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The application of the angular overlap model in the calculation of paramagnetic principal susceptibilities for fn-electron systems
Volume 45, number 3
CHEMICAL PHYSICS LETTERS
15
MarlAt 1977
THE APPLlCATlON OF THE ANGULAR OVERLAP MODEL IN THE CALCULATION OF PARAMAGNETIC PRINCIPAL SUSCEPTIBILITIES FOR f”-ELECTRON SYSTEMS W. URLAND Instltut fir anorganlsche und analytiscire Chemie.63 Giessen, West Gtwnan>p Rcccivcd 24 November 1976
A description of the calculation of the paramagnctrc principal suweptlbdltles hgand-fields i$ gvcn.
for f”-clectron systems with arbitrary
1. Introduction There arc many publications on the interpretation of paramagnctic susceptibilities of lanthanide and acrinide ions in various ligand fields, e.g. rcfs. [ l-1 0] . The models for the calculation of the susceptibilities differ mainly in the way how the influence of the Iigand field has been considered. In most cases the electrostatic concept has been used. It has been shown by several authors however that the angular overlap model (AOM) is more suitable than the electrostatic model to interpret ligand fields in f”-systems [ 11,121. ‘I’hc purpose of this letter is to describe the calculation of the paramagnetic principal susccpttbilities for fn-systerns in any kind of ligand fields by applying the ligand field potential in the AOM which has been derived in a previous publication [ 131.
2. The model 2.1. ?YZze complete
ligated field problem
The hamiltonian appropriate electronic repulsion, spin-orbit
for the total &and field calculation consists of the operators of Coulomb intcrcoupling and the ligand field potential in the AOM
This and the following equations follow generally the nomenclature and conventions of Brink and Satchlcr [14] _ The eigcnvalues and eigcnfunctions for the computation of the susceptibility tensor are obtained by diagonaliration of a given IJ, MJ) basis set under the hamiltonian (1). The Coulomb interaction perturbation-matrix eIcments can be written as
where the matrix elements (OLCSIXi>j e2/r&a’LS) are tabulated as Z,k +Yk are linear combinations of Slatcr Fk integrals also given in ref. [IS] .
in Nielson and Kostcr [ Is]_ The E”‘s
457
CffEMiCAL PHYSICSLfXfXRS
Volume 46, number 3
15 Mrtrch 1977
The spin-orbit coupling matrix are evaluated appiying the equation
t
f&~)-J-s-L’
841’26 (J , J’)6 (MJ, M;)
GldSII v~%wS’~.
(3)
The reduced matrix elements of this uperatar are listed in ref. f 15Jf The expressron for the Iigand field potential is:
The evaluation of the Rkckq-coeffcients The &and field matrtx requires now
has been described extensively in ref. [I3]. the calculation of the matrix elements: (~LWIMJIYkQla’t’.V‘M’)given by
the relation
The reduced matrix eiements of the unit tensor Ufk) arc tabulated in ref. [15]. ~at~~cs for the eIectron Cou101nbinteraction operator (three for f orbitals~, one spin-orbit matrix, and one matrix for each sphcricaf harmonic Yz or coefficient Rkckq are constructed for tfie (LUM’) basis set and stored. To establish the complete perturbation matrix for the opcmtor (1) WCproceed in the following way: First the coefficients R~CQ arc computed, introducing the angular overlap parameters eA as described in ref. [ 13). Tbcn cop~cs of the stored matrices are mu!tiplied by appropriate Ek, f ti , and kkckqparameters and finally summed. The diagonalization of the contpletc matrix gives eigenvalues and eigenvcctors for the interpretation of optical spectra and for the c~m?utation of the susceptibility.
The paramagnetic susceptibility is a second-rank tensor quantity expressing the relation between ~agnc~~~a~ion and a r~a~~etIc field: A& = xopHp *
(61
Here we USCthe standard tensor notation whcrc Greek letters as non-repeating suffices refer to the components of a set (x,y, z), but whcrc Greek letters as repeating suffices imply summation over the whole set. The defining cquatlon for magnetization in direction cr, for a system of identical molecules is given by: (7)
(8s)
458
Volume46, numbcr 3
CiIEMiCALPHYSICSLETTERS
with iu, = (I&, f 2S&30 (PO= Bohr magneton, k = Stevens’ orbital reduction factor). Differentiating (8) with respect to components of the magnetic field leads to:
~~~~a~*={~~~~~j~~ CI~il~oli,rjl~~lo+{ii~~li,{jl~~in)H,I~~~ i Following van Vleck’s treatment
-l$).
[I61 , considering only small magnetic fields, we expanded exp(-E#T)
exp(-E’,lkO = e~p(-E~~~H-)(1.- Gllr,liMfpjkT)
6) as: 001
and obtain after substitution and application of cq. (6) for the ~o~~unents of the susceptibility tensor:
Relation (11) reduces to van Vie&s expression when applied to a diagonal clement. If we consider that the trace of a matrix is invariant under a unitary transformation we can easily show that the susceptibility tensor is symmetric and valid for rilt kind of basis sets not only for those diagonal in Ha. The relations for the matrix elements of the operator fl* are given by GXLSJMJI (Id,
X
!
+ zs&?,
1
3
-4fJ
.J’
M.SJ’MJ>
= (- pQ(2J+
I[
0 i%fJ
(-t)L*S+J’+g k fL(L
i
)‘925
t 1) (2L + l)]
f 1)1’2
It2
J
i
1
LSL
3
1
3’
X
--II+ 1 M; )I[
(4
J
I
+‘+“+’ k [L(L t 1) (211f i)] “’
&LS.M~I (kL, + 2S,,)& IoUJ’M;>= (-$)*/2i(-l)‘-hzJ(2i
+‘+“+’
X
J’
* l)“‘(W
k [La
+ ljl”
f l)(ZL + I)] If2
-Mj $
(_~)LeS+J+l
a [S&s + 1) (2s + 1)) It2
(14)
After the computation of all six independent xnp components in a molecule-fured frame we then diagondlize the 453
Volume 46. number 3
CHBKlCAL PHYSICS LETTERS
susceptibility tensor to obtain the principal susceptibilities xu, x,,,,, xzz and their orientations 7%~mean susceptibility is calculated to:
15 March 1977 in
the molecule.
3. Conclusion Calculations of paramagnctic susceptibilities of fn-clcctron systems have hitherto been generally limited to moleC&X either with high degree of symmetry or with small deviations from high symmetry, in the latter case the dcviations have been ignored. In this letter a model is presented which allows one to calculate the paramagnetic susccptibility tcnsar for compounds oflanthanidcs and actinidcs with any cooordination number and geometry. There is no I-~~gera necessity for approximations of co-ordmation spheres and there arc no d~f~~u~tiesto tackle tow symmetry systems. lath the descrfbed model we wdl interpret the rn3~etjc properties of compounds with ~~nth~nides in the near future.
Acknowledgement I am grateful to Drs. M. Gcrloch and R.F. McMeeking from the University of Cambridge (England) for discussions. I thank the “Dcutschc Forschungsgemcinschaft” for financial support.
References [I] L&f. En~~!hardt and 3-N. Fig& 3, Chem. Sot. A (1968) 1258. 12) L.M. Engctb~rdt and 3-N. Figgs, 3. Chem. Sot. A (1970) 415. [3] M. Cerloch and D.J. hfsckcy, 1. Chcm. Sot. A (1970) 3030,304O; (197X) 2605,26I2,3372; (1972) 37,42,410,415. [4 j E. Bucher, H.J. Guggenheim, K. Andws, G.W. Hull Jr. and AS. Coopef, Phys. Rev. B 10 (1974) 2945. [S) H.-D. Ambcrgcr, R.D. Fischer and B. Kanellakopulos. Thcorct. Chim. Acta 37 (1975) 105. 16j B.D. Dunlap and G.K. Shcnoy, Phys. Rev. B 12 (1975) 27 16. (71 IL-D. Ambergcr, R.D. Fischer and G.G. Rosenbaucr, Z. Naturforsch. 31b (1976) 1 [8J H.-D. Ambcrger, R.D. Fischer and B. Kancllakopulos, 2, Naturforsch. 31b (1976) 12. 191 N. Edclstcin, A. Streitwicser Jr., D.G. Mom11and R. Waker, Jnorg, Chcm. 15 (1976) 1397. [ 101 E. Soufi6 and G. Goodman, Thcorct. Chim. Acta 41 (1976) 17. fl I j C.K. &rgcnscn, R. Pappafxdo and H.-H. Schmidtkc, J. Chem. Phys. 39 (1963) 1422. (12 j M.&f. Eilis and D-J. Newman, .I. Ckem. Phys. 49 (1968) 4037. fl31 W. Urland, Chem. Phys. 14 (1976j 393. 1141 D.M. Brink and G-R. Satchlcr, Annular rnorn~~~tnrn (Cl~rendon Prees, oxford, 3968). {151 C. W. Nicfson :snd G-F. Kostcr, Spcktroscopic coefficients for the pa, d” and f” conf~umtions (KfT Press, Cambridge, 1963). f 161 J,H. van Meek. The theory ofclectric and magnetic susceptibilities (Oxford Univ. Press, London, 1932).
CHEMICAL PHYSICS LETTERS
15
MarlAt 1977
THE APPLlCATlON OF THE ANGULAR OVERLAP MODEL IN THE CALCULATION OF PARAMAGNETIC PRINCIPAL SUSCEPTIBILITIES FOR f”-ELECTRON SYSTEMS W. URLAND Instltut fir anorganlsche und analytiscire Chemie.63 Giessen, West Gtwnan>p Rcccivcd 24 November 1976
A description of the calculation of the paramagnctrc principal suweptlbdltles hgand-fields i$ gvcn.
for f”-clectron systems with arbitrary
1. Introduction There arc many publications on the interpretation of paramagnctic susceptibilities of lanthanide and acrinide ions in various ligand fields, e.g. rcfs. [ l-1 0] . The models for the calculation of the susceptibilities differ mainly in the way how the influence of the Iigand field has been considered. In most cases the electrostatic concept has been used. It has been shown by several authors however that the angular overlap model (AOM) is more suitable than the electrostatic model to interpret ligand fields in f”-systems [ 11,121. ‘I’hc purpose of this letter is to describe the calculation of the paramagnetic principal susccpttbilities for fn-systerns in any kind of ligand fields by applying the ligand field potential in the AOM which has been derived in a previous publication [ 131.
2. The model 2.1. ?YZze complete
ligated field problem
The hamiltonian appropriate electronic repulsion, spin-orbit
for the total &and field calculation consists of the operators of Coulomb intcrcoupling and the ligand field potential in the AOM
This and the following equations follow generally the nomenclature and conventions of Brink and Satchlcr [14] _ The eigcnvalues and eigcnfunctions for the computation of the susceptibility tensor are obtained by diagonaliration of a given IJ, MJ) basis set under the hamiltonian (1). The Coulomb interaction perturbation-matrix eIcments can be written as
where the matrix elements (OLCSIXi>j e2/r&a’LS) are tabulated as Z,k +Yk are linear combinations of Slatcr Fk integrals also given in ref. [IS] .
in Nielson and Kostcr [ Is]_ The E”‘s
457
CffEMiCAL PHYSICSLfXfXRS
Volume 46, number 3
15 Mrtrch 1977
The spin-orbit coupling matrix are evaluated appiying the equation
t
f&~)-J-s-L’
841’26 (J , J’)6 (MJ, M;)
GldSII v~%wS’~.
(3)
The reduced matrix elements of this uperatar are listed in ref. f 15Jf The expressron for the Iigand field potential is:
The evaluation of the Rkckq-coeffcients The &and field matrtx requires now
has been described extensively in ref. [I3]. the calculation of the matrix elements: (~LWIMJIYkQla’t’.V‘M’)given by
the relation
The reduced matrix eiements of the unit tensor Ufk) arc tabulated in ref. [15]. ~at~~cs for the eIectron Cou101nbinteraction operator (three for f orbitals~, one spin-orbit matrix, and one matrix for each sphcricaf harmonic Yz or coefficient Rkckq are constructed for tfie (LUM’) basis set and stored. To establish the complete perturbation matrix for the opcmtor (1) WCproceed in the following way: First the coefficients R~CQ arc computed, introducing the angular overlap parameters eA as described in ref. [ 13). Tbcn cop~cs of the stored matrices are mu!tiplied by appropriate Ek, f ti , and kkckqparameters and finally summed. The diagonalization of the contpletc matrix gives eigenvalues and eigenvcctors for the interpretation of optical spectra and for the c~m?utation of the susceptibility.
The paramagnetic susceptibility is a second-rank tensor quantity expressing the relation between ~agnc~~~a~ion and a r~a~~etIc field: A& = xopHp *
(61
Here we USCthe standard tensor notation whcrc Greek letters as non-repeating suffices refer to the components of a set (x,y, z), but whcrc Greek letters as repeating suffices imply summation over the whole set. The defining cquatlon for magnetization in direction cr, for a system of identical molecules is given by: (7)
(8s)
458
Volume46, numbcr 3
CiIEMiCALPHYSICSLETTERS
with iu, = (I&, f 2S&30 (PO= Bohr magneton, k = Stevens’ orbital reduction factor). Differentiating (8) with respect to components of the magnetic field leads to:
~~~~a~*={~~~~~j~~ CI~il~oli,rjl~~lo+{ii~~li,{jl~~in)H,I~~~ i Following van Vleck’s treatment
-l$).
[I61 , considering only small magnetic fields, we expanded exp(-E#T)
exp(-E’,lkO = e~p(-E~~~H-)(1.- Gllr,liMfpjkT)
6) as: 001
and obtain after substitution and application of cq. (6) for the ~o~~unents of the susceptibility tensor:
Relation (11) reduces to van Vie&s expression when applied to a diagonal clement. If we consider that the trace of a matrix is invariant under a unitary transformation we can easily show that the susceptibility tensor is symmetric and valid for rilt kind of basis sets not only for those diagonal in Ha. The relations for the matrix elements of the operator fl* are given by GXLSJMJI (Id,
X
!
+ zs&?,
1
3
-4fJ
.J’
M.SJ’MJ>
= (- pQ(2J+
I[
0 i%fJ
(-t)L*S+J’+g k fL(L
i
)‘925
t 1) (2L + l)]
f 1)1’2
It2
J
i
1
LSL
3
1
3’
X
--II+ 1 M; )I[
(4
J
I
+‘+“+’ k [L(L t 1) (211f i)] “’
&LS.M~I (kL, + 2S,,)& IoUJ’M;>= (-$)*/2i(-l)‘-hzJ(2i
+‘+“+’
X
J’
* l)“‘(W
k [La
+ ljl”
f l)(ZL + I)] If2
-Mj $
(_~)LeS+J+l
a [S&s + 1) (2s + 1)) It2
(14)
After the computation of all six independent xnp components in a molecule-fured frame we then diagondlize the 453
Volume 46. number 3
CHBKlCAL PHYSICS LETTERS
susceptibility tensor to obtain the principal susceptibilities xu, x,,,,, xzz and their orientations 7%~mean susceptibility is calculated to:
15 March 1977 in
the molecule.
3. Conclusion Calculations of paramagnctic susceptibilities of fn-clcctron systems have hitherto been generally limited to moleC&X either with high degree of symmetry or with small deviations from high symmetry, in the latter case the dcviations have been ignored. In this letter a model is presented which allows one to calculate the paramagnetic susccptibility tcnsar for compounds oflanthanidcs and actinidcs with any cooordination number and geometry. There is no I-~~gera necessity for approximations of co-ordmation spheres and there arc no d~f~~u~tiesto tackle tow symmetry systems. lath the descrfbed model we wdl interpret the rn3~etjc properties of compounds with ~~nth~nides in the near future.
Acknowledgement I am grateful to Drs. M. Gcrloch and R.F. McMeeking from the University of Cambridge (England) for discussions. I thank the “Dcutschc Forschungsgemcinschaft” for financial support.
References [I] L&f. En~~!hardt and 3-N. Fig& 3, Chem. Sot. A (1968) 1258. 12) L.M. Engctb~rdt and 3-N. Figgs, 3. Chem. Sot. A (1970) 415. [3] M. Cerloch and D.J. hfsckcy, 1. Chcm. Sot. A (1970) 3030,304O; (197X) 2605,26I2,3372; (1972) 37,42,410,415. [4 j E. Bucher, H.J. Guggenheim, K. Andws, G.W. Hull Jr. and AS. Coopef, Phys. Rev. B 10 (1974) 2945. [S) H.-D. Ambcrgcr, R.D. Fischer and B. Kanellakopulos. Thcorct. Chim. Acta 37 (1975) 105. 16j B.D. Dunlap and G.K. Shcnoy, Phys. Rev. B 12 (1975) 27 16. (71 IL-D. Ambergcr, R.D. Fischer and G.G. Rosenbaucr, Z. Naturforsch. 31b (1976) 1 [8J H.-D. Ambcrger, R.D. Fischer and B. Kancllakopulos, 2, Naturforsch. 31b (1976) 12. 191 N. Edclstcin, A. Streitwicser Jr., D.G. Mom11and R. Waker, Jnorg, Chcm. 15 (1976) 1397. [ 101 E. Soufi6 and G. Goodman, Thcorct. Chim. Acta 41 (1976) 17. fl I j C.K. &rgcnscn, R. Pappafxdo and H.-H. Schmidtkc, J. Chem. Phys. 39 (1963) 1422. (12 j M.&f. Eilis and D-J. Newman, .I. Ckem. Phys. 49 (1968) 4037. fl31 W. Urland, Chem. Phys. 14 (1976j 393. 1141 D.M. Brink and G-R. Satchlcr, Annular rnorn~~~tnrn (Cl~rendon Prees, oxford, 3968). {151 C. W. Nicfson :snd G-F. Kostcr, Spcktroscopic coefficients for the pa, d” and f” conf~umtions (KfT Press, Cambridge, 1963). f 161 J,H. van Meek. The theory ofclectric and magnetic susceptibilities (Oxford Univ. Press, London, 1932).