eigenfunctions with Ms= (2, (1, 0,
the Hamiltonian of eq 14 takes the following matrix form
HZFS
j?2æ
j?1æ
0
j0æ
p
j1æ
0
3E
0
j2æ
0
0
ffiffiffi
2D þ 12B40
0
ffiffiffi
0
ffiffiffi
6
0
E
?D ?48B40
0
3E
0
6
0
p
E
?2D þ 72B40
0
ffiffiffi
6
0
p
E
?D ?48B40
06
p
E2D þ 12B40
2
6666664
3
7777775
ð15Þ
From the separations between the energies of the computed E((2 f
(1)andE((2f0)levels(cf.diagonalenergiesofeq15)DandB40are
given by
D ¼ ?1
7½Eð ( 2 f ( 1Þ þ Eð ( 2 f 0Þ?
1
140Eð ( 2 f 0Þ ?4
B40¼
3Eð ( 2 f ( 1Þ
??
ð16Þ
and |E| is just 1/6 the off-axial splitting of (1.
The parameters of the SH (eq 14) ?D, B40, E, and additionally g
have been fitted to the reported field-dependent magnetization data
of complexes 1?4 using a nonlinear optimization tool provided by
MatLab that minimizes the maximum deviation between the calcu-
lated and the experimental values of the magnetization (see Support-
ing Information for details regarding the fitting). The resulting
values of D, B40, and E can then be compared with the ab-initio-
computed ones.
II.3. Multimode5EXε Jahn?Teller Effect. The5E ground state
oftrigonalFe(II)isvibronicallyunstableandsubjecttodistortionsalong
ε vibrations which can lower the energy and lift the orbital degeneracy.
Restricting our attention to linear5EXε Jahn?Teller coupling, the
ground state adiabatic potential energy is represented by the matrix of
eq 17,76?78where Kεand Fεare the force constant and linear vibronic
coupling parameters, respectively.
2
ð1=2ÞKεðQ2
xþ Q2
?FεQy
yÞ þ FεQx
?FεQy
xþ Q2
ð1=2ÞKεðQ2
yÞ ?FεQx
4
3
5
ð17Þ
In the complexes under consideration, focusing on the N-donor atoms
involved in bonding to Fe(II), there are three types of symmetrized
localized modes of ε symmetry involved: two-bending (in- and out-of-
plane,εdandεo)andonestretching(εs)whosehigh-(xorQx,C3vfCs)
and low- (y or Qy, C3vf C1) symmetry components are visualized in
Figure 4. In the [Fe(tpaR)]?complex these modes are mixed with
vibrational coordinates with contributions from all atoms of the tpa
macrocycle to give rise to numerous normal modes of the same
symmetry.
Thus, the simple eq 17, valid for a single active ε vibration (ideal
vibronic system), has to be extended with a number of similar
symmetry-related linear vibronic and restoring force terms resulting
in a multimode Jahn?Teller problem of the5EX(ε(1) + ε(2) + ...)
type. To this end, we apply the formalism for treating such problems
described in ref 78. More specifically, it was shown that the adiabatic
potential surface of eq 17 extended with all contributing terms can be
represented in configurational space in terms of a displacement along
a single interaction mode ε(qx,qy) (eq 18).
j5E,dxzæ
xþ q2
?Fqy
j5E,dyzæ
ð1=2Þðq2
yÞ þ Fqx
?Fqy
xþ q2
ð1=2Þðq2
yÞ þ Fqx
2
4
In the above expression qε, qζ, F, and the Jahn?Teller stabilization
energy EJTare given in eqs 19?21. The linear vibronic coupling Fεi
and the force constant Kεiparameters are characteristic of each of the
normal modes involved.
3
5
ð18Þ
qγ¼1
F∑
i¼1
Nε
FεiQγi;γ ¼ x,y
ð19Þ
F ¼ ∑
Nε
i¼1
F2
Kεi
εi
!1=2
ð20Þ
EJT¼1
2∑
i¼1
Nε
F2
Kεi
εi
ð21Þ
The force constant parameters Kεiare directly accessible from the
Hessian matrix, resulting from an ab initio or DFT calculation, while
Fεican be calculated as follows. Focusing on a given normal mode,
described by the set of mass-weighted displacement vectors qγi;(γ = x,y),
wenotethatbecauseofthelocalizednatureofthe3d6groundstateofthe
Fe(II)complexonlymovementsofthefirstcoordinationsphere,i.e.,the
local modes (Figure 4), contribute toFεi. This leads to the expression of
FεiintermsofthelocalvibroniccouplingconstantsFεofeq22,weighted
withthevanVleckcoefficientsaiε,79i.e.,theprojectionofagivennormal
mode qγi;(γ = x,y) on the local mode Qγ(eq 23). Focusing on the
in-plane bending modes εd(Figure 4, see Figure 2 for ligand number-
ing and coordinate orientation) with dominant contributions to the
Jahn?Teller activity, aiεhave been obtained from the scalar product
between the vectors 1/(6)1/2(2,?1,?1) and 1/(2)1/2(0,1,?1) repre-
senting the local modes of eqs 24 and 25 and the ab-initio-computed
normal modes.
Figure 4. Symmetrized displacements contributing to the Jahn?Teller
splitting of the5E ground state of the FeN4complexes.
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ARTICLE
Fεi¼
5E,dxzj∂V
?
∂qxij5E,dxz
??
?∂Qx
¼
5E,dxzj∂V
∂Qxj5E,dxz
∂qxi
¼ aiεFε
ð22Þ
Qγ¼∑
Nε
i¼1
aiεqγi;ðγ ¼ x,yÞð23Þ
Qx¼
1ffiffiffi
1ffiffiffi
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
6
p Rð2δR34?δR23?δR24Þð24Þ
Qy¼
2
p RðδR23?δR24Þð25Þ
Fε¼
EFC=ð2∑
Nε
i
a2
iε=KεÞ
s
ð26Þ
Within the approximation of a linear
(eq 18), the Jahn?Teller stabilization energy EJT(eq 21)is just 1/4 of
the5E energy splitting EFC, the energy of the Franck?Condon (FC)
transition from the lower to the upper sheet of adiabatic potential
energy surface taken at the ground state equilibrium geometry.
Adopting a value of EFCas given by the CASSCF(NEVPT2) results,
the value of the unknown parameter Fεhas been fixed using eq 26, as
derived from combination of eqs 21 and 22.
II.4. Ligand Field Interpretations of the ab Initio Data.
Significantinsightintermsoffamiliarchemicalconceptscanbeobtained
by mapping the ab initio results onto ligand field theory. This might be
viewed as another effective Hamiltonian treatment. While above we
restricted our attention to the magnetic sublevels arising from the5E
manifold,we here focusontheentire subsetofligandfieldexcited states
and their parametric representation in terms of the angular overlap
model (AOM)80?83variant of ligand field theory.84This results in
bonding parameters that are to a large extent transferable between
systems. Here, multiplet energies are expressed in terms of one-electron
matrix elements between molecular orbitals (MOs) with dominant d
5EXε Jahn?Teller coupling
character, thus defining an effective (5 ? 5) ligand field matrix that
describes the anisotropic interaction of the central metal orbitals with the
ligandorbitals.InterelectronicrepulsionandSOCareaccountedfor,asusual,
bycovalentlyreducedatom-likeparametersB,C(Racahparameters),andζ.
The AOM introduces perturbations on the metal d electrons from
well-aligned ligand σ and π orbitals, which are described by energy
increment parameters eσand eπ, respectively. These parameters are
specific to the chemical nature of the ligand and also depend on the
metal?liganddistance.Inaddition,factors thatsolely reflect the angular
distribution of the ligands around the metal (e.g., the symmetry of the
ligandfield)areintroduced.Theyarecalculatedfromtheactualstructure
ofthesystem,withoutanyfittingorarbitrariness.Forcomplexes1?4 with
geometriesclosetotrigonalandaplanarFeN3moiety,the(5?5)AOM
matrix takes a block diagonal form given by eqs 27 and 28 for the (dyz,
dx2?y2;dxz, dxy) and dz2orbital sets of e and a1symmetry, respectively.
The parameters eσ
equatorial pyrrolide and axial amine ligands, while eπs
due to the out-of-plane orbitals of the pyrrolide ligand. With ψ = 0?
(in this case the pyrrolide ring makes a dihedral angle γ of 90? with the
FeN3plane, ψ = 90 ? γ) the antibonding effect of eπs
orbitals is minimal, whereas on the dx2?y2(dxy) orbitals it is maximal.
e dyzðdxzÞ
ð3=2Þðsin2ψÞee
ð3=4Þsinð2ψÞee
eand eσ
adescribe σ-antibonding interactions with the
ereflects the effect
eon the dyz(dxz)
dx2?y2ðdxyÞ
ð3=4Þsinð2ψÞee
ð3=8Þ½4ðcos2ψÞee
πs
πs
πs
πsþ 3ee
σ
2
4
3
5
ð27Þ
a1dz2: ð3=4Þee
Thesetofligandfieldparameterseσ,eπ,B,C,andζareusuallyadjustedwith
respect to high-resolution spectroscopy.81?83However, quite frequently,
this is a seriously underdetermined problem; usually there are more para-
metersthanobservables.Herewefollowtheoppositerouteandtakethe
ab initio results as a much more comprehensive numerical database for
the AOM parametrization.
The AOM model is applied in a stepwise procedure as follows: We
start with the energies of the four spin-allowed ligand field excitations
following their assignment. These transition energies are not directly
σþ ea
σ
ð28Þ
Figure 5. SA-CASSCF orbital shapes and energies and the ground state configuration of [Fe(tpa)Mes]?.
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Inorganic Chemistry
ARTICLE
affectedby the parameters B andC. Weadjust the parameters eσ
eπs
step, we adopt these one-electron parameters without changes and
obtain the parameters B and C from the computed energies of the spin-
forbidden transitions. In the present case these are the transitions from
the5E ground state to the triplet ligand field excited states. Finally,
adopting the values of eσ, eπ, B, and C and switching on the SOC, we fix
the value of ζ from the computed energies of the5E ground state SOC
split sublevels. These calculations have been done by interfacing the
ORCA program with the well-established ligand field program
AOMX.85The use of artificially high symmetry was allowed for the
sake of simplifying the assignments.
e, eσ
a, and
eof the Fe?N bonds from a best fit to these transitions. In the second
III. RESULTS AND DISCUSSION
III.1. Chemical Bonding, Geometric Structures, and Vibro-
nic Activity. The ground state structures and properties of the
[Fe(tpaR)]?series are governed by the Fe?tpa bonding inter-
actions, which are reflected in the shapes and energies of the
MOs dominated by 3d functions of Fe(II). This is illustrated in
Figure5,taking[Fe(tpaMes)]?asanexample.Withthepurelyπ-
type dxz,yzand the σ-antibonding dz2and dxy,dx2?y2orbitals
(eqs 27 and 28), an orbital energy ordering typical for a trigonal
pyramidal complex results. The absence of a second axial ligand,
due to steric hindrance of the bulky tpaRligands, leads to con-
siderable stabilization (included in eq 28 in the parameter eσ
of the dz2orbital due to 3dz2?4s mixing with respect to the
dxy,dx2?y2orbitals. However, this stabilizing effect is not strong
enough to place dz2below the dxz,dyzpair of orbitals. Thus, a
ground state of5E symmetry with an extra electron of π-type
results. The underlying Fe?tpa interaction will be quantified
following a ligand field analysis of the ab initio results in section
III.4.
The coordination geometry around Fe(II) in all four com-
pounds is close to trigonal with one longer bond to the axial
N-ligandfromtheaminogroup(mean valueover reported X-ray
structures of 2.17 Å) and three shorter bonds to the pyrrolyl N
donors in the equatorial plane (2.03 Å). The Fe ion is displaced
significantly (by 0.26 Å) from the plane defined by the three
pyrrolyl N ligands. Bond distances and bond angles relevant for
the following discussion are defined in Figure 2, and their values
from X-ray data are listed in Table 2; these are the Fe?N bond
distances (ri), N?Fe?N bond angles Rijbetween the equatorial
a)
Fe?Nbonds,andtheβijanglesformedbetweentheaxialandthe
equatorial Fe?N bonds.
DFT geometry optimizations of the entire complexes yield
geometric parameters in good agreement with values reported
from X-ray data (Table 2).35,2Calculations of the Hessian and
the complete vibrational spectrum show that optimized struc-
tures forcomplexes 3and4correspond to minima of theground
state potential energy surface. For complex 2 two imaginary
frequencies are obtained. The latter are found to be due to
skeletal vibrations with main participation from the pyrrolyl
fragments. No vibrational spectrum could be calculated for
complex 1, for which the optimized geometry is found to be
closest to trigonal. The reason is that for nearly orbitally
degenerate systems the numerical second-derivative treatment
becomes unstable as tiny distortions change the electronic
ground state and hence drastically change the directions of the
obtained forces.
For convenience, the geometric parameters can be repre-
sented in terms of their deviations from axial symmetry. Such
deviations are clearly discernible, both in the experimental
structures and, to a lesser extent, in the optimized structures as
well (Table 2). For 1 a regular trigonal structure has been
reported,2whereas the DFT-optimized structure is significantly
distorted.Thepresenceofsuchadistortionisconsistentwiththe
magnetic behavior of 1 as will be discussed in section III.3.
Thelow-symmetry distortions of the firstcoordination sphere
of Fe(II) are rather complex and can originate from the
Jahn?Teller activity of the5E ground state as well as from steric
effects imposed by the rigid ligand backbone and from the
counterions in the solid. In order to shed more light on these
issues, a hypothetical complex was calculated that involved
the non-Jahn?Teller active Mn(II) ion instead of Fe(II). The
corresponding optimized structure was found to be nearly
perfectly axial (Table S1 in the Supporting Information). Thus,
we conclude that the off-axial distortions are largely due to
Jahn?Teller activity. These distortions can be quantified using
the Jahn?Teller radii F corresponding to the local modes of ε
symmetry, two bending and one stretching mode (defined in
Figure 4 and listed in Table 3a).
They show that displacements along the FeN3 bending
mode εdare dominant. Further analysis of these distortions
shows, in agreement with the epikernel principle,86,87that the
Table 2. Fe?N Bond Distances (in Å) and N?Fe?N Bond Angles (ino) of the FeN4Chromophore from X-ray Diffraction Data
and DFT Geometry Optimizationa
[FetpaTbu]?
[FetpaMes]?
[FetpaPh]?
[FetpaDfp]?
exp. geometryDFT geometryexp. geometryDFT geometryexp. geometryDFT geometryexp. geometryDFT geometry
r1
r2
r3
r4
R23
R24
R34
β12
β13
β14
2.144
2.031
2.031
2.031
118.35
118.35
118.35
82.55
82.55
82.55
2.187
1.989
1.995
1.996
117.93
118.72
117.33
81.86
81.83
81.68
2.172
2.008
2.041
2.024
117.35
115.28
122.38
82.45
82.01
83.21
2.218
1.992
2.014
2.015
116.02
118.77
119.48
82.46
81.89
81.62
2.161
2.013
2.019
2.016
120.22
115.56
120.27
83.84
83.37
82.88
2.229
1.989
1.993
1.995
118.64
119.31
116.53
82.27
82.36
81.80
2.196
2.042
2.038
2.037
115.82
121.57
116.63
81.19
82.30
82.01
2.229
2.005
2.009
2.013
116.99
119.88
116.35
81.27
80.73
81.78
aSee Figure 2 for ligand numbering and definitions.
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high-symmetry (Cs) distortion Qx(x) prevails over the low-symme-
try(C1) oneQy(y) (Table3b) andthusdeterminestoalargeextent
the magnitude of the total F = (Qx2+ Qy2)1/2. Following the
formalismofsectionII.3weanalyzedthemultimode5EX(ε(1)+
ε(2)...) Jahn?Teller effect using truncated model complexes
[FeN4C15H15]?(I, Figure 6a) and [MnN4C15H15]?(II). The
firstonewasused inordertoobtain theparametersFεi,whilethe
second one was used to obtain a reliable Hessian, avoiding the
complications from additional quadratic vibronic coupling terms
present in I.88In these complexes we replaced the bulky
substituents of the pyrrolyl fragments by hydrogen atoms
(Figure 6a). From the normal modes with ε symmetry only four
contribute to a very small total Jahn?Teller stabilization energy
EJT(total)= 28.4 cm?1(Table 3c). We can assume that the clear
local distortions along the Qxcoordinate must originate from
low-symmetryperturbations(strains)stemmingfromthedistant
ligand substituents which become vibronically enhanced in the
way specified in refs 76?78 and 89.
To study this point, geometry optimizations on complexes
1?4 have been performed, where FeII(d6) has been replaced by
MnII(d5), thus eliminating vibronic forces. Geometrical param-
eters ri, Rij, and βijalong with values of F (defined in Figure 4)
are listed in Table S1 (Supporting Information). For the stereo-
chemically inactive Mn(II) the latter values reflect the net effect
of the substituents on the geometry. As for Fe(II) the tiny
distortions are dominated by displacements along the εdmode
but are now of purely elastic origin. Values of F(εd) for Mn(II)
follow a clear trend increasing from complex 1 to 4 as shown in
Figure 7, where for the sake of comparison the corresponding
values of Fe(II) are presented. The results demonstrate in an
impressive way the vibronic amplification of the distortions
caused by the substituents R. The latter modify the bonding
properties of the tpaRligand and therefore the vibronic coupling
parameter Fε. This will be the subject of the analysis in section
III.4. In addition to the substituents, counterions and packing
forces may also affect the geometry. Treatment and analysis of
these effects is, however, beyond the scope of the present work
and will require further theoretical development.
Apart from the effect of the substituents, our results clearly
show that vibronic coupling in the complexes under study is
weak.Asemergesfromconsiderationofasmallermodelcomplex
[FeN4H9]?(III) (Figure 6b) with axial NH3and three unlinked
equatorial NH2?amido groups, we can attribute the rather weak
Jahn?Telleractivitytotherigidtpaligandbackbone;ageometry
optimization starting from the FeN4coordination geometry,
identical to (I), shows that the axial Fe?NH3bond is unstable.
Upon geometry optimization the axial ligand tends to dissociate
leading to [FeN3H6]?(complex IV, Figure 6c). A much larger
Jahn?Teller activity in (IV) compared to (I) is obtained
(Table 3c). However, in this system5E is an excited state as
the dz2orbital falls below the e set and hence5A1becomes the
lowest state. Weconcludefromtheseanalysesthat the tpaligand
plays a crucial role in regulating the electronic and steric
Figure 6. Truncated model clusters adopted for the study of the
vibronic effects within the5E ground state manifold: (a) Truncated
model complex [FeN4C15H15]1?employed in the study of the EXε
Jahn?Tellereffect;(b)[FeN4H9]?modelcomplexwiththreeunlinked
equatorial NH2?amido groups; (c) [FeN3H6]?model fragment
resulting upon dissociation of [FeN4H9]?.
Table 3. Geometrical Distortions (in Å) of the FeIIN4Cores
in tpa Complexes As Quantified by the Jahn?Teller Radii G
Deduced from Experimental X-ray Data and DFT Geometry
Optimization (a); Decomposition of the Distortions of FeN4
As Given by X-Ray Diffraction Structures along the High-
Symmetry (Qx, Cs) and Low-Symmetry (Qy, C1) Components
of the in-Plane Bending Mode εd(Figure 4) (b); Contribu-
tions of the ε Normal Modes to the Vibronic Coupling
Constants Fεiand the Jahn?Teller Stabilization Energies
EJT(i) in the Multimode5EX(ε(1) + ε(2)...) Jahn?Teller
Effect in [FeN4C15H15]?and [FeN3H6]?Truncated Model
Complexes (c)
[FetpaTbu]?
[FetpaMes]?
[FetpaPh]?
[FetpaDfp]?
(a)X-rayDFTX-rayDFT X-rayDFT X-rayDFT
F(εs)
F(εd)
F(εo)
0
0
0
0.005
0.034
0.005
0.023
0.182
0.031
0.018
0.090
0.022
0.004
0.134
0.024
0.004
0.071
0.015
0.004
0.158
0.029
0.006
0.093
0.027
(b)[FetpaMes]?
[FetpaPh]?
[FetpaDfp]?
Qx
Qy
0.175
0.052
?0.134
0.001
0.157
0.018
(c)
h 9ωi(cm?1)Fεi(cm?1/Å)c
EJT(i) (cm?1)
[FeN4C15H15]?
38a
121a
193a
226a
?87.5
?278.0
?168.4
156.7
11.4
11.3
1.6
1.0
EJT(total)28.4
[FeN3H6]?
57b
86b
97b
530b
?229.6
202.9
?124.7
?390.4
73.6
25.3
7.4
2.4
EJT(total)111.3
aOn the basis of a force field of a geometry-optimized, nondistorted
[MnN4C15H15]?model complex.bOn the basis of a force field of a
geometry-optimized, nondistorted [MnN3H6]?model complex.cCal-
culatedadoptingvaluesofFε(eq22)of1133cm?1/Åand2653cm?1/Å
and the NEVPT25E splittings of [FeN4C15H15]?and [FeN3H6]?
model complexes, respectively.
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ARTICLE
propertiesofthe[Fe(tpaR)]?series.Theeffectis2-fold:first,the
stiffness of the ligand suppresses at least to some extent the
Jahn?Teller coupling of the5E ground state. Second, the axial
Fe?N is imposed by the ligand, and thus, the5E (e3a11e2)
ground state is stabilized over the
otherwise would be the ground state.
The weak Jahn?Teller effect in the complexes under study
leads to a situation in which SOC dominates over vibronic
coupling (Figure 3); here the range of variation of δ is distinctly
smaller than the value of ζ. The energy dependence of the two
lowest states (A1, A2) and E on δ shows that the Jahn?Teller
coupling in these states is completely quenched. This is just the
opposite in the second excited state (E(|MJ| = 1/2), Figure 3),
which is not affected by SOC (Ms= 0 in this case). While the
latter state is only weakly populated at the temperatures of the
magnetic measurement, it may affect the structure at room
temperature, thus supporting the distortions.90
III.2. Multiplet Structure and Effect of Spin?Orbit Cou-
pling. The six d electrons of Fe(II) (Figure 5) give rise to 5
5A1 (a12e2e2) state that
quintet states. Their SA-CASSCF and NEVPT2 energies ob-
tained on the basis of the X-ray structures are listed in Table 4.
The energies of all quintet states are computed to be lower
than those of the triplet states (see section III.4 and Table 6a)
with a sizable energy gap of 13000 cm?1between the5E ground
stateandthelowesttripletstate.Therathersmalldeviationsfrom
axial symmetry (vide supra) lead to a ground state splitting 2δ
which increases progressively from 0 to 375 cm?1across the
series1<2<3<4.Fe(tpaTbu)complex1,whichwasreportedto
be strictly trigonal,possessesa2δofzero atthe CASSCFlevel of
theorywhichincreasesto2δ=15cm?1inNEVPT2.Clearly,this
isanartifactofthecontractionprocedureusedtodefinethefirst-
order interacting space and is shared by all internally contracted
electron correlation methods. Dynamical correlation effects
accounted for at the NEVPT2 level of theory introduce energy
shifts of the5E f5A1,5E f5E transitions by 2000?2500 cm?1
(to higher energy) and of the quintet to triplet transitions by
4000?5000 cm?1(to lower energies) (Table 6a). The
Jahn?Teller effect in the electronic ground state is rather weak
because thedegenerate orbital set involved onlyparticipates inπ
bonding. The excited5E state that has an uneven occupation in
the degenerate, σ- and π-antibonding dxyand dx2?y2set of
orbitals shows a much larger low-symmetry splitting (into5A0
and5A00, Cssymmetry) than the ground5E state (cf. Table 4).
Accounting for SOC leads to the energies of the sublevels
included in Table 4. The energies of the lowest five sublevels
follow on the trends obtained for δ, as depicted in Figure 3. As
follows by a comparison of various sets of model calculations
(Figure 8), the orthorhombic splitting of the first excited term E
(Figure3,5E,E(|MJ|=3/2))increaseswhenextendingtheSOC
matrix with the5A1and5E excited states. It increases even more
when the triplet excited states are also included. Moreover, the
topmost level (Figure 3,5E, E(|MS| = 0)) of the zero-field split
S = 2 ground state drops down in energy when including excited
quintet and triplet states in the CI treatment. It follows from
Figure 8 that accounting for surrounding effects and extending
Figure 7. Extent of distortions as quantified by F(εd) of [Mn(tpaR)]?
incomparisonwiththoseoftheirFe(II)congenersfromDFTgeometry
optimization; the value of F(εd) resulting from the DFT geometry
optimization of the truncated model complex [MnN4C15H15]1?is
shown by a horizontal dotted line.
Table4. Energies(incm?1)ofS=2Statesand5E(C3parentsymmetrynotations)ComponentsSplitoutbySpin?OrbitCoupling
from CASSCF and NEVPT2 Calculations of the Four FeN4Complexes with Geometries from X-ray Diffraction Data and
Accounting for the Complete Set of the 5 Quintet and 45 Triplet Electronic Statesa
[FetpaTbu]?
[FetpaMes]?
[FetpaPh]?
[FetpaDfp]?
electronic stateb
CASSCFNEVPT2CASSCFNEVPT2 CASSCF NEVPT2CASSCF NEVPT2
5E0
0.3
5011.4
6701.8
6702.1
0
0.001
80.6
80.6
170.7
171.7
250.4
300.8
389.7
389.8
0
15.1
7429.2
8759.5
8760.1
0
0.001
83.4
83.5
168.2
183.2
258.2
300.1
389.1
389.2
0
112.4
5103.4
6426.4
7365.8
0
0.054
72.1
76.2
124.4
236.2
277.2
323.6
406.9
407.8
0
118.2
7487.9
8373.3
9611.5
0
0.034
74.8
78.2
126.8
244.5
286.3
325.0
407.8
408.6
0
123.7
5102.7
6773.7
7209.3
0
0.056
70.4
74.8
120.1
243.1
281.7
327.9
410.2
411.2
0
139.8
7440.6
8836.4
9432.1
0
0.047
71.6
75.8
119.3
258.2
295.8
333.8
414.7
415.6
0
316.8
5386.2
5972.1
6502.2
0
0.605
40.1
52.2
67.2
380.9
396.6
433.4
491.0
493.4
0
374.8
7579.1
7791.3
8511.1
0
0.605
37.3
49.0
61.8
433.9
447.7
477.8
529.7
531.7
5A1
5E
A1,A2
E
E
A1
A2
E
aThe state of lowest energy has been taken as energy reference.bTerm notations are given for the D3holohedrized symmetry.
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ARTICLE
the set of active orbitals with ligand lone pairs does not alter the
energy level structure. Finally, the effect of the remote substit-
uents on the ground state spin sublevels emerges when compar-
ing the results for the truncated model complex (I) with those
including the entire ligand (exemplified in Figure 8 using
complex 2).
III.3. Magnetic Anisotropy. Compounds 1?4 have pre-
viously beenmagnetically characterized and were found, by both
direct (dc) and alternating current (ac) susceptibility measure-
ments, to be highly anisotropic.1,2Specifically, for 1 a large D
of ?48 cm?1and a much smaller yet non-negligible |E|max=
0.4 cm?1were deduced from a fit of the magnetization data
employing a SH of eq 13. Aside from the approximations
inherent in this particular choice of SH (see below), both (dc)
and (ac) data show the presence of an orthorhombic anisotropy
leading to an efficient tunneling pathway in all four complexes.
Since these ions are non-Kramers systems, this implies low-
symmetrysplittingofthe (Mspairs (describedbytheparameter
E) and deviations from the apparent crystallographic symmetry
inthe128KX-raystructure.Thisislargelyinagreementwiththe
subtle distortions found by DFT for this and all other complexes
investigatedhere(Table2).AdoptingtheDFTstructure andthe
ab initio results for this compound we plot in Figure 9 field-
dependent magnetizations obtained directly from the ab initio
calculations by using all SOC split sublevels of the5E ground
state and their wave functions. There is reasonable agreement
between computed and experimental data points with calcu-
lated values of the magnetization M being systematically
higherthantheexperimentalones.Aplotofthemagnetization
with a magnetic field oriented parallel (||) and perpendicular
(^) to the pseudo C3axis nicely illustrates the almost Ising-
type behavior of the anisotropy with a large and maximal M||
and almost zero M^ (Figure 9a). With a magnetic field
oriented parallel to the x or y axis (see Figure 2 for their
definitions)within the FeN3plane we further obtain Mx> My,
implying a substantial transversal anisotropy and a negative
value of the parameter E (see Figure 9b and eq S40, Support-
ing Information).
The geometric and electronic structures of the investigated
systems imply that a SH of the form of eq 14 is best suited for a
comparison between theoretical and simulated data and to
allowforanexplorationofmagneto-structuralcorrelations.The
parameters D, B40, and |E|, obtained from a best fit to
experimentally reported magnetizations and from the calcu-
lated lowest five SOC split sublevels, are presented in Table 5a.
The sign of E could not be fixed from these considerations.
However, based on calculations (e.g., Figure 9b) E is found to
benegativeinallcomplexesconsidered(seeabove).Wenotein
passing that the values of these parameters are not equally well
constrained by the two sets of data. While all three parameters
D, B40, and |E| are apart from the sign of E (see above)
accurately determined in the ab initio calculation they are
subjectto large error-bars whenadjusted tofitthe experimental
magnetization data. In particular, for complexes 1 and 2, which
according to the ab initio results possess the largest |D|, no
accurate fits could be achieved. For example, D is mainly
determined from the third, fourth, and fifth levels, which are
almost completely depopulated at the temperatures used in the
experiments. D gets smaller when going to complexes 3 and 4,
and so the error bars of the experimentally fitted D, B40, and E
(σ = 0.013 (4) to be compared with σ = 0.024 (1), 0.029 (2),
and 0.030 (3), see Table 5a). The more accurate best fit values
of D, B40, and E for complex 4 (see their error bars given in
Table 5a) compare very well with the theoretically predicted
ones (Table 5a). As illustrated in Figure 10, the low-field values
Figure 8. Effects of the spin states (S), number of roots (NR, i.e.,
number of nonrelativistic eingenvectors and eigenvalues used in con-
struction of the SOC matrix) for each spin, adopted cluster model (NA,
number of atoms), inclusion of solvent (COSMO), and space of active
orbitalsonthelowestthreeexcitedspinlevelsof[Fe(tpa)Mes]?withthe
geometry from reported X-ray data.
Figure 9. Theoretical (DFT-optimized geometry, NEVPT2) vs experimental field-dependent magnetizations for [FetpaTbu]?(experimental data
points are adopted from ref 2 and plotted using numerical data provided by the authors of this reference).
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Inorganic Chemistry
ARTICLE
of M (B = 1 T) are found to be mostly affected by B40(to be
compared with Figure S4 showing much less variation of M
(B=1T)withD,E,andgbutseethechangesofthehigh-fieldM
(B = 7 T) with D, B40, E, and g). Experimental12and simulated
(using best-fit parameters from Table 5a)M data for complexes
1?4aregiveninthe Supporting Information(Figures S5?S9).
In spite of the uncertainty in determination of D, B40for
complexes 1 and 2 and to a lesser extent for complex 3 as well,
there is good agreement between the ab initio and best fit to M
data sets (Table 5a). In particular, the trend that |D| decreases
alongtheseriesiswellreproduced.Onthebasisofthesetrendsit
was concluded in ref 2 that Drises with increasingligand basicity
tuned by the tpa substituents. A ligand field analysis (see section
III.4) lends support to this proposal.
The comparison between the CASSCF and the NEVPT2
results shows that dynamical correlation effects do not play a
leading role for the anisotropy; the values of D, B40, and E
obtained in the two treatments, the simple CASSCF and the
more sophisticated NEVPT2, are quite similar (Table 5a). Good
agreement between SA-CASSCF calculations and experiment
was found in a number of previous studies.13By contrast, as
pointed out above, the geometry that is adopted is of crucial
importance. For example, for 2 D changes significantly from
?29.1 cm?1, obtained with the X-ray geometry, to ?23.7 cm?1,
obtained with a DFT-optimized structure. Not unexpectedly,
values of D resulting from calculations restricted to truncated
modelclusters (Table5b) ofthetypeof(I, seeFigure 6a)donot
display significant variations, which is in disagreement with both
experiment and calculations on nontruncated models.
Importantly, it should be noted that there is an essential
contribution of the fourth-order parameter B40to U, the energy
barrierforthermalrelaxationofthemagnetization(eq29),which
is found to be
U ¼ ? 4D þ 60B40
ð29Þ
This value is positive for complexes 1?4 and increases the value
of U significantly by 24.6, 10.8, 9.0, and 2.0 cm?1, respectively
(NEVPT2,X-raygeometry).Thevaluescloselyfollowthetrends
in |D|. On the basis of on the comparison between the param-
eters D, B40(Table 5a), and the values of the5E ground state
Table5a. Theoretical andExperimental ValuesaoftheParameters oftheSpinHamiltonian D,B40,and|E|(in cm?1)oftpaComplexes withGeometries from X-rayDiffraction
Data and DFT Geometry Optimizationc
[FetpaTbu]?
[FetpaMes]?
[FetpaPh]?
[FetpaDfp]?
CASSCF
expb
NEVPT2
CASSCF
expb
NEVPT2
CASSCF
expb
NEVPT2
CASSCF
expb
NEVPT2
D
?35.89
?47.5(+7.5,?11.4)
?35.95[?29.71]
?28.36
?37.5(+5.0,?7.6)
?29.06[?23.67]
?27.54
?27.5(+1.6,?2.6)
?27.57 [?26.49]
?16.20
?10.3 (+0.27,?0.77)
?15.00[?15.05]
B40
0.45
0.30(+0.37,?0.57)
0.41[0.20]
0.18
0.30(0.25,?0.39)
0.18 [0.09]
0.17
0.46(+0.08,?0.13)
0.15[0.14]
0.04
0.00(+0.01,?0.03)
0.03[0.03]
|E|
0.00
0.18(+4.78)
0.00[0.40]
0.69
0.18(+4.27)
0.57[0.99]
0.72
0.08(+3.54)
0.69[0.75]
2.01
2.67(+0.30)
1.95[1.91]
aDeducedfromafittoexperimentalfield-dependentmagnetizationdata1?2witherrorbarslistedinparentheses(seeSupportingInformationfordetailsregardingthefitandsimulations).bNonlistedgvalues
and standard deviation (σ) between calculated and experimental magnetizations are [FetpaTbu]?g = 2.28((0.02), σ = 0.024; [FetpaMes]?g = 2.20(+0.03,?0.02), σ = 0.029; [FetpaPh]?g = 2.31(+0.025,
?0.02), σ = 0.030; [FetpaDfp]?g = 2.19(+0.01,?0.02), σ = 0.013.cNEVPT2 results pertaining to geometric structures from DFT geometry optimizations are given in square brackets.
Figure 10. Experimental magnetization (M, B = 1 T, open circles) for
complex 4 (adopted from ref 2 and plotted using numerical data
provided by the authors of the cited work) and its values calculated
using SH with D = ?10.30 cm?1, B40= 0.0 cm?1, E = 2.67 cm?1, and
g = 2.19 (solid line, see Supporting Information for details regarding the
fit and Table 5a for standard deviations and parameter error bars). The
variation of M with B40(broken lines) is illustrated.
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ARTICLE
splitting ?2δ (Table 4) it follows that the magnetic anisotropy
persists up to quite large values of δ. It stems from the
unquenched ground state orbital momentum |Lz| (Figure 10),
which is reduced only slowly with increasing δ but depends on
|Ms| (Figure 11).91However, with increasing δ there is a drastic
increase of the parameter E, which induces a tunneling splitting
ΔE of the Ms= (1 pair, ‘shortcutting’ the thermal relaxation
barrier in this series to effective values that are far smaller
than those predicted theoretically (eq 29). The interaction of
theMs=(2magneticpairwiththetopmostMs=0sublevelleads
to a similar splitting given by perturbation theory according
to eq 30.92It dominates the magnetic behavior at cryogenic
temperature and explains the lack of blocking of the magnetiza-
tion and hysteresis reported for all four systems.
3E2
?D þ 15B40
Thus,boththesecond (E,D)andfourth-orderterm(B40)can
contribute to increasing ΔE and thus lead to a reduction of the
anisotropy. Already small distortions δ lead to an increase of ΔE
and thus to a low-temperature loss of magnetization dominated
by quantum tunneling.
Inarecentpublication93itwasclaimedthat“thesplittingofthe5E
term cannot be described by a conventional zero-field splitting
Hamiltonian proving the irrelevance of the spin-Hamiltonian
ΔE ¼
ð30Þ
formalism for FeN4”. As far as all sublevels of the ground state
term
However, because of the large sensitivity of the5E level splitting
with respect to δ and because at the temperature of the
experiments only the lowest five levels are thermally populated,
we can still apply the SH of eq 14 in a slightly extended form
compared to the usual form that involves only D and E (eq 13).
Nevertheless, the SH of eq 14 is still fairly conventional.
III.4. Ligand Field Analysis and Magneto-Structural Corre-
lations. III.4.1. Ligand Field Analysis of the ab Initio Results. The
X-ray structure of 1 is trigonal and, due to the degenerate
irreducible representations in a perfect 3-fold symmetry, allows
for an unambiguous assignment of the electronic transitions.
Their values resulting from CASSCF and NEVPT2 calculations
are listed in Table 6a. Neglecting first metal?ligand π overlap
(eπ= 0, however, see below), ligand field matrix elements canbeen
5E are concerned this statement is certainly correct.
Figure 11. Expectation value of the orbital angular momentum opera-
tor within the5E state manifold as a function of the departure from C3
symmetry(quantifiedbyδdefinedbyone-halfthe5Esplitting)):broken
lines,in-state5ESOConlyincluded;solidlines,SOCcalculationwithall
210 S = 2, 1, and 0 states taken into account. The plot has been
constructedusingtheAOMXprogrampackage,85alongwithligandfield
parameters obtained from a fit to CASSCF results for (1, Table 6b)
allowing for a variation of δ.
Table5b. Spin-HamiltonianParametersObtainedUsingTruncated[FeN4C15H15]1?ModelComplexeswiththeSameGeometry
As the Corresponding Nontruncated Complex
[FetpaTbu]?
[FetpaMes]?
[FetpaPh]?
[FetpaDfp]?
exp. geometryDFT geometryexp. geom.DFT geometry exp. geometryDFT geometryexp. geometryDFT geometry
D
B40
|E|
?36.56
0.46
0
?31.96
0.28
0.17
?27.96
0.17
0.57
?31.53
0.27
0.26
?30.56
0.23
?0.38
?30.08
0.23
?0.35
?27.16
0.16
0.64
?30.34
0.24
0.32
Table 6. Energy Transitions (in cm?1) of [FetpaTbu]?from
CASSCFandNEVPT2calculationsinComparisonwithTheir
Values As Resulting from Ligand Field theory (a) Calculated
Using Best-Fit Ligand Field Parameters (in cm?1, b)a
(a) termCASSCFAILFTNEVPT2AILFT
5E(1) f5A1
5E(1) f5E(2)
5E(1) f3A2(1)
5E(1) f3E(1)
5E(1) f3E(2)
5E(1) f3E(3)
5E(1) f3A2(2)
5E(1) f3A1(1)
5E(1) f3E(4)
5011
6702
17698
20380
21390
21805
22693
22740
23896
5011
6702
17626
18994
21649
21903
22780
23184
24487
7429
8760
12929
17502
18509, 18512
19296, 19304
20305
20358
20763,20764
7429
8760
13083
14928
19154
19532
20348
20447
22258
5E(1)
A1A2f E(1)
A1A2f E(2)
A1A2f A1(2)
A1A2f A2(2)
A1A2f E(3)
80.6
171.0
250.4
300.8
389.8
75.4
163.9
236.5
309.9
391.5
83.4,83.5
168.2,183.2
258.2
300.1
389.1, 389.2
80.1
171.2
247.0
308.2
390.0
(b) parameterCASSCFNEVPT2Fe2+(free ion)
eσ
eσ
B
C
ζ
e
5768
1110
1785
3459
496
7540
2330
1213
3372
494
a
1058
3901
410
aParameters for the free ion are taken from Griffith, J. S. The Theory of
Transition-Metal Ions; University Press: Cambridge, 1971; p 437.
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Inorganic Chemistry
ARTICLE
expressed in terms of only two eσparameters describing the
interaction between Fe(II) and the axial amino (eσ
equatorial pyrrolyl (eσ
oped in section II.4 has been used to fit the ab initio data,
resultingincomputed (denoted by“AILFT”inTable 6a) energy
levels and best-fit parameter values listed in Table 6b. Only five
parameters afforded by LFT allow one to reproduce all the ab
initio numerical data with remarkably consistency.94
Onthebasisof acomparisonbetweentheparametersB, C,and
ζforthefreeFe(II)ionwiththebest-fitvaluestotheCASSCFand
NEVPT2, we conclude that the dynamical correlation introduced
by NEVPT2 leads to a significant improvement for B and C
compared to the values obtained from CASSCF. Even so,
NEVPT2 values for B, C, and ζ are still larger than those reported
for the free Fe2+ion (Table 6b). This is a feature of the wave
function contraction and the lack of electronic relaxation in
second-orderperturbationtheoryandhencecannotbecuredwith
the present methodology. As elaborated in the Supporting In-
formationusingempiricallycorrectedvaluesofBandC(affording
a reduction by 20%) does not affect D and B40. However, both
values get smaller by 30% when the same reduction of ζ is
performed (relativistic nephelauxetic effect). CASSCF and
NEVPT2 values of the parameters eσ
Table 6b) are in good agreement with the ones deduced from
highly resolved optical spectra of tetrahedral complexes of 3d
metals, while eσ
value of eσ
largelyreducesthedestabilizingantibondingeffectduetotheaxial
amine N ligand. Itfollows from a comparisonof theCASSCF and
NEVPT2 data that computed higher energy levels using the
second approach are artificially split, a drawback that cannot be
avoided while keeping to the perturbational approach.
Adopting the parameters B, C, and ζ obtained for complex (I)
and continuing to use the CASSCF method for the aforemen-
tionedreasons,welistinTable7thebest-fitvaluesofeσ
the whole series. Here, due to the low-symmetry distortions
smallbutpositivevaluesofeπs
3d orbitals and the out-of-plane π orbitals of tpa were deduced.
Focusing further on eσ
series 4 < 3 < 2 < 1. It is remarkable that this trend correctly
reproduces the Lewis basicity and nicely fits with the increase
of |D| in the same direction (Figure 12), as postulated in a
previous study.2
Following the same concept one should also expect that
increasing π-donor basicity will act in the opposite direction.
This is supported by the values of eπs
from 2 to 3 and 4 (Table 7). This behavior will be analyzed in
section III.4.2
a) and the
e) ligands. The three-step procedure devel-
e(5768 and 7540 cm?1,
aappears to be too low.95The anomalously low
ais due to the stabilizing effect of s?d mixing which
eandeσ
afor
eduetointeractionsbetweentheFe
e, it is found that it increases along the
ethat are found to increase
The set of ligand field parameters deduced from the ab initio
dataprovideatoolforthesystematicsearchofnewligandswhich
are better σ donors and thus expected to display enhanced
magnetic anisotropies. To this end, we would like to stress the
symbiosis between the theoretically rigorous ab initio approach
and the approximate but intuitive and chemically more readily
intelligible ligand field model.
III.4.2. Magneto-Structural Correlations. The parameters of
the SH are complex and sensitive functions of small distor-
tions of the geometry of [Fe(tpa)]?as discussed in sections
II.2.2 and III.3 and of the chemical nature of the tpaRligands
that is inturnfine tuned by the substituents R. The parametric
structure of the AOM is ideally suited to separate these two
effects and to study the effects of variations of the Lewis
basicity of tpaRinduced by the substituents R as revealed by
variations of the parameters eσ
of the analysis, we here adopt a FeN4geometry as given by the
X-ray structure of [Fe(tpaMes)]?, the most distorted complex
within the series. Taking the geometry as fixed we explore the
dependence of the parameters δ, D, B40, and E on eσ
Note that according to the sum rule, the trace of the ligand field
matrix (here the sum over all ligands, ∑ = 3(eσ
is approximately invariant for complexes that have related
coordination environments.96The ligand field analysis of
d?d spectra for a variety of 3d ions in the +II oxidation state
have shown that the trace ∑exp≈ 20000 cm?1.97?101This is
pleasingly confirmed for the present systems as a value of
∑AILF≈ 19135 ( 706 cm?1is obtained from the fitting of
the AOM parameters to the ab initio data (Table 7). Upon
inspection of Table 7 it is observed that not only ∑AILFbut
also the sum eσ
is (within (300 cm?1) constant (5760 cm?1) along the
series. Following this constraint, we plot in Figure 13a the
dependence of δ on eσ
Lewis basicity (i.e., an increase eσ
tion of δ.
This results (opposite to Figure 12 where purely electronic
effects interfere with geometric distortion effects) in a smooth δ
vs eσ
(Figure 13c) increase with eσ
the increasing Lewis basicity on the ground state magnetic
eand eπs
ein Table 7. For the sake
eand eπs
e.
e+ eπs
e) + eσ
a))
e+ eπs
epertaining to a given Fe?N(tpa) bond
e. It follows that an increase in the
e) is accompanied by a reduc-
edependence. As expected, both D (Figure 13b) and B40
e. We can subdivide the net effect of
Table 7. Angular Overall Model Parameters (in cm?1)
Deduced from a Fit of the Energies of the d?d Transitions
(CASSCF results)a
[FetpaTbu]?
[FetpaMes]?
[FetpaPh]?
[FetpaDfp]?
eσ
eσ
eπs
ψ1
ψ2
ψ3
e
5768
1110
5374
1588
570
8.6
10.9
3.8
4820
1941
1142
1.7
1.6
2.2
4246
2784
1119
6.8
3.2
3.3
a
e
11.9
11.9
11.9
aData are based on the geometry as given by the X-ray structures.
Figure 12. Correlation between the σ-donor ability of the tpaRligand
(described by the parameter eσ
parameter D; data are based on the geometry as given by the X-ray
structure.
e) and the axial zero-field splitting
Page 15
7474
dx.doi.org/10.1021/ic200196k |Inorg. Chem. 2011, 50, 7460–7477
Inorganic Chemistry
ARTICLE
anisotropy into electronic effects evolving within the5E electro-
nic ground state (in-state SOC) and contributions from config-
urational mixing via SOC with excited state multiplets. Model
calculations show (Figures 13a?c) that the bulk of the effect of
the Lewis basicity originates from configuration interaction
between the5E(1) ground state with the5A1and5E(2) excited
states. As follows from Figure 13a?c, inclusion of these excited
statesintotheSOCmatrixleadstodrasticreductioninδandthis
effect increases with eσ
increase in D and B40. By contrast, the extension of the SOC
matrix with the S = 1 and 0 ligand field excited states induces
smallerchanges inthese parameters. Asmalldecreasein?Dand
negligible change in B40are calculated. Resolving δ into separate
contributions from eπs
e. This is accompanied by a corresponding
eand eσ
eis possible by choosing the first set
of values of eσ
(Figure 13d, left) and the last one - with eσ
vanishing eπs
Focusingonthefirstsetofparameters,afirst-ordersplitting2δ
(Figure 13d, left, a diagonal effect described by the parameter
δ1in eqs 9 and 10) of5E is observed. This is induced by weak
dxz, yz?ligandinteractionofπtype.Becausethepyrrolideringsin
these complexes are nearly (but not completely) perpendicular
to the FeN3plane (cf. eq 27 and the values of ψ, Table 7) this
effect is small. Allowing further for off-diagonal mixing between
the
(parametrized by the parameter δ2in eqs 9 and 10) leads to a
further increase in 2δ (Figure 13d, left, middle), an effect
dominated by eσ
eand eπs
e; eσ
e= 4000 cm?1and eπs
e= 1760 cm?1
e=5750 cm?1and
e(Figure 13d, right).
5A0(xz) and
5A00(yz) sublevels of the
5E ground state
e. Finally, extending the SOC matrix with excited
Figure 13. Dependence of δ (a), D (b), and B40(c) on eσ
excited states; (d) contributions to δ for two limiting cases of values for eσ
ewith and without configuration interaction between the5E ground term with ligand field
eand eπs
e: large eπs
e(left) and vanishing eπs
e(right).
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