- Email: [email protected]
THEORETICAL METHOD OF LIGAND FIELD THEORY AND ITS APPLICATION
THEORETICAL METHOD OF LIGAND FIELD THEORY AND ITS APPLICATION
Tang Auchin, Sun Chiachung, Li Befu (Institute of Theoretical Chemistry, Jilin University, Changchun, China)
It has been more than fifty years since the ligand field theory [1-5] made its appearance
. It was initiated in 1931 when Bethe's cry
stal field theory was proposed. The theory of complex spectra for atoms contributed by Racah
has had an important effect on the deve
lopment of ligand field theory as it has affected that of nuclear and elementary particle theories. In this article, we shall try to give a brief review on the extension of irreducible tensor method to ligand field theory, which has been performed by Tang and his collabor[7-14] ators . As an illustration, the analysis of the spectra of Ή ^ Ο ^ will be presented. I. THE V-COUPLING COEFFICIENTS FROM S0(3) TO OCTAHEDRAL GROUP Let us consider the octahedral group 0 to which a molecule be longs. Since the group 0 is not only a finite group, but also a sub group of the continuous group S0(3) which is further a subgroup of a larger continuous one, as a direct consequence, the continuous group can be employed to give a unified treatment of the ligand field theory with point group symmetry. It seems that the gap between the contin uous group S0(3) and the finite group 0 should be filled in order to make sufficient use of the achievements of both atomic molecular theories in the ligand field theory. For this purpose, the V-coupling
241
coefficients from S0(3) to group 0 have been well defined
[7-9 11] ' as
follows: K
(r U la
v
r )=(-ΐ)*™> Σ P'
h
12b
1 3c
m2
χν
ζ\
l·
m\m2mi\m\ 717273
T
y\
x
'
h z
Πίτ, J|
^^.r^S&r^Sftr*,,
(1)
where the first and the second factors on the right hand side stand for the 3-j symbol and the V-coupling coefficient of the group 0, respectively, and where the transformation coefficient from S0(3) to group 0 can be expressed in terms of inner product of the orbitals between the ligand and central ion <Γγ|jm> and the group overlap integral G J
Cl5] S
m Γγ
=
Ybm>/Gjl(m)
(2)
Under the action of time reversal operator, we have
s£i>=[-i] r -n-iy- m sw
(3)
The V-coupling coefficients from S0(3) to group 0 satisfy the symmetry property such that
K
fc
i2 r2
i3) y= vlj2 rjr {r2
h
r
^33
h
ι,. = ΓΓ|/,
(-ιν+-+--,+Γ2+Γ3
Χ^Γ,Γ,ΓΛΚ^
^
^
(4)
where the factor Θ(Γ 1 Γ 2 Γ3). always equals unity, except 6(T2U'Uf)£ =-1. For practical application to rare earth ions in the field with point group symmetry, the numerical values of the V-coupling coefficients 1 1 from S0(3) to group 0 in the range j = — to j = 12— were evaluated by the use of Elliott-503 computer. Similarly, the V-coupling coefficients from the octahedral group 0 to its subgroup can be defined similarly as we have done in defining the V-coupling coefficients from S0(3) to the group 0, but the former which are quite different from the latter are involved in the classi fication
of the irreducible representation of point groups into A, B
and C species L8—9] #
242
II. LIE COMMUTATIVE RELATION CHARACTERIZED BY OCTAHEDRAL SYMMETRY Let us begin to discuss Lie commutative relation characterized [8-14] by octahedral symmetry
. We use Γ to denote an irreducible re
presentation of the group 0 and S to denote the spin quantum number \ . The creation and the annihilation operators associated with atomic or molecular orbitals with octahedral symmetry are defined in the follow ing forms air(Sr) 10> - I STar)
0„r(Sr)
a n d
| STar) - 10>
(5 )
respectively. By means of the conventional second quantization method, the quantum number q=i with respect to quasispin group SU (2) can be introduced to define
a,ar(,sn - {a_iariqSO ^ ^ - - [ - - i r ^ c s r )
(6)
and consecutively to define a triple tensor operator
JW(rro -
UG0»(«)i(O]*(-D'(-Dp[-H'"e
Σ ββ'αα'ΎΥ'
(q q X\fS S ω\
/Γ Γ
v
κ\
■ W X ° ρϊ W r· ζΐ^^ST)w^n
(7)
As a direct result, Lie commutative relation with octahedral symmetry can be established by writing
V3P3C3
' ~
1
kl
fXlX2Xl\
. {- V r ( - ιγ>+χ<- 0"·+"<+
*Γ,Γ,(- iy>
+
(ωι ω2 ωι\[ΧιΧ2Χι\
(ω1 ω2 ωΛ
Ο Γ , + Γ > ·*(Γ 1 Γ'Λ),»' (Κ; Κ; Κ;)
°<- ο·ι+·3+Γί+Γί e(rirI.1)ie(rlr;«I)iH' ("' * *3)
+ *r,rsC— 0*»+"> ( - ι)''+'·+Γ'·+Γ' θπτ^,ινί'1
*\"')
/^ κ2 κ3\
w ^ ^ m w%e&KrjV>
η^νΥσΛ)
\ i 2 * 1 i 2 7 kill
- 5r;r5(- I / ' ^ C - 1 )»·+"<- i)"+r;+r«0(r;r2Ka);w (** "3 * ) X
243
i 2 i 1 * 2 7 *//7
vSfcftW./)} J
(β)
where (IV.) and {**' } are the 3-j and the b-j symbols, and the V and the W factors are used to denote the V-and the W-coefficients of the octahedral group. For a given electron configuration ( Γ ^ ^ ' ' # r m ) n of octahedral symmetry, we can find out the corresponding group chains using the Lie commutative relation. For illustration, we take (et2) n and (a2t 1 t 2 ) n for example. For configuration (et 2 ) , there are three group chains under the leading group S0(21) L ' S0(2l)
J
D SO(20) Ώ SUQ(2) X \Sp(\0) D SOs(2) X [S0(5) D 50(3) D 0)\
(9)
50(21) D 50(20) D5C/<2(2) X \Sp(\0) D SUS(2) [S0(5) D 0(2) X 0(3) D 0]\
(10 )
50(21) D 50(20) D 50(8) X 50(12) D 50
X (5/7(4) X Sp(6) D SUS(2) X [0(2) X 0(3) D 0]}
{ 1 1 )
For the three group chains, the corresponding wave functions with octa hedral symmetry constitute the bases of three equivalent irreducible representations of SO(21) and therefore the three bases are connected by unitary transformation. It should be pointed out that whether or not the ligands participate in forming chemical bonds with the d-orbitals of the central ion, all of the three group chains hold true. Cer tainly, the group chain (9) describes the original weak-field scheme if the atomic orbitals are used to serve as the irreducible bases. The group chain (11) corresponds to the strong-field scheme. The group chain (10) is indeed an intermediate one between the group chains (9) and (10), so that an intermediate field scheme is proposed, but it must be examined by experimental facts to varify its existence. For configuration (a 2 t^t 2 ) , there are a great number of group chains arising from the Lie commutative relation (8). For brevity, we Ά * 4-u C8-14] only consider one of them SO(29)^SO(28)r>SUQ(2)x{Sp(m)DSUS(2)x[SO(7)r>G2r>SO(3)^>0]}
(12)
This group chain is the well-known one of Racah adapted to octahedral symmetry in quasispin scheme. It should be noted that when the ligands form chemical bonds with the orbitals of the central rare-earth ion, group chain (13) still holds good. 244
III.
MATRIX ELEMENTS OF THE SLJTx SCHEME IN OCTAHEDRAL SYMMETRY
The SLJtx scheme which is regarded as the extension of the Russell-Saunders coupling in atomic theory links with the following group , . [9-14] J chain |50(29)Ι3 50(28)ΐ3 5υ Ω (2)χ{5ρ(14)^5υ 3 (2)χ[50(7)^0 2 -50(3)]^50 α (3)^θ} S
L
J
τ (13)
We shall make a brief discussion on the matrix elements with respect to this group chain in treating rare-earth ions in the field with oc tahedral symmetry. By means of the linear transformation of the triple tensor oper ator given in Eq.(7), a new triple tensor operator W x t (SLJ)
(1Ό
ητ can be obtained to adapt the symmetry of group chain (13) as listed in Table 1. Furthermore, various kinds of interactions in the SLJtx scheme in the ligand field theory can be expressed in terms of the corresponding triple tensor operators as listed in Table 1. For brevity, only the interactions such as ligand field H c and relativistic effect H r are givenC 8 "" 1 ^ 16 ] Hr = Σ B(OKK) W^ A 1(0KK), (15) c
k
Η~ = Σ
0a
B(1KK ! ) W^ A l(lkk f ).
(16)
Application of generalized Wigner-Eckart theorem to the matrix elemen ts of H c and H r in SLJtx scheme gives = <5ss* ott* C
δττ'Σ B(okk) / ( 2 J + 1 ) ( 2 J ' + 1 ) k ^ (2S+l)X(t)
' { % l \ V l S : ^ ( -^ Q " M Q -M Q l S'.)<«SLi|w.
(17)
^ a S L J t T l H r l f V s ' L ' J ' t · · ^ = 6tt' δττ' Σ B(l K ' K ) V ( 2 J + 1 ) ( 2 J ' + 1 )
k'k 245
x(t)
Table 1
The
SO(28)
SUQ(2)
Sp(14)
SUS(2)
SO(7)
(00. . . 0 )
0
[00. . . 0 ]
0
(000)
2 (00)
0
[20. . . 0 ]
1
(000)
W (033) (110...0)
0
[20. . . 0 ]
0
W ot ( OLL) ( 1 1 0 . . . 0 )
0
[20. . .0]
0
W
oA,
(000)
Wot ( 1 0 1 ) ( 1 1 0 . . . 0 ) ot
0>
Group Theoretical Classification of W
(SLJ) SO(3)
G
S0
0
.7 0
(00)
0
1
(110)
(10)
3
3
(110)
(11)
1
1
5 W ot ( 1 L J ) ( 1 1 0 . . . 0 ) IA
0
[20. . .0]
1
(200)
(20)
'(000)
(110...0)
1
[00. . .0]
0
(000)
(00)
W'* (OLL)
(110...0)
1
[110...0]
0
(200)
(20)
(1LJ) ( 1 1 0 . . . 0 )
1
[110...0]
1
(110)
(11)
W(t ( 1 3 J ) ( 1 1 0 . . . 0 )
1
[110...0]
1
(110)
(10)
W W
jt
2, 2,
4, 6
J
0
0
4, 6
J
1,
5 3
J J
S S1 1
'L L 1 K 1 \ f
J J
K
J, A\)
<-1)Q-^(_QMQ
J QM.) <«SL |W.(1 K.)| a.S.L.> (18)
k
It is especially noteworthy that the V-coupling coefficients from SO (3) to the group 0 play an important role in the evaluation of the matrix elements in Eqs. (17) and (18), since they extend the results of continuous group to point group symmetry. From Eq. (17), we know that the reduced matrix element introduced by Racah can be further split into the product of 3-j symbol of quasispin group SU^(2) and the reduced matrix elements of W T (ok) which is independent of the number of electrons. It is known that by taking advantage of the 3-j symbol of quasispin group, the complementary rule of matrix elements can be followed directly. IV. THE SPECTRA OF T b P 5 0 1 4 The symmetry of crystal structure of T b P ^ O ^ is of the distorted 0 4 ν . Since C 4 v is equivalent to D^, the matrix elements in Eqs. (17) and (18) can be modified by means of the V-coupling coefficients from oc tahedral group to group D^ to adapt the lower symmetry of TbP^O^^. For the purpose of comparision, the results of theoretical calculations and the data of fluorescent spectra are listed in Table 2[16].
247
Table 2 The Splitting of Terms of 7 Fj for Tb 3 + in the Field 3_ of PsO-^ with Cj^y Symmetry _ _ _ _— Expt. (cm ) Calc. (cm ) Expt. (cm ) Calc. (cm ) 7F0
5914
5907
7F-L
5778
5776
Ai A2
5642
5632
5341
7F2
7F
3
7F4
2502
2489
E
2315
2318
Bl
E
2304
2297
E
5337
E
2239
2245
A2
5324
5332
Bl
2219
2212
5190
5192
Al
2188
2199
Al B2
5073
5079
B
2174
2179
E
4547
4559
2 A2
2170
2156
A2
4515
4510
E
451
449
E
4492
4487
E
414
B
4433
4430
B2
310
A
4413
4420
3691
3707
3640
?F5
7F6
l
Bl
292
293
l A2
166
170
E
3648
Al A2
121
117
3600
3597
E
20
Bl B2
3570
3574
8
E
3524
3528
Al B2
0
Al
3444
3446
E
3426
3410
Bl
248
0
REFERENCES [1] H.A., Bethe, Ann. Phys., 3, 133 (1929). [2] Y. Tanabe, and S. Sugano, J. Phys. Soc. Japan, 9, 753, 766 (1954). [3] J.S. Griffith, "The Theory of Transition-Metal Ions", Cambridge University Press, (1961). [4] B.G. Wybourne, Int. J. Quantum Chem., 7, 1117 (1973). [5] M.K. Kibler and G. Grenet, Int. J. Quantum Chem., 2, 359 (1977). [6] G. Racah, Phys. Rev., 76, 1352 (1949). [7] Research Group on Structure of Matter, Scientiarium Naturalium (Kirin University), 3, 79 (1964); Scientia Sinica, 15, 610 (1966). [8] Research Group on Structure of Matter, "The Summer Physics Collo quium of the Peking Symposium", 1966. [9] Tang Auchin et al., "Theoretical Method of the Ligand Field Theory" Science Press, Peking, (1979). [10] Tang Auchin, Sun Chiachung,and Li Befoo, Int. J. Quantum Chem., 18, 545 (1980). [11] Tang Auchin, Cho Chingyu, and Sun Chiachung, Int. J. Quantum Chem., 18, 557 (1980). [12] Sun Chiachung, Li Beifoo, and Tang Auchin, Int. J. Quantum Chem., 14, 521 (1981). [13] Sun Chiachung, Li Beifoo, and Tang Auchin, Int. J. Quantum Chem., 15, 305 (1981). [14] Sun Chiachung, Han Yande, Li Beifoo and Li Qianshu, Int. J. Quan tum Chem., 23, 169 (1983). [15] Zhang Qianer, Universitatis
Amoiensis Acta Scientiarum Naturalium,
19, 117 (1980). [16] Li Zesheng, M.S., Thesis, Jilin University, (1984).
249
Tang Auchin, Sun Chiachung, Li Befu (Institute of Theoretical Chemistry, Jilin University, Changchun, China)
It has been more than fifty years since the ligand field theory [1-5] made its appearance
. It was initiated in 1931 when Bethe's cry
stal field theory was proposed. The theory of complex spectra for atoms contributed by Racah
has had an important effect on the deve
lopment of ligand field theory as it has affected that of nuclear and elementary particle theories. In this article, we shall try to give a brief review on the extension of irreducible tensor method to ligand field theory, which has been performed by Tang and his collabor[7-14] ators . As an illustration, the analysis of the spectra of Ή ^ Ο ^ will be presented. I. THE V-COUPLING COEFFICIENTS FROM S0(3) TO OCTAHEDRAL GROUP Let us consider the octahedral group 0 to which a molecule be longs. Since the group 0 is not only a finite group, but also a sub group of the continuous group S0(3) which is further a subgroup of a larger continuous one, as a direct consequence, the continuous group can be employed to give a unified treatment of the ligand field theory with point group symmetry. It seems that the gap between the contin uous group S0(3) and the finite group 0 should be filled in order to make sufficient use of the achievements of both atomic molecular theories in the ligand field theory. For this purpose, the V-coupling
241
coefficients from S0(3) to group 0 have been well defined
[7-9 11] ' as
follows: K
(r U la
v
r )=(-ΐ)*™> Σ P'
h
12b
1 3c
m2
χν
ζ\
l·
m\m2mi\m\ 717273
T
y\
x
'
h z
Πίτ, J|
^^.r^S&r^Sftr*,,
(1)
where the first and the second factors on the right hand side stand for the 3-j symbol and the V-coupling coefficient of the group 0, respectively, and where the transformation coefficient from S0(3) to group 0 can be expressed in terms of inner product of the orbitals between the ligand and central ion <Γγ|jm> and the group overlap integral G J
Cl5] S
m Γγ
=
Ybm>/Gjl(m)
(2)
Under the action of time reversal operator, we have
s£i>=[-i] r -n-iy- m sw
(3)
The V-coupling coefficients from S0(3) to group 0 satisfy the symmetry property such that
K
fc
i2 r2
i3) y= vlj2 rjr {r2
h
r
^33
h
ι,. = ΓΓ|/,
(-ιν+-+--,+Γ2+Γ3
Χ^Γ,Γ,ΓΛΚ^
^
^
(4)
where the factor Θ(Γ 1 Γ 2 Γ3). always equals unity, except 6(T2U'Uf)£ =-1. For practical application to rare earth ions in the field with point group symmetry, the numerical values of the V-coupling coefficients 1 1 from S0(3) to group 0 in the range j = — to j = 12— were evaluated by the use of Elliott-503 computer. Similarly, the V-coupling coefficients from the octahedral group 0 to its subgroup can be defined similarly as we have done in defining the V-coupling coefficients from S0(3) to the group 0, but the former which are quite different from the latter are involved in the classi fication
of the irreducible representation of point groups into A, B
and C species L8—9] #
242
II. LIE COMMUTATIVE RELATION CHARACTERIZED BY OCTAHEDRAL SYMMETRY Let us begin to discuss Lie commutative relation characterized [8-14] by octahedral symmetry
. We use Γ to denote an irreducible re
presentation of the group 0 and S to denote the spin quantum number \ . The creation and the annihilation operators associated with atomic or molecular orbitals with octahedral symmetry are defined in the follow ing forms air(Sr) 10> - I STar)
0„r(Sr)
a n d
| STar) - 10>
(5 )
respectively. By means of the conventional second quantization method, the quantum number q=i with respect to quasispin group SU (2) can be introduced to define
a,ar(,sn - {a_iariqSO ^ ^ - - [ - - i r ^ c s r )
(6)
and consecutively to define a triple tensor operator
JW(rro -
UG0»(«)i(O]*(-D'(-Dp[-H'"e
Σ ββ'αα'ΎΥ'
(q q X\fS S ω\
/Γ Γ
v
κ\
■ W X ° ρϊ W r· ζΐ^^ST)w^n
(7)
As a direct result, Lie commutative relation with octahedral symmetry can be established by writing
V3P3C3
' ~
1
kl
fXlX2Xl\
. {- V r ( - ιγ>+χ<- 0"·+"<+
*Γ,Γ,(- iy>
+
(ωι ω2 ωι\[ΧιΧ2Χι\
(ω1 ω2 ωΛ
Ο Γ , + Γ > ·*(Γ 1 Γ'Λ),»' (Κ; Κ; Κ;)
°<- ο·ι+·3+Γί+Γί e(rirI.1)ie(rlr;«I)iH' ("' * *3)
+ *r,rsC— 0*»+"> ( - ι)''+'·+Γ'·+Γ' θπτ^,ινί'1
*\"')
/^ κ2 κ3\
w ^ ^ m w%e&KrjV>
η^νΥσΛ)
\ i 2 * 1 i 2 7 kill
- 5r;r5(- I / ' ^ C - 1 )»·+"<- i)"+r;+r«0(r;r2Ka);w (** "3 * ) X
243
i 2 i 1 * 2 7 *//7
vSfcftW./)} J
(β)
where (IV.) and {**' } are the 3-j and the b-j symbols, and the V and the W factors are used to denote the V-and the W-coefficients of the octahedral group. For a given electron configuration ( Γ ^ ^ ' ' # r m ) n of octahedral symmetry, we can find out the corresponding group chains using the Lie commutative relation. For illustration, we take (et2) n and (a2t 1 t 2 ) n for example. For configuration (et 2 ) , there are three group chains under the leading group S0(21) L ' S0(2l)
J
D SO(20) Ώ SUQ(2) X \Sp(\0) D SOs(2) X [S0(5) D 50(3) D 0)\
(9)
50(21) D 50(20) D5C/<2(2) X \Sp(\0) D SUS(2) [S0(5) D 0(2) X 0(3) D 0]\
(10 )
50(21) D 50(20) D 50(8) X 50(12) D 50
X (5/7(4) X Sp(6) D SUS(2) X [0(2) X 0(3) D 0]}
{ 1 1 )
For the three group chains, the corresponding wave functions with octa hedral symmetry constitute the bases of three equivalent irreducible representations of SO(21) and therefore the three bases are connected by unitary transformation. It should be pointed out that whether or not the ligands participate in forming chemical bonds with the d-orbitals of the central ion, all of the three group chains hold true. Cer tainly, the group chain (9) describes the original weak-field scheme if the atomic orbitals are used to serve as the irreducible bases. The group chain (11) corresponds to the strong-field scheme. The group chain (10) is indeed an intermediate one between the group chains (9) and (10), so that an intermediate field scheme is proposed, but it must be examined by experimental facts to varify its existence. For configuration (a 2 t^t 2 ) , there are a great number of group chains arising from the Lie commutative relation (8). For brevity, we Ά * 4-u C8-14] only consider one of them SO(29)^SO(28)r>SUQ(2)x{Sp(m)DSUS(2)x[SO(7)r>G2r>SO(3)^>0]}
(12)
This group chain is the well-known one of Racah adapted to octahedral symmetry in quasispin scheme. It should be noted that when the ligands form chemical bonds with the orbitals of the central rare-earth ion, group chain (13) still holds good. 244
III.
MATRIX ELEMENTS OF THE SLJTx SCHEME IN OCTAHEDRAL SYMMETRY
The SLJtx scheme which is regarded as the extension of the Russell-Saunders coupling in atomic theory links with the following group , . [9-14] J chain |50(29)Ι3 50(28)ΐ3 5υ Ω (2)χ{5ρ(14)^5υ 3 (2)χ[50(7)^0 2 -50(3)]^50 α (3)^θ} S
L
J
τ (13)
We shall make a brief discussion on the matrix elements with respect to this group chain in treating rare-earth ions in the field with oc tahedral symmetry. By means of the linear transformation of the triple tensor oper ator given in Eq.(7), a new triple tensor operator W x t (SLJ)
(1Ό
ητ can be obtained to adapt the symmetry of group chain (13) as listed in Table 1. Furthermore, various kinds of interactions in the SLJtx scheme in the ligand field theory can be expressed in terms of the corresponding triple tensor operators as listed in Table 1. For brevity, only the interactions such as ligand field H c and relativistic effect H r are givenC 8 "" 1 ^ 16 ] Hr = Σ B(OKK) W^ A 1(0KK), (15) c
k
Η~ = Σ
0a
B(1KK ! ) W^ A l(lkk f ).
(16)
Application of generalized Wigner-Eckart theorem to the matrix elemen ts of H c and H r in SLJtx scheme gives
δττ'Σ B(okk) / ( 2 J + 1 ) ( 2 J ' + 1 ) k ^ (2S+l)X(t)
' { % l \ V l S : ^ ( -^ Q " M Q -M Q l S'.)<«SLi|w.
(17)
^ a S L J t T l H r l f V s ' L ' J ' t · · ^ = 6tt' δττ' Σ B(l K ' K ) V ( 2 J + 1 ) ( 2 J ' + 1 )
k'k 245
x(t)
Table 1
The
SO(28)
SUQ(2)
Sp(14)
SUS(2)
SO(7)
(00. . . 0 )
0
[00. . . 0 ]
0
(000)
2 (00)
0
[20. . . 0 ]
1
(000)
W (033) (110...0)
0
[20. . . 0 ]
0
W ot ( OLL) ( 1 1 0 . . . 0 )
0
[20. . .0]
0
W
oA,
(000)
Wot ( 1 0 1 ) ( 1 1 0 . . . 0 ) ot
0>
Group Theoretical Classification of W
(SLJ) SO(3)
G
S0
0
.7 0
(00)
0
1
(110)
(10)
3
3
(110)
(11)
1
1
5 W ot ( 1 L J ) ( 1 1 0 . . . 0 ) IA
0
[20. . .0]
1
(200)
(20)
'(000)
(110...0)
1
[00. . .0]
0
(000)
(00)
W'* (OLL)
(110...0)
1
[110...0]
0
(200)
(20)
(1LJ) ( 1 1 0 . . . 0 )
1
[110...0]
1
(110)
(11)
W(t ( 1 3 J ) ( 1 1 0 . . . 0 )
1
[110...0]
1
(110)
(10)
W W
jt
2, 2,
4, 6
J
0
0
4, 6
J
1,
5 3
J J
S S1 1
'L L 1 K 1 \ f
J J
K
J, A\)
<-1)Q-^(_QMQ
J QM.) <«SL |W.(1 K.)| a.S.L.> (18)
k
It is especially noteworthy that the V-coupling coefficients from SO (3) to the group 0 play an important role in the evaluation of the matrix elements in Eqs. (17) and (18), since they extend the results of continuous group to point group symmetry. From Eq. (17), we know that the reduced matrix element introduced by Racah can be further split into the product of 3-j symbol of quasispin group SU^(2) and the reduced matrix elements of W T (ok) which is independent of the number of electrons. It is known that by taking advantage of the 3-j symbol of quasispin group, the complementary rule of matrix elements can be followed directly. IV. THE SPECTRA OF T b P 5 0 1 4 The symmetry of crystal structure of T b P ^ O ^ is of the distorted 0 4 ν . Since C 4 v is equivalent to D^, the matrix elements in Eqs. (17) and (18) can be modified by means of the V-coupling coefficients from oc tahedral group to group D^ to adapt the lower symmetry of TbP^O^^. For the purpose of comparision, the results of theoretical calculations and the data of fluorescent spectra are listed in Table 2[16].
247
Table 2 The Splitting of Terms of 7 Fj for Tb 3 + in the Field 3_ of PsO-^ with Cj^y Symmetry _ _ _ _— Expt. (cm ) Calc. (cm ) Expt. (cm ) Calc. (cm ) 7F0
5914
5907
7F-L
5778
5776
Ai A2
5642
5632
5341
7F2
7F
3
7F4
2502
2489
E
2315
2318
Bl
E
2304
2297
E
5337
E
2239
2245
A2
5324
5332
Bl
2219
2212
5190
5192
Al
2188
2199
Al B2
5073
5079
B
2174
2179
E
4547
4559
2 A2
2170
2156
A2
4515
4510
E
451
449
E
4492
4487
E
414
B
4433
4430
B2
310
A
4413
4420
3691
3707
3640
?F5
7F6
l
Bl
292
293
l A2
166
170
E
3648
Al A2
121
117
3600
3597
E
20
Bl B2
3570
3574
8
E
3524
3528
Al B2
0
Al
3444
3446
E
3426
3410
Bl
248
0
REFERENCES [1] H.A., Bethe, Ann. Phys., 3, 133 (1929). [2] Y. Tanabe, and S. Sugano, J. Phys. Soc. Japan, 9, 753, 766 (1954). [3] J.S. Griffith, "The Theory of Transition-Metal Ions", Cambridge University Press, (1961). [4] B.G. Wybourne, Int. J. Quantum Chem., 7, 1117 (1973). [5] M.K. Kibler and G. Grenet, Int. J. Quantum Chem., 2, 359 (1977). [6] G. Racah, Phys. Rev., 76, 1352 (1949). [7] Research Group on Structure of Matter, Scientiarium Naturalium (Kirin University), 3, 79 (1964); Scientia Sinica, 15, 610 (1966). [8] Research Group on Structure of Matter, "The Summer Physics Collo quium of the Peking Symposium", 1966. [9] Tang Auchin et al., "Theoretical Method of the Ligand Field Theory" Science Press, Peking, (1979). [10] Tang Auchin, Sun Chiachung,and Li Befoo, Int. J. Quantum Chem., 18, 545 (1980). [11] Tang Auchin, Cho Chingyu, and Sun Chiachung, Int. J. Quantum Chem., 18, 557 (1980). [12] Sun Chiachung, Li Beifoo, and Tang Auchin, Int. J. Quantum Chem., 14, 521 (1981). [13] Sun Chiachung, Li Beifoo, and Tang Auchin, Int. J. Quantum Chem., 15, 305 (1981). [14] Sun Chiachung, Han Yande, Li Beifoo and Li Qianshu, Int. J. Quan tum Chem., 23, 169 (1983). [15] Zhang Qianer, Universitatis
Amoiensis Acta Scientiarum Naturalium,
19, 117 (1980). [16] Li Zesheng, M.S., Thesis, Jilin University, (1984).
249