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First- and second-order contributions to the effective Hamiltonian parameters for Gd3+(8S) and Mn2+(6S)
Journal of Magnetism and Magnetic Materials 54 57 (1986) 1481 1482 FIRST- AND SECOND-ORDER CONTRIBUTIONS PARAMETERS F O R G d 3 + (8S) A N D M n 2 + (6S). J.A.
1481 TO
THE
EFFECTIVE
HAMILTONIAN
TUSZYI'~!SKI
Department of Physics, Memorial Uniuersity of Newfoundland, St. John's, Nenfoundland, Canada A I B 3X7
The effective operator formalism is used to calculate the values of effective Hamiltonian parameters which arise from higher order perturbation corrections due to the relativistic crystal field and spin-orbit interactions. The method is illustrated on two examples: the Gd 3+ ion in the C3h-symmetry and the Mn 2+ ion in the C3,-symmetry. Transitions to all excited orbitals for / = s, p, d, f and n = l + 1..... 10 are included in calculations. The role of second-order perturbation corrections is significant for both excitations within the ground configuration and outside it.
1. Introduction The effective operator technique was developed by Sandars a n d Beck [1] and recently systematized as a succession of three equivalence transformations of the molecular H a m i l t o n i a n of a transition ion complex [2]: (i) relativistic molecular orbitals ~ are replaced by relativistic atomic orbitals [nOm ) and the molecular H a m i l t o n i a n (two-body) is replaced by a symmetrya d a p t e d atomic H a m i l t o n i a n (one-body) which includes the crystal field term
F= E
= Z r*c C ,
kq
(1)
kq
where Tqk are Racah tensor operators [3] ~nd Cqk are tesseral harmonics [4]. (ii) relativistic wavefunctions I nljm) are replaced by non-relativistic wavefunctions ] nljm) according to [5]
[ F,(r)lnljm ) Inljm ) = r I iG,(r)lnl+_ l j m ) 1
(2)
A p a r t from the s t a n d a r d terms such as SO which corresponds to d [ l l 0 0 ] a n d C F ( k ) which corresponds to d[OkkO] new terms corresponding to d[1 k_+ 1 k0] arise due to relativistic crystal field effects (henceforth denoted by RCF).
2. Applications The outlined m e t h o d has been applied to investigate the role of the various microscopic mechanisms in the formation of the phenomenological crystal field constants. The necessary orbital energies e(nl) a n d relativistic radial integrals Rk(nO; n T j ' ) have been adopted from self-consistent D i r a c - H a r t r e e - F o c k c o m p u t a t i o n s [7-9]. We used n = l + 1 . . . . . 10, l = s, p, d, f.
Example 1. G d 3 + i n La(C2HsSO4) 3" 9 H 2 0 . The symmetry of this e n v i r o n m e n t is C3h a n d the crystal field parameters are [10]: C02 = 200 cm-I; C04 = - 5 5 0 cm l; C 6 = - 5 5 0 cm 1; C6 = + 5 5 0 cm -1 a n d the spin - o r b i t
simultaneously with a transition from Racah tensor operators to double tensor operators [6]
E b( k + l, OWq'',*+-',* /=0,+l
(3)
(iii) inclusion of higher order perturbations, some of which corresponding to configuration mixing effects, into an effective H a m i l t o n i a n acting on the non-relativistic states of the ground configuration Heft = Ho + H1+ Y~, Hl l n'I ) ( n T I H1 + . . . . .'r
~0-~("l')
(4)
where H o is the spherically symmetric part, H i includes crystal field terms (henceforth denoted by C F ( k ) ; k is rank) and the s p i n - o r b i t term (henceforth denoted by SO), eo is the ground state energy, e(nl) is the energy of excited states of the ground configuration, nl, a n d two types of excited configurations: (nl)N-ln'l ' a n d ( n ' / ' ) 4 r + l ( n l ) N+ 1. The resultant effective H a m i l t o n i a n is a linear c o m b i n a t i o n of double tensor operators
Heft:
E
d[k,kzkq]Wq (k'k:>*.
(5)
Table 1 Percentage contributions to the effective spin-orbit and crystal field parameters of Gd 3+ according to mechanisms Mechanism:
d[ll00]
d[0220l
d[04401
d[0660]
First order:
+120.5%
+65.8%
+79.5%
+84.4%
Second order: 1.4f~nf SO × SO SO × CF(2) SO × CF(4) SO × CF(6) CF(2) × CF(2) CF(2) × CF(4) CF(2) × CF(6) CF(4) × CF(4) CF(4) x CF(6) CF(6) × CF(6) 2.4f-~ np CF(2) × CF(2) CF(2) × CF(4) CF(4) × CF(4)
klk2kq
0304-8853/86/$03.50
© E l s e v i e r S c i e n c e P u b l i s h e r s B.V.
-20.5% -0.3% -0.6% +0.3% +0.7% -0.6% +5.4% -10.8% + 29.7%
+0.5% -0.4% -1.3% +3.5% -7.0% + 19.3%
+0.5% +1.7% +7.9%
+0.3% +1.1% +5.1%
-0.4% -1.4% +3.8% -7.7% + 21.0%
1482
J.A. Tuszvhski / Effi>ctive Hamiltonian parameterv for Gd s* and Mn: "
"Fable 2 Percentage contributions to the effective spin-orbit and crystal field parameters of Mn-' ' according to mechanisms SITE 1
SITE II
Mechanism:
d [1100]
d [0220]
d [0440]
d [1100]
d [0220]
d [0440]
First order:
157.9~
118.4%,
104.2%
157.9%
158.5%
199.9%
Second order: 1.3d ~ np ('F(I)XCF(1) CF(1)×CF(3) 2.3d - ' n f CF(I )×CF( 1) CF(1)xCF(3) CF(3)×CF(3) 3.3d ~ nd CF(2) × CF(2) CF(2) × CF(4) CF(4) × CF(4) SO × CF(4) CF(0) × CF(2) CF(0) × CF(4) SOxCF(0) SO × SO
+11.1% 0.4%
1.6%
+ 3.2% -0.2%
0.6%
+ 7.0% + 1.3% + 0.3%
27.7% + 5.1% 1.1%
4.59,' + 0.6~ 0.1%
40.7%
54.5~ 25.9%
54.3% -3.6%
c o n s t a n t is [11] ~ . = 1 4 8 0 c m L. T h e results o f our calculations for d [ k i k 2 k q ] are s u m m a r i z e d in table 1. This table p r e s e n t s the m a g n i t u d e o f the c o n t r i b u t i o n s to the s t a n d a r d spin orbit a n d crystal field coefficients. T h e value of 100% c o r r e s p o n d s to that e x t r a p o l a t e d from e x p e r i m e n t . C o n t r i b u t i o n s from 4f--+ n s a n d 4 f - , nd excitations are less than 0.1% and are thus neglected. T h e m a g n i t u d e of the R C F coefficients is small by c o m p a r i s o n with the s t a n d a r d ones (less t h a n 1%) but they are still e x p e c t e d to play an i m p o r t a n t role in the zero field splitting since G d 3+ is an S-state ion,
Example 2. M n 2+ in L a 2 M g 3 ( N O 3 ) I 2 . 2 4 H 2 0 . T h e s y m m e t r y of this e n v i r o n m e n t is C~, a n d there are two inequivalent sites for M n 2+ resulting in two different sets of crystal field p a r a m e t e r s [12]. A t site I: C~~ = 1834 c m i: C(1=2345 c m - i ; C ~ = 1157 cm-]: C{~=72 c m -1 a n d C~ = 331 c m -1. At site II: C~~= 1834 c m - l : <~ = - 5 5 7 c m l: (-~4= 115 c m 1. T h e s p i n - o r b i t c o n s t a n t is [12] X = 300 c m -1. T h e results of our calculations for d [ k l k 2kq] are s u m m a r i z e d in table 2. T h e c o n s t r u c t i o n o f this table is a n a l o g o u s to that of table 1. 3. Conclusions In both e x a m p l e s the first- as well as s e c o n d - o r d e r effects are i m p o r t a n t . W i t h i n the latter class the most d o m i n a n t c o n t r i b u t i o n s are due to ( n l ) m ~ (nl) x ln'l excitations (i.e. 4f 7 ~ 4f 6 n f for G d 3+ a n d 3d -~~ 3 d 4 n d for M n 2+ ). T h e role o f c o n t i n u u m states a n d o d d - p a r i t y c o n f i g u r a t i o n s is negligible. T h e s e c o n d - o r d e r m e c h a n i s m s are crucial in d e t e r m i n i n g R C F p a r a m e t e r s which
35.1 ¢i 4.7% 0.8 q/r
68.7q' 54.3<2/ 3.6~
has recently been d e m o n s t r a t e d for M n 2~ [13]. T h e first-order m e c h a n i s m s d e t e r m i n e the SO and C F ( k ) c o n s t a n t s within a 15-50% accuracy which is rather p o o r . Hence, the relativistic higher o r d e r p e r t u r b a t i o n s are necessary to p r o p e r l y d e s c r i b e crystal field effects. M o r e o v e r , b o t h G d 3+ a n d M n 2+ have in the lowest o r d e r only spin d e g e n e r a c i e s a n d thus should be insensitive to electrostatic e n v i r o n m e n t s . Zero-field splittings, however, have been o b s e r v e d [10]. In o r d e r to explain these spectra, one must, therefore, include relativistic effects in the form o f spin orbit c o u p l i n g a n d relativistic crystal fields. [1] P.G.H. Sandars and J. Beck, Proc. R. Soc. A259 (1970) 147. [2] J.A. Tuszyflski, An effective operator method for relativistic perturbation calculations involving crystalline electric fields, submitted to J. Chem. Phys. [31 G. Racah, Phys. Rev. 62 (19421 438. [4] J i . Prather, Atomic Energy Levels in Crystals (US National Bureau of Standards, Washington. 19611. [5] L. Armstrong Jr., Theory of the Hyperfine Structure of Free Atoms (Wiley-lnterscience, New York, 1971 ). [6] B.G. Wybourne, J. Chem. Phys. 43 (19651 4506. [7] A.J. Freeman and J.P. Desclaux, J. Magn. Magn. Mat. 12 (19791 11. [8] J.P. Desclaux, Comput. Phys. Commun. 9 (19751 31. [9] J. Andriessen, private communication (19821. [10] B.G. Wybourne, Phys. Rev. 148 (19661 317. [11] B.G. Wybourne, Spectroscopic Properties of Rare Earths (lnterscience, New York, 19651. [12] R. Chanerjee and D. Van Ormondt, Phys. Lett. 33A (19701 147. (131 J.A. Tuszynski and R. Chatterjee. Phys. Lett. 104A (19841 267.
1481 TO
THE
EFFECTIVE
HAMILTONIAN
TUSZYI'~!SKI
Department of Physics, Memorial Uniuersity of Newfoundland, St. John's, Nenfoundland, Canada A I B 3X7
The effective operator formalism is used to calculate the values of effective Hamiltonian parameters which arise from higher order perturbation corrections due to the relativistic crystal field and spin-orbit interactions. The method is illustrated on two examples: the Gd 3+ ion in the C3h-symmetry and the Mn 2+ ion in the C3,-symmetry. Transitions to all excited orbitals for / = s, p, d, f and n = l + 1..... 10 are included in calculations. The role of second-order perturbation corrections is significant for both excitations within the ground configuration and outside it.
1. Introduction The effective operator technique was developed by Sandars a n d Beck [1] and recently systematized as a succession of three equivalence transformations of the molecular H a m i l t o n i a n of a transition ion complex [2]: (i) relativistic molecular orbitals ~ are replaced by relativistic atomic orbitals [nOm ) and the molecular H a m i l t o n i a n (two-body) is replaced by a symmetrya d a p t e d atomic H a m i l t o n i a n (one-body) which includes the crystal field term
F= E
= Z r*c C ,
kq
(1)
kq
where Tqk are Racah tensor operators [3] ~nd Cqk are tesseral harmonics [4]. (ii) relativistic wavefunctions I nljm) are replaced by non-relativistic wavefunctions ] nljm) according to [5]
[ F,(r)lnljm ) Inljm ) = r I iG,(r)lnl+_ l j m ) 1
(2)
A p a r t from the s t a n d a r d terms such as SO which corresponds to d [ l l 0 0 ] a n d C F ( k ) which corresponds to d[OkkO] new terms corresponding to d[1 k_+ 1 k0] arise due to relativistic crystal field effects (henceforth denoted by RCF).
2. Applications The outlined m e t h o d has been applied to investigate the role of the various microscopic mechanisms in the formation of the phenomenological crystal field constants. The necessary orbital energies e(nl) a n d relativistic radial integrals Rk(nO; n T j ' ) have been adopted from self-consistent D i r a c - H a r t r e e - F o c k c o m p u t a t i o n s [7-9]. We used n = l + 1 . . . . . 10, l = s, p, d, f.
Example 1. G d 3 + i n La(C2HsSO4) 3" 9 H 2 0 . The symmetry of this e n v i r o n m e n t is C3h a n d the crystal field parameters are [10]: C02 = 200 cm-I; C04 = - 5 5 0 cm l; C 6 = - 5 5 0 cm 1; C6 = + 5 5 0 cm -1 a n d the spin - o r b i t
simultaneously with a transition from Racah tensor operators to double tensor operators [6]
E b( k + l, OWq'',*+-',* /=0,+l
(3)
(iii) inclusion of higher order perturbations, some of which corresponding to configuration mixing effects, into an effective H a m i l t o n i a n acting on the non-relativistic states of the ground configuration Heft = Ho + H1+ Y~, Hl l n'I ) ( n T I H1 + . . . . .'r
~0-~("l')
(4)
where H o is the spherically symmetric part, H i includes crystal field terms (henceforth denoted by C F ( k ) ; k is rank) and the s p i n - o r b i t term (henceforth denoted by SO), eo is the ground state energy, e(nl) is the energy of excited states of the ground configuration, nl, a n d two types of excited configurations: (nl)N-ln'l ' a n d ( n ' / ' ) 4 r + l ( n l ) N+ 1. The resultant effective H a m i l t o n i a n is a linear c o m b i n a t i o n of double tensor operators
Heft:
E
d[k,kzkq]Wq (k'k:>*.
(5)
Table 1 Percentage contributions to the effective spin-orbit and crystal field parameters of Gd 3+ according to mechanisms Mechanism:
d[ll00]
d[0220l
d[04401
d[0660]
First order:
+120.5%
+65.8%
+79.5%
+84.4%
Second order: 1.4f~nf SO × SO SO × CF(2) SO × CF(4) SO × CF(6) CF(2) × CF(2) CF(2) × CF(4) CF(2) × CF(6) CF(4) × CF(4) CF(4) x CF(6) CF(6) × CF(6) 2.4f-~ np CF(2) × CF(2) CF(2) × CF(4) CF(4) × CF(4)
klk2kq
0304-8853/86/$03.50
© E l s e v i e r S c i e n c e P u b l i s h e r s B.V.
-20.5% -0.3% -0.6% +0.3% +0.7% -0.6% +5.4% -10.8% + 29.7%
+0.5% -0.4% -1.3% +3.5% -7.0% + 19.3%
+0.5% +1.7% +7.9%
+0.3% +1.1% +5.1%
-0.4% -1.4% +3.8% -7.7% + 21.0%
1482
J.A. Tuszvhski / Effi>ctive Hamiltonian parameterv for Gd s* and Mn: "
"Fable 2 Percentage contributions to the effective spin-orbit and crystal field parameters of Mn-' ' according to mechanisms SITE 1
SITE II
Mechanism:
d [1100]
d [0220]
d [0440]
d [1100]
d [0220]
d [0440]
First order:
157.9~
118.4%,
104.2%
157.9%
158.5%
199.9%
Second order: 1.3d ~ np ('F(I)XCF(1) CF(1)×CF(3) 2.3d - ' n f CF(I )×CF( 1) CF(1)xCF(3) CF(3)×CF(3) 3.3d ~ nd CF(2) × CF(2) CF(2) × CF(4) CF(4) × CF(4) SO × CF(4) CF(0) × CF(2) CF(0) × CF(4) SOxCF(0) SO × SO
+11.1% 0.4%
1.6%
+ 3.2% -0.2%
0.6%
+ 7.0% + 1.3% + 0.3%
27.7% + 5.1% 1.1%
4.59,' + 0.6~ 0.1%
40.7%
54.5~ 25.9%
54.3% -3.6%
c o n s t a n t is [11] ~ . = 1 4 8 0 c m L. T h e results o f our calculations for d [ k i k 2 k q ] are s u m m a r i z e d in table 1. This table p r e s e n t s the m a g n i t u d e o f the c o n t r i b u t i o n s to the s t a n d a r d spin orbit a n d crystal field coefficients. T h e value of 100% c o r r e s p o n d s to that e x t r a p o l a t e d from e x p e r i m e n t . C o n t r i b u t i o n s from 4f--+ n s a n d 4 f - , nd excitations are less than 0.1% and are thus neglected. T h e m a g n i t u d e of the R C F coefficients is small by c o m p a r i s o n with the s t a n d a r d ones (less t h a n 1%) but they are still e x p e c t e d to play an i m p o r t a n t role in the zero field splitting since G d 3+ is an S-state ion,
Example 2. M n 2+ in L a 2 M g 3 ( N O 3 ) I 2 . 2 4 H 2 0 . T h e s y m m e t r y of this e n v i r o n m e n t is C~, a n d there are two inequivalent sites for M n 2+ resulting in two different sets of crystal field p a r a m e t e r s [12]. A t site I: C~~ = 1834 c m i: C(1=2345 c m - i ; C ~ = 1157 cm-]: C{~=72 c m -1 a n d C~ = 331 c m -1. At site II: C~~= 1834 c m - l : <~ = - 5 5 7 c m l: (-~4= 115 c m 1. T h e s p i n - o r b i t c o n s t a n t is [12] X = 300 c m -1. T h e results of our calculations for d [ k l k 2kq] are s u m m a r i z e d in table 2. T h e c o n s t r u c t i o n o f this table is a n a l o g o u s to that of table 1. 3. Conclusions In both e x a m p l e s the first- as well as s e c o n d - o r d e r effects are i m p o r t a n t . W i t h i n the latter class the most d o m i n a n t c o n t r i b u t i o n s are due to ( n l ) m ~ (nl) x ln'l excitations (i.e. 4f 7 ~ 4f 6 n f for G d 3+ a n d 3d -~~ 3 d 4 n d for M n 2+ ). T h e role o f c o n t i n u u m states a n d o d d - p a r i t y c o n f i g u r a t i o n s is negligible. T h e s e c o n d - o r d e r m e c h a n i s m s are crucial in d e t e r m i n i n g R C F p a r a m e t e r s which
35.1 ¢i 4.7% 0.8 q/r
68.7q' 54.3<2/ 3.6~
has recently been d e m o n s t r a t e d for M n 2~ [13]. T h e first-order m e c h a n i s m s d e t e r m i n e the SO and C F ( k ) c o n s t a n t s within a 15-50% accuracy which is rather p o o r . Hence, the relativistic higher o r d e r p e r t u r b a t i o n s are necessary to p r o p e r l y d e s c r i b e crystal field effects. M o r e o v e r , b o t h G d 3+ a n d M n 2+ have in the lowest o r d e r only spin d e g e n e r a c i e s a n d thus should be insensitive to electrostatic e n v i r o n m e n t s . Zero-field splittings, however, have been o b s e r v e d [10]. In o r d e r to explain these spectra, one must, therefore, include relativistic effects in the form o f spin orbit c o u p l i n g a n d relativistic crystal fields. [1] P.G.H. Sandars and J. Beck, Proc. R. Soc. A259 (1970) 147. [2] J.A. Tuszyflski, An effective operator method for relativistic perturbation calculations involving crystalline electric fields, submitted to J. Chem. Phys. [31 G. Racah, Phys. Rev. 62 (19421 438. [4] J i . Prather, Atomic Energy Levels in Crystals (US National Bureau of Standards, Washington. 19611. [5] L. Armstrong Jr., Theory of the Hyperfine Structure of Free Atoms (Wiley-lnterscience, New York, 1971 ). [6] B.G. Wybourne, J. Chem. Phys. 43 (19651 4506. [7] A.J. Freeman and J.P. Desclaux, J. Magn. Magn. Mat. 12 (19791 11. [8] J.P. Desclaux, Comput. Phys. Commun. 9 (19751 31. [9] J. Andriessen, private communication (19821. [10] B.G. Wybourne, Phys. Rev. 148 (19661 317. [11] B.G. Wybourne, Spectroscopic Properties of Rare Earths (lnterscience, New York, 19651. [12] R. Chanerjee and D. Van Ormondt, Phys. Lett. 33A (19701 147. (131 J.A. Tuszynski and R. Chatterjee. Phys. Lett. 104A (19841 267.