Magnetic Functions Beyond the Spin-Hamiltonian

264 Pages·2006·11.038 MB·English

by  Roman Boča (auth.), D. M. P. Mingos (eds.)

StructBond(2006)117:1–264
DOI10.1007/b136907
© Springer-VerlagBerlinHeidelberg2005
Publishedonline:3December2005
MagneticParametersandMagneticFunctions
inMononuclearComplexesBeyond
theSpin-HamiltonianFormalism
RomanBoˇca
DepartmentofInorganicChemistry,SlovakTechnicalUniversity,SK-81237Bratislava,
Slovakia
[email protected]
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 MagneticParameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 MagneticFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 EnergyLevelsofMultitermSystems. . . . . . . . . . . . . . . . . . . . . . 20
2.1 PathsforEvaluatingEnergyLevels . . . . . . . . . . . . . . . . . . . . . . . 20
2.2 MatrixElements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 ZeemanLevels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3 ModelingtheMagneticParameters . . . . . . . . . . . . . . . . . . . . . . 41
3.1 Computer-AidedCalculationsofSpin-HamiltonianParameters . . . . . . . 41
3.2 PartitioningTechnique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3 ManualCalculationsofSpinHamiltonianParameters . . . . . . . . . . . . 46
3.4 MagnetochemicalFormulae . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4 CalculationsofEnergyLevelsandMagneticParameters . . . . . . . . . . 62
4.1 d1-OctahedralReference . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.2 d2-OctahedralReference . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.3 d3-OctahedralReference . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.4 d4-OctahedralReference . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.5 d5-OctahedralReference . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
4.6 d6-OctahedralReference . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
4.7 d7-OctahedralReference . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
4.8 d8-OctahedralReference . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
4.9 d9-OctahedralReference . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
5 EmpiricalMagneticParameters . . . . . . . . . . . . . . . . . . . . . . . . 181
5.1 Jahn–TellerEffect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
5.2 Jahn–TellerEffectinCrystal-FieldModel . . . . . . . . . . . . . . . . . . . 186
5.3 PrincipalMagneticParameters . . . . . . . . . . . . . . . . . . . . . . . . . 193
5.4 OrbitalReductionFactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
A SpectroscopicConstants,Coefficients,andMatrixElements . . . . . . . . 206
B IrreducibleTensorsandTensorOperators . . . . . . . . . . . . . . . . . . 220
2 R.Boˇca
C ClassificationofCrystal-FieldTermsandMultiplets. . . . . . . . . . . . . 233
D CalculatedEnergyLevelsandMagneticParameters . . . . . . . . . . . . . 244
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
Abstract Usingthespin-Hamiltonianformalismthemagneticparametersareintroduced
through the components of the Λ-tensor involving only the matrix elements of the an-
gularmomentumoperator.Theenergylevelsforavarietyofspinsaregenerated,andthe
modelingofthemagnetization,themagneticsusceptibility,andtheheatcapacityisdone.
Theoretical formulae necessary for performing the energy level calculations for a mul-
titerm system are prepared with the help of the irreducible tensor operator approach.
Thegoaloftheprogrammingistoevaluatetheentirerelevantmatrixelement(electron
repulsion, crystalfield, spin–orbit interaction, orbital-Zeeman,andspin-Zeeman opera-
tors)inthebasissetoffree-atomterms.Themodelingofthezero-fieldsplittingisdone
at three levels of sophistication: (i) the differences in the crystal-field multiplets when
the high-dimensional matrix in the complete space is diagonalized; (ii) the differences
in the energy levelsof a model subspace when the partitioning technique is applied in
the first iteration; (iii) the differences in the energy levels as they are produced by the
second-order perturbation theory for the spin Hamiltonian. The spin-Hamiltonian for-
malismofferssimpleformulaeforthemagneticparameters(theg-factors,D-parameter,
χTIP)byevaluatingthematrixelementsoftheangularmomentumoperatorinthebasis
set of thecrystal-fieldterms. Themagneticfunctionsfor dn complexesaremodeledfor
awiderangeofcrystal-fieldstrengths.
Keywords Magnetism·Magneticparameters·Magneticsusceptibility·
Multitermsystems·Transitionmetalcomplexes
Abbreviations
AT atomicterm
CI configurationinteraction
CF crystalfield
CFM crystal-fieldmultiplet
CFP coefficientoffractionalparentage
CFT crystal-fieldterm
CSC completespacecalculation
IR irreduciblerepresentation
ITO irreducibletensoroperator
JT Jahn–Teller
MAM magneticangularmomentum
MO-LCAO molecularorbitals–linearcombinationofatomicorbitals
MP magneticparameter
SH spinHamiltonian
TIP temperature-independentparamagnetism
ZFS zero-fieldsplitting
MagneticParametersandMagneticFunctions 3
1
Introduction
1.1
Motivation
Magnetochemical/physical experiments are done at thermal equilibrium: at
sufficiently stabilized temperatures the thermodynamic response functions
are monitored, i.e., the temperature dependence of the magnetic suscepti-
bility (or magnetization with the SQUID apparatus) at a constant magnetic
field, the field dependence of the magnetization at constant temperatures,
and eventually the temperature dependence of the heat capacity at a con-
stant(orzero)magneticfield.Theaboveresponsefunctionsreflectthemag-
netic energy levels (the Zeeman levels) whose separation is on the order
of a fraction to several reciprocal centimeters. Complementary information
couldbeobtainedfromspectroscopicmethods:electronspinresonance,far-
infrared spectroscopy or its modern version called the frequency domain
spectroscopy,magneticcirculardischroism,andinelasticneutronscattering.
Eachofthesemethods,however, hasitsownlimitationssothatthebestway
istocombinetheminordertohaveasmuchinformationaspossible.
The monographs on magnetism incorporate theoretical aspects to a dif-
ferent extent and at a different level of complexity (from almost nothing to
a very broad presentation) [1–24]. However, most of the initial progress in
this area was made in electron spin resonance (ESR) [25–33]. In treating
zero-fieldsplitting(ZFS)systems,themagnetochemicalandESRsourcesuti-
lizedextensivelytheoperatorequivalentapproachalongwiththecrystalfield
(CF) of lowered symmetry. Suchan approach, however, is limited to ground
electrontermsobeyingHund’srules.TheZFSparametersareconsideredpa-
rametersthatarefixedbyfittingtheexperimentaldatatoeachsystemunder
investigation. To date there has been no rational approach that could corre-
lateZFSparameterstosomeother,morefundamentaldata(integrals).Sofar,
tailoring and tuning of ZFS parameters has only been an unrealized dream.
What wouldfacilitaterealizationofthisdreamisadeep theoreticalanalysis.
This is the main goal of the present paper. However, a sufficient descrip-
tion of the situation requires much more effort, and we will see later that
there is a need for combining information from the electronic structure of
atoms [34–42], crystal/ligandfieldtheory [43–51], fundamentals ofangular
momentum [52–55], and, finally, the irreducible tensor operator (ITO) ap-
proach [56–63] as a real working tool. Recent monographs utilize the ITO
approachasacommonstandard[22,23,32,49].
The next paragraph introduces briefly the set of magnetic parameters
(MPs) that determine the spacing of the lowest energy levels in metal com-
plexes. Then the magnetic response functions can be reconstructed using
statisticalthermodynamics.
4 R.Boˇca
Section 2 prepares all the theoretical formulae necessary for performing
energy level calculations for a multiterm system. The spectral parameters
enteringtheformulaearetabulatedinAppendixA.Thegoaloftheprogram-
ming is to evaluate the entire relevant matrix element (electron repulsion,
CF, spin–orbit interaction, orbital-Zeeman, and spin-Zeeman operators) in
the basis set offree-atom terms. The transition to crystal-fieldterms (CFTs)
is considered a redundant operation that can be omitted as such a process
represents a unitary transformation that leaves the eigenvalues invariant.
In this way a difficult manipulation with the point-group coupling coeffi-
cients(3Γ-,6Γ-,9Γ-,orV-,W-,andX-coefficients)isavoidedentirely.This
partuses theITOapproachextensively. Sincethislanguageisavailable from
alimitednumber ofsources,acomprehensive reviewisofferedtoreadersin
AppendixB.
ThemodelingoftheZFS(Sect.3)isdoneatthreelevelsofcomplexity:
(a)The differences in the crystal-field multiplets (CFMs) when the high-
dimensionalmatrixinthecompletespaceisdiagonalized;
(b)The differences in the energy levels of a modelsubspace when the parti-
tioningtechniqueisappliedinthefirstiteration;
(c)The differences in the energy levels as they are produced by the second-
orderperturbationtheoryforthespinHamiltonian(SH).
The labeling oftheCFMsisbased onthedouble-groupconcept,whichisfar
from common. Therefore, the most relevant items of the double groups are
collected in Appendix C. The SH formalism offers simple formulae for the
MPs (g-factors, D-parameter, χ ) by evaluating the matrix elements of the
TIP
angular momentum operator in the basis set of the CFTs. The values of the
temperature-independent paramagnetism(TIP)areabyproductofthetheor-
eticalmodeling.
In Sect.4 detailed calculations are performed for d1 to d9 systems in the
geometry of a regular octahedron and a compressed/elongated tetragonal
bipyramid. Calculated energy levels and MPs are presented in Appendix D
alongwiththeenergyleveldiagrams.
Section 5 introduces the Jahn–Teller (JT) effect and its treatment within
theCFmodel.Briefinsightintotheelectronic-structureparametersisgiven.
Notationsused:
(a)SI units are used consistently throughout: χ [SI] = 4π×106χ
mol mol
[cgs&emu].
(b)EnergyquantitiesE(likeε,J,D,E,a,F,etc.)arepresentedascorrespond-
ingwavenumbers, i.e.,E/hc,andgiveninunitsofcm–1.
(c)The angular momentum operators translate the reduced Planck constant
(cid:1)
duringoperationintoacorrespondingwavefunction(aket).
(d)The fundamental physical constants (c, ε , µ , N , k=k , R, µ , e, h,
0 0 A B B
(cid:1)
) adopt their usual meaning—they enter the reduced Curie constant
C =N µ µ2/k.
0 A 0 B
MagneticParametersandMagneticFunctions 5
1.2
MagneticParameters
Transition-metal(dn)complexeswithopenshellsbelongtotheclassofpara-
magneticmaterials:theirmagneticsusceptibilityispositive(thesampleisat-
tractedtothemagneticfield)andistemperature dependent. Athighenough
temperatures andinsmallfields,themolarmagneticsusceptibility normally
obeystheCurielaw
χ =C/T. (1)
mol
The Curie constant C can be considered a magnetic parameter (MP) asso-
ciated with the sample. Theory, however, tells us that such a phenomeno-
logical parameter could be made of fundamental physical constants and the
magnetogyric-ratioparameterg inthefollowingway:
(cid:1) (cid:2)
C= N µ µ2/k g2S(S+1)/3=C g2S(S+1)/3, (2)
A 0 B 0
where S is the spin of a Curie paramagnet. Thus we arrive at another MP—
the g-factor—and a reduced Curie constant C . The hyperbolic decay of the
0
susceptibility with increasing temperature isbetter followedwhen the prod-
uct function P=χ T/C (dimensionless) is plotted as a function of tem-
mol 0
perature, and the perfect fulfillment of the Curie law is now represented by
a straightline P vs. T withazeroslope. Inan olderrepresentation, the tem-
perature dependence oftheeffective magneticmoment,µ giveninunitsof
eff
Bohrmagneton,isfollowedwhere
(cid:3)
(cid:1) (cid:2) √
µ /µ = 3k/ N µ µ2 χ T= 3P. (3)
eff B A 0 B mol
The Curie paramagnets [e.g., octahedral Fe(III) complexes] are rather rare,
andthetemperaturedependenceofthemagneticsusceptibilityrequiresmore
parameters, depending on the actual spacing of the low-lying energy levels.
LetusenumeratetheMPsassociatedwiththeSHformalism:
(a)g-tensorcomponents,g(cid:2) andg⊥,orgx,gy,gz;
(b)AxialZFSparameterDandrhombicZFSparameterE;
(c)TIPχ ;
TIP
(d)Eventually,biquadraticspin–spininteractionparametersaandF.
Under the SH formalism it is understood that the energy levels are recon-
structed by considering only the formal spin kets |S,M (cid:4) that are under the
S
action of the SH involving the spin–spin interaction and the spin-Zeeman
term
(cid:5) (cid:6) (cid:7)(cid:8)
(cid:1) (cid:2)
H(cid:4)S=(cid:1)–2 D (cid:4)S2–(cid:4)S2/3 +E (cid:4)S2–(cid:4)S2 +(cid:1)–1µ Bg (cid:4)S (4)
a z x y B a a
(a=x,y,z). Such a Hamiltonian is appropriate for ZFS systems. (When the
spin–spin interaction term vanishes, the SH collapses to the pure spin-
6 R.Boˇca
Zeeman term appropriate to the Curie paramagnet.) The MPs g , g , g , D,
x y z
andEareunderstoodasconstantsthatcharacterizethesampleunderstudy.
The Hamiltonian that describes the interaction of the single magnetic
center with the external magnetic field involves the spin-Zeeman term, the
orbitalZeemanterm,andtheoperatorofthespin–orbitcoupling,i.e.,
H(cid:4)(cid:5)=(cid:1)–1µ g (S(cid:6)·B(cid:6))+(cid:1)–1µ (L(cid:6)·B(cid:6))+(cid:1)–2λ(L(cid:6)·S(cid:6)). (5)
B e B
The aboveHamiltonianactsasaperturbationoperator,sothatperturbation
theoryyieldsthefirst-orderandsecond-ordercorrections
(cid:9) (cid:10) (cid:10) (cid:11)
H(cid:4)(1)= 0(cid:10)H(cid:4)(cid:5)(cid:10)0 , (6)
(cid:9) (cid:10) (cid:10) (cid:11)(cid:9) (cid:10) (cid:10) (cid:11)
(cid:12) (cid:10)(cid:4)(cid:5)(cid:10) (cid:10)(cid:4)(cid:5)(cid:10)
0 H K K H 0
H(cid:4)(2)=– . (7)
E –E
K(cid:7)=0 K 0
It is assumed that the state vectors are represented by orthonormal kets
of spatial variables, viz., |K(cid:4)=|α,L,M (cid:4). The first-order correction can be
L
rewrittenasfollows:
(cid:13) (cid:10) (cid:10) (cid:14)
(cid:10) (cid:10)
H(cid:4)(1)=(cid:1)–1µ g (B(cid:6)·S(cid:6))(cid:8)0|0(cid:4)+((cid:1)–1µ B(cid:6)+(cid:1)–2λS(cid:6))· 0(cid:10)L(cid:6)(cid:10)0 , (8)
B e B
wherethelasttermvanishesintheabsenceofthefirst-orderangularmomen-
tum,andthuswearriveatthepurespin-Zeemaninteraction:
H(cid:4)(1)=(cid:1)–1µ g (B(cid:6)·S(cid:6)). (9)
B e
Thesecond-ordercorrectioncontainstheterms
(cid:9) (cid:10) (cid:10) (cid:11) (cid:13) (cid:10)(cid:10) (cid:10)(cid:10) (cid:14)
0(cid:10)H(cid:4)(cid:5)(cid:10)K =(cid:1)–1(µ B(cid:6)+(cid:1)–1λS(cid:6))· 0(cid:10)L(cid:6)(cid:10)K +(cid:1)–1µ g B(cid:6)·S(cid:6)(cid:8)0|K(cid:4) , (10)
B B e
where the last contribution vanishes owing to the orthogonalityof the state
vectors.Thecompletesecond-ordercorrectionis
(cid:5) (cid:13) (cid:10) (cid:10) (cid:14)(cid:8)(cid:5) (cid:13) (cid:10) (cid:10) (cid:14)(cid:8)
(cid:10) (cid:10) (cid:10) (cid:10)
(cid:12) (µ B(cid:6)+(cid:1)–1λS(cid:6))· 0(cid:10)L(cid:6)(cid:10)K (µ B(cid:6)+(cid:1)–1λS(cid:6))· K(cid:10)L(cid:6)(cid:10)0
B B
H(cid:4)(2)=–(cid:1)–2 ,
E –E
K(cid:7)=0 K 0
(11)
andafterintroducingtheΛ-tensor
(cid:9) (cid:10) (cid:10) (cid:11)(cid:9) (cid:10) (cid:10) (cid:11)
(cid:12) (cid:10)(cid:4) (cid:10) (cid:10)(cid:4) (cid:10)
0 L K K L 0
Λ =–(cid:1)–2 a b , [energy–1], (12)
ab
E –E
K(cid:7)=0 K 0
itcanberewrittenintheform
H(cid:4)(2)=(µ B(cid:6)+(cid:1)–1λS(cid:6))·Λ·(µ B(cid:6)+(cid:1)–1λS(cid:6)). (13)
B B
MagneticParametersandMagneticFunctions 7
(OneshouldbecarefulinthedefinitionofthesignoftheΛ-tensorsinceone
encounterstheoppositesignintheliterature.)Theoverallresultofperturba-
tiontheoryuptothesecondordercanbewrittenasfollows:
H(cid:4)S=– 1(B(cid:6)·κ·B(cid:6))+(cid:1)–1µ (B(cid:6)·g·S(cid:6))– 1(cid:1)–2(S(cid:6)·∆·S(cid:6)), (14)
B
2 2
whereweintroducetheκ-tensor (reduced, temperature-independent param-
agneticsusceptibilitytensor)
1
– κpara=µ2Λ , [energy×induction–2], (15)
2 ab B ab
theg-tensor(magnetogyricratiotensor)
g =g δ +2λΛ , [dimensionless], (16)
ab e ab ab
andtheD-tensor(spin–spin interactiontensor)
– 1∆ =λ2Λ =D(cid:5) , [energy]. (17)
2 ab ab ab
(Thenumericalprefactors,like–1/2,areamatterofconvention.)
The SHH(cid:4)S actsonlyonthespin kets |S,M (cid:4)yielding eigenvalues that are
S
(cid:4)(cid:5)
identical withthoseproduced bythe perturbationoperator H acting onthe
fullsetofspin–orbit variables|α,L,M ,S,M (cid:4).Thissituationisexplained in
L S
Table1:thetruncatedSHmatrixinvolvesintegralsovertheangularmomen-
tumviatheperturbationtheory.
On concluding, MPsare comprised of three Cartesian tensors, each com-
posed of the Λ-tensor and the spin–orbit splitting parameter λ. In this way,
just the Λ-tensor is responsible for the shift of the g-components relative to
thefree-electronvalueg ,fortheappearanceoftheZFStensor∆,andforthe
e
existenceoftheTIP.
Table1 ExplanationofthemagneticHamiltonianandthespinHamiltonian
8 R.Boˇca
The temperature-independent paramagnetic term is omitted hereafter
(this can be included in the empirical correction of the experimental data,
togetherwiththediamagneticterm)sothatwearriveattheZFSHamiltonian:
(cid:5)
H(cid:4)zfs=(cid:1)–2(S(cid:6)·D ·S(cid:6))+(cid:1)–1µ (B(cid:6)·g¯¯·S(cid:6)). (18)
B
As far as the D-tensor is concerned, it contains nine Cartesian components.
Assumingthatthecoordinateaxesareidenticalwiththeprincipalaxesofthe
D-tensor, onlythediagonalelements contribute. Byintroducingnewparam-
eters
D=(–D(cid:5) –D(cid:5) +2D(cid:5) )/2, (19)
xx yy zz
E=(D(cid:5) –D(cid:5) )/2, (20)
xx yy
J=(D(cid:5) +D(cid:5) +D(cid:5) )/3, (21)
xx yy zz
theequivalentformoftheZFSHamiltonianis
H(cid:4)zfs=(cid:1)–2[D((cid:4)S2–(cid:4)S2/3)+E((cid:4)S2 –(cid:4)S2)+J(cid:4)S2]+(cid:1)–1µ (B(cid:6)·g¯¯·S(cid:6)). (22)
z x y B
The constant term J(cid:4)S2 can be removed from further consideration since it
uniformly shifts all the energy levels. The omission of the constant term, in
fact,correspondstothesubtractionononethirdofthetraceoftheD-tensor
fromthediagonalelements,andinthiswaythe(diagonal)D-tensorbecomes
traceless.NotethattheDandEparametersremainunchangedwhenthesame
constant term is added (subtracted) to (from) the diagonal elements of the
D-tensor.NormallyitisassumedthattheZFSparametersobeyarelationship
|D|≥3E≥0. (23)
One caninterchange the Cartesian axes (which cannot influence the proper-
tiesofthesystem),buttheaboverelationshipwillstillhold.
Theinvolvementoftheangularmomentumthroughtheperturbationthe-
ory is not permitted when it is contained in the ground-stateket. The triply
degenerateelectronterms2S+1Tbelongtosuchacaseofthemagneticangular
momentum(MAM).ThentheinteractionHamiltonianinvolves(1)thespin–
orbitinteraction,(2)theorbitalZeemanterm,(3)thespin-Zeemanterm,and
(4)eventuallythesymmetry-loweringterm
H(cid:4)=(cid:1)–2λsf(γsfAκL(cid:6) ·S(cid:6))+(cid:1)–1µ B(cid:6)·(γsfAκL(cid:6) +g S(cid:6))
p B p e
+∆ [(cid:1)–2(cid:4)L2–L(L+1)/3] (24)
ax z
(whereL=1andγsf=–1shouldbeapplied).TheZeemantermcanbetaken
asaperturbation,whereastheothertermsneedtobeincludedintheunper-
(cid:4)
turbedHamiltonianH .ThisapproachhasbeenoutlinedbyKotani[64]and
0
thengraduallyextendedbyGriffith[44],Figgis[45],andFiggisetal.[65–68].
In this case the only MPs are the spin–orbit splitting parameter λ, the or-
bitalreductionfactorκ≤1,theFiggisCI-mixingparameter1≤A≤3/2,and
MagneticParametersandMagneticFunctions 9
theaxialsplittingparameter∆ (whichispositivewhentheA-term,arising
ax
fromthesplittingoftheT-term,isthegroundstate).
Wehaveseenthatmononuclearcomplexeswithorwithoutthefirst-order
angularmomentumcanbetheoreticallytreatedinadifferentdegreeofcom-
plexity. Thus we can speak of some magnetotheoretical hierarchy (Table2).
Dependingonthebasicpostulateabouttheextentoftheactivespace(aspace
of kets included in the zero-order Hamiltonian) various levels of the theory
canbedistinguished.
In the most complete treatment of this approach (such as the theory of
König and Kremer [69–71]), the whole spectrum of the electron energy
Table2 Magneto-theoreticalhierarchy
Level Description Activespace Freeparameters Magnetic
parameters
7 Notavailable Alltermsoffkdnsmpl ξ ,ξ,...
d f
6a K¨onig& Alltermsofdn, B,C, ∆g=0
Kremer kets|vSLJΓγa(cid:4) Dq(Dt,Ds) κpara=0
[62,69–71] orF4 (F2,...), D=0
ξ ,κ
d
6b Schilder& Alltermsofdnorfn,
Lueken[78] kets—atomic
microstates
6c [Thiswork] Alltermsofdn,
kets|v,L,ML,S,MS(cid:4)
5a Figgisetal. LimitedCI, λ,κ,A,∆ax ∆g=0
[65–68] ketsoflowered (orv=∆ax/λ) κpara(cid:7)=0
symmetry
5b Weissbluth LimitedCIforspin
[59] admixedstates
kets|Γ,γ,S,MS(cid:4)
4 Figgis[45] LimitedCIfor λ,κ,A ∆g=0
2S+1T1 terms κpara(cid:7)=0
3 Kotani[64], Groundterm2S+1T, λ,κ ∆g=0
Griffith[44] ket(cid:10)s|L=(cid:11)1,ML,S,MS(cid:4) κpara(cid:7)=0
(cid:10)
or J,MJ
2 Zero-field Groundterm2S+1A, gaa,D,E,(a,F), ∆g(cid:7)=0
splitting kets|S,MS(cid:4) χTIP,(orΛaa) κpara(cid:7)=0
D(cid:7)=0
1 Effectivespin LowestK(cid:10)ramers(cid:11) gaefaf ∆g(cid:7)=0
doublet(cid:10)(cid:15)S,±1/2 κpara(cid:7)=0
10 R.Boˇca
levels is determined by the electronic-structure theory: the Racah parame-
ters B and C, the CF parameters F (L) and F (L) for each ligand L (or the
4 2
conventional Dq, Ds, Dt, ... parameters), and the effective spin–orbit coup-
ling constant ξ ; eventually the orbitalreduction factorκ canbeconsidered.
d
There is almost nothing to be included from the perturbation theory, ex-
cept the Zeeman term. Therefore, the differential g-tensor, the temperature-
independent paramagnetic tensor, and the spin–spin interaction tensor are
blank:∆g=g–g 1=0,κpara=0,D=0.TheZFSparameterD(E,a)isdirectly
e
read off as the splitting of the lowest CFMs, and the g-factors are deter-
mined bytheevolutionoftheZeeman levelsinthemagneticfield.Thereare
nolongeranyphenomenologicalMPsasthemagnetism iscompletelydeter-
minedbytheelectronicstructureandthemagneticfieldstrength(induction).
By contrast, when the active space is restricted to the spin-only kets, the
influence of all attainable excited states manifests itself in the filling of the
MPs (tensors). In such a case the g-tensor deviates considerably from the
free-electronvalue,theTIPappearssubstantial,andthespin–spininteraction
tensortransformstohighvaluesoftheZFSparameters(DandE).
1.3
MagneticFunctions
Having determined the magnetic energy levels ε (B) (as eigenvalues of the
a,i
interactionHamiltonian)wecanproceedwiththeapparatusofthestatistical
thermodynamicbydefiningthe(magnetic)partitionfunction
(cid:12)N
Z (T,B)= exp[–ε (B)/kT] (25)
a a,i
i=1
andthefree(Helmholtz)energyperparticle
F(T,M)=–kTlnZ. (26)
Thentheobservablethermodynamicfunctionsaredeterminedasfollows:
1. Molarmagnetization
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
∂F ∂lnZ 1 ∂Z
M =–N =–N kT =–RT ; (27)
mol A ∂B A ∂B Z ∂B
T T T
2. Molarisothermalmagneticsusceptibility
(cid:16) (cid:17) (cid:16) (cid:17)
∂M ∂2F
χ =µ =–µ ; (28)
T 0 ∂B 0 ∂B2
T T
3. Molarisofieldheatcapacity(E–ethalpy,Q–heat,S–entropy)
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
∂Q ∂E(S,H) ∂S
C = = =T , (29)
H ∂T ∂T ∂T
H H H