# 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.

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- Nature Chemistry 06/2013; 5(7):556-557. · 21.76 Impact Factor
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**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 - 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

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Published:July 11, 2011

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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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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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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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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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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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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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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Inorganic Chemistry

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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Inorganic Chemistry

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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