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7 Application: Spin Hamiltonian for the Octahedral Quartet State

7 Application: Spin Hamiltonian for the Octahedral Quartet State

Tải bản đầy đủ - 0trang

7.7 Application: Spin Hamiltonian for the Octahedral Quartet State

185

Table 7.7 Representation matrices for the spinor basis in O ∗

DΓ6 (C4z ) =

√1

2

DΓ8 (Cˆ 4z ) = √1

2

1−i

0

0

1+i

−1 − i

0

0

0

0

1−i

0

0

0

0

1+i

0

0

0

DΓ8 (Cˆ 3 ) =

xyz

1

4

xyz

DΓ6 (C3 ) =

−1 − i

⎜√

⎜ 3(−1 − i)

⎜√

⎜ 3(−1 − i)

−1 − i

1

2

1−i

−1 − i

1−i

1+i

−1 + i

3(−1 + i)

3(1 + i)

0

−1 + i

−1 − i

1−i

3(1 − i)

−1 − i

3(1 + i)

1−i

3(−1 + i) ⎟

3(1 − i) ⎟

−1 + i

that define the components of the Γ8 . In Griffith’s notation, these four components

are defined in the following way:

3

2

3

|U λ ∼

2

3

|U μ ∼

2

3

|U ν ∼

2

|U κ ∼

3

2

1

+

2

1

2

3

2

+

(7.70)

Knowing the symmetries of the components, we can now turn to the coupling

coefficients that describe their interactions. The coupling coefficients that we need

are determined by the Zeeman Hamiltonian, which can be written as

HZe =

μB

B · (L + 2.0023S)

(7.71)

The electronic part of this operator contains the orbital angular momentum and the

spin operator. In octahedral symmetry the overall electronic operator transforms

as T1g . Applying the Wigner–Eckart theorem to the interaction elements in this operator yields

μB

Γ8 i|(L + 2.0023S)|Γ8 j = Γ8 ||T1 ||Γ8

a

+ Γ8 ||T1 ||Γ8

Γ8 i|T1 j Γ8 k

b

a

Γ8 i|T1 j Γ8 k

b

(7.72)

Here, we have introduced the extra labels a and b in order to distinguish that

there are two coupling channels. The coupling coefficients that are required are

of type Γ8k |Γ8i T1j , while the coefficients, as given in Appendix G, are of type

186

7

Spherical Symmetry and Spins

Γ8i T1j |Γ8k and describe the spin-orbit coupling coefficients for the spin-orbit levels of a 4 T1 state. However, since all coefficients in the table are real, turning them

around does not make any difference. Hence, we can directly use the spin-orbit

tables to obtain the Zeeman matrix. Only one of the coupling channels is seen to

link the κ and ν levels. In spherical symmetry this requires a jump of 3 spin units,

which can be bridged only by an = 3 operator. The coupling coefficients of this

channel are thus of spherical octupole parentage ( = 3), while the other set is of

dipole parentage ( = 1). We shall parameterize the corresponding reduced matrix

elements√

as Jf and Jp , respectively. The constant μB / , as well as a common factor of 1/ 15, is also absorbed into these parameters. The electronic operator will

be represented as |Ti |, and the Zeeman Hamiltonian is then recast in complex form

as

HZe = Bx |Tx | + By |Ty | + Bz |Tz |

1

1

= Bz |T0 | − √ (Bx − iBy )|T+1 | + √ (Bx + iBy )|T−1 |

2

2

(7.73)

This expression follows the convention of Eq. (7.39). The operator part is defined

by

|T0 | = |Tz |

1

|T+1 | = − √ |Tx | + i|Ty |

2

1

|T−1 | = √ |Tx | − i|Ty |

2

(7.74)

The elements of the interaction matrix are then given by

Hij = Bz Jp Γ8 i|T0 Γ8 j

p

+ Jf Γ8 i|T0 Γ8 j

1

− √ (Bx − iBy ) Jp Γ8 i|T+1 Γ8 j

2

1

+ √ (Bx + iBy ) Jp Γ8 i|T−1 Γ8 j

2

f

p

+ Jf Γ8 i|T+1 Γ8 j

f

p

+ Jf Γ8 i|T−1 Γ8 j

f

(7.75)

The resulting interaction matrix is given in Table 7.8. Since the Zeeman interaction leads to a splitting of the levels that conserves the barycentre, the secular

equation does not contain odd powers in the energy:

aE 4 + bE 2 + c = 0

(7.76)

7.7 Application: Spin Hamiltonian for the Octahedral Quartet State

187

Table 7.8 Spin Hamiltonian matrix for the octahedral quartet irrep. A common factor of 1/ 15

is absorbed into the J -parameters

|Γ8 κ

Bz

|Γ8 λ

|Γ8 μ

|Γ8 ν

(3Jp − Jf )

Γ8 κ|

(Jp + 3Jf )

Γ8 λ|

−(Jp + 3Jf )

Γ8 μ|

−(3Jp − Jf )

Γ8 ν|

Bx − iBy

|Γ8 κ

|Γ8 λ

Γ8 κ|

|Γ8 μ

|Γ8 ν

3(Jp + 12 Jf )

(2Jp − 32 Jf )

Γ8 λ|

Γ8 μ|

3(Jp + 12 Jf )

− 52 Jf

Γ8 ν|

Bx + iBy

|Γ8 κ

Γ8 κ|

|Γ8 λ

|Γ8 μ

− 52 Jf

3(Jp + 12 Jf )

Γ8 λ|

(2Jp − 32 Jf )

Γ8 μ|

|Γ8 ν

Γ8 ν|

3(Jp + 12 Jf )

The parameters are identified as

a=1

b = −10 Bx2 + By2 + Bz2 Jp2 + Jf2

c = Bx4 + By4 + Bz4 9Jp4 + 48Jp3 Jf + 46Jp2 Jf2 − 48Jp Jf3 + 9Jf4

(7.77)

+ Bx2 By2 + Bx2 Bz2 + By2 Bz2

× 18Jp4 − 144Jp3 Jf + 32Jp2 Jf2 + 24Jp Jf3 + 63Jf4

The parameter b in this expression is isotropic, i.e., it does not depend on the orientation of the magnetic field in the octahedron. On the other hand, the parameter c

contains an anisotropic contribution. It depends on the orientation of the magnetic

field in the octahedron, but symmetry-equivalent orientations must, of course, yield

the same splitting. This means that c is certainly an octahedral invariant and may

thus be written as the sum of the familiar scalar L = 0 and hexadecapolar L = 4

cubic invariants that we derived in Sect. 7.2. We thus write

c = c1 Bx2 + By2 + Bz2

2

+ c2 Bx4 + By4 + Bz4 − 3Bx2 By2 − 3Bx2 Bz2 − 3By2 Bz2

(7.78)

188

7

Spherical Symmetry and Spins

Fig. 7.5 Possible isotropic Zeeman splittings of the octahedral Γ8 spinor irrep

The coefficients in this equation are identified as

c1 = 9Jp4 + 34Jp2 Jf2 − 24Jp Jf3 + 18Jf4

(7.79)

c2 = Jf (4Jp + 2Jf )2 (3Jp − 9/4Jf )

The c2 coefficient is of special interest since it controls the only octahedral term

in the linear Zeeman effect. If this coefficient vanishes, the splitting will be completely isotropic and does not depend on the orientation of the magnetic field in the

octahedron. There are three possible isotropies [13].

• For Jf = 0, the spin operator is strictly dipolar, and the Zeeman Hamiltonian will

induce a regular splitting of the quartet level, which is proportional to the spin

quantum number, MS . This case is illustrated in the left-hand panel of Fig. 7.5.

Such cases will arise for an octahedral 4 A1 state and also for the quartet spin-orbit

level of a 2 T1 state.

• For 4Jp + 2Jf = 0, the matrix splits into two separate 2 × 2 blocks, which

have the same eigenvalues. The splitting pattern is thus as in the central panel

of Fig. 7.5. Such a case can occur for a 2 E state. The orbital part of this state has

no angular momentum, since the corresponding operator is not included in the

direct square: T1 ∈

/ E × E. As a result, the magnetic moment of such a state is

due only to the doublet spin part. Such a state behaves as a pseudo-doublet.

• Finally, for 3Jp − 9/4Jf = 0, the splitting again resembles the Zeeman splitting

of a regular spherical quartet, but the spin labels are interchanged, as compared

with the standard quartet splitting: the ±3/2 levels become the inner levels of

the split manifold, while the ±1/2 levels form the outer levels. This inverted

quartet behavior is observed for 2 A2 and 2 T2 states. The orbital part in these

states reverses the assignment of the fictitious spin levels, as can be seen, for

instance, from the A2 × Γ8 coupling table in Appendix G.

7.8 Problems

189

To conclude, we present the eigenenergies in the notation of Satten [14], who used

parameters g1 and g2 . The reduced matrix elements are expressed as

μB g1 + 9g2

2

10

μB 3(g1 − g2 )

Jf =

2

10

Jp =

(7.80)

Furthermore, the magnetic field is represented by directional cosines as Bz =

Bnz , Bx = Bnx , By = Bny . The four eigenvalues then become

E=±

μB

1 2

1

B

g + 9g22 ± √ (g1 + 3g2 ) 9(g1 − 9g2 )(g1 − g2 )F

2

2 1

4 2

− g12 − 42g1 g2 + 9g22

1/2

1/2

(7.81)

The three isotropic cases are reflected by the zeroes of the three factors preceding

the function F = (n4x + n4y + n4z ).

7.8 Problems

7.1 Find a relationship between the crystal-field potentials of an octahedron and a

cube.

7.2 The product of two rotations is a rotation. Obtain an expression for the Cayley–

Klein parameters of the product as a function of the parameters of its factors.

Is the product commutative? The SU(2) matrices may also be identified as normalized quaternions.

7.3 Work out the group multiplication table for the D3∗ double group and derive the

class structure.

7.4 Consider a set of three eigenlevels transforming as A1 + E in D3 symmetry.

A general matrix for the interaction between the states can be written as

H

A1 |

Ex |

Ey |

|A1

|Ex

|Ey

−2

a − ib

c − id

a + ib

c + id

e − if

e + if

Introduce a fictitious spin operator S˜ that recognizes these states as the components of a triplet spin, S˜ = 1, and consider the spin Hamiltonian

HZe =

μB

g Bz S˜z + g⊥ (Bx S˜x + By S˜y )

Express the a, . . . , f parameters as functions of the two g-parameters.

(7.82)

190

7

Spherical Symmetry and Spins

7.5 In octahedral symmetry the fictitious S˜ operator follows the T1 irrep. Its third

symmetrized power transforms as the components of the f -harmonics and also

subduces a T1 irrep, as indicated in Table 7.1. Rewrite the p- and f -parts of

the |Ti | operators for the Γ8 quartet state as a spin Hamiltonian of the fictitious

spin.

References

1. Altmann, S.L.: Rotations, Quaternions, and Double Groups. Clarendon Press, Oxford (1986)

2. Butler, P.H.: Point Group Symmetry Applications, Methods and Tables. Plenum Press, New

York (1981)

3. Edmonds, A.R.: Angular Momentum in Quantum Mechanics, 4th edn. Princeton University

Press, Princeton (1996)

4. Gilmore, R.: Lie Groups, Physics, and Geometry: An Introduction for Physicists, Engineers

and Chemists. Cambridge University Press, Cambridge (2008)

5. Ceulemans, A., Compernolle, S., Lijnen, E.: Hiatus in the spherical shell model of fullerenes.

Phys. Chem. Chem. Phys. 6, 238 (2004)

6. Troullier, N., Martins, J.L.: Structural and electronic properties of C60 . Phys. Rev. B 46, 1754

(1992)

7. Condon, E.U., Odaba¸si, H.: Atomic Structure. Cambridge University Press, Cambridge

(1980)

8. Griffith, J.S.: The Theory of Transition-Metal Ions. Cambridge University Press, Cambridge

(1961)

9. Opechowski, W.: Sur les groupes cristallographiques “doubles”. Physica 7, 552 (1940)

10. Haake, F.: Quantum Signatures of Chaos, 3rd edn. Springer Series in Synergetics. Springer,

Berlin (2010)

11. Wigner, E.P.: Group Theory. Academic Press, New York (1959)

12. Abragam, A., Bleaney, B.: Electron Paramagnetic Resonance of Transition Ions p. 656.

Clarendon Press, Oxford (1970)

13. Ceulemans, A., Mys, G., Walỗerz, S.: Symmetries of time-odd electronphonon coupling in

the cubic Γ8 system. New J. Chem. 17, 131 (1993)

14. Satten, R.A.: Vibronic splitting and Zeeman effect in octahedral molecules: odd-electron systems. Phys. Rev. A 3, 1246 (1971)

Appendix A

Character Tables

Contents

A.1

A.2

Finite Point Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

C1 and the Binary Groups Cs , Ci , C2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The Cyclic Groups Cn (n = 3, 4, 5, 6, 7, 8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The Dihedral Groups Dn (n = 2, 3, 4, 5, 6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The Conical Groups Cnv (n = 2, 3, 4, 5, 6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The Cnh Groups (n = 2, 3, 4, 5, 6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The Rotation–Reflection Groups S2n (n = 2, 3, 4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The Prismatic Groups Dnh (n = 2, 3, 4, 5, 6, 8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The Antiprismatic Groups Dnd (n = 2, 3, 4, 5, 6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The Tetrahedral and Cubic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The Icosahedral Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Infinite Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Cylindrical Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Spherical Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

192

192

192

194

195

196

197

198

199

201

202

203

203

204

Character tables were introduced to chemistry through the pioneering work of

Robert Mulliken [1]. The book on “Chemical Applications of Group Theory” by

F. Albert Cotton has been instrumental in disseminating their use in chemistry [2].

Atkins, Child, and Phillips [3] produced a handy pamphlet of the point group character tables.1

1 In the tables the columns on the right list representative coordinate functions that transform according to the corresponding irrep. The symbols Rx , Ry , Rz stand for rotations about the Cartesian

directions.

A.J. Ceulemans, Group Theory Applied to Chemistry, Theoretical Chemistry and

Computational Modelling, DOI 10.1007/978-94-007-6863-5,

191

192

A Character Tables

A.1 Finite Point Groups

C1 and the Binary Groups Cs , Ci , C2

C1

A

1

Cs

σˆh

A

A

1

1

1

−1

Ci

ıˆ

Ag

Au

1

1

1

−1

C2

Cˆ 2z

A

B

1

1

1

−1

x 2 , y 2 , z2 , xy

yz, xz

x, y, Rz

z, Rx , Ry

Rx , Ry , Rz

x, y, z

x 2 , y 2 , z2 , yz, xz, xy

x 2 , y 2 , z2 , xy

yz, xz

z, Rz

x, y, Rx , Ry

The Cyclic Groups Cn (n = 3, 4, 5, 6, 7, 8)

C3

Cˆ 3

Cˆ 32

A

1

1

1

1

1

¯

C4

Cˆ 4

Cˆ 2

Cˆ 43

A

B

1

1

1

1

1

−1

i

−i

1

1

−1

−1

1

−1

−i

i

E

E

¯

= exp(2πi/3)

z, Rz

x 2 + y 2 , z2

(x, y)(Rx , Ry )

(x 2 − y 2 , xy)(yz, xz)

z, Rz

x 2 + y 2 , z2

x 2 − y 2 , xy

(x, y)(Rx , Ry )

(yz, xz)

A.1 Finite Point Groups

193

C5

Cˆ 5

Cˆ 52

Cˆ 53

Cˆ 54

A

1

1

1

1

1

2

1

¯2

1

¯

¯2

2

E1

¯

¯

2

= exp(2πi/5)

1

1

C6

Cˆ 6

Cˆ 3

Cˆ 2

Cˆ 32

Cˆ 65

A

B

1

1

1

1

1

−1

1

−1

−1

−1

1

1

−¯

1

−1

¯

¯

1

1

−¯

E2

1

1

−¯

−¯

1

1

−¯

−¯

C7

Cˆ 7

Cˆ 72

Cˆ 73

Cˆ 74

Cˆ 75

Cˆ 76

A

1

1

1

1

1

1

1

¯2

1

¯

2

E1

¯

¯2

2

3

1

¯3

¯2

¯3

3

2

¯3

¯

¯2

3

¯

E2

1

1

E3

1

1

¯3

C8

Cˆ 8

Cˆ 4

A

B

1

1

1

1

1

−1

1

1

1

1

i −1

−i −1

E2

1

1

i

−i

−1

−1

E3

1

1

−¯

i

−i

E1

3

¯

x 2 + y 2 , z2

(x, y)(Rx , Ry )

(yz, xz)

¯2

E2

E1

z, Rz

¯

¯

2

¯2

¯2

2

Cˆ 2

Cˆ 43

3

¯2

¯3

2

= exp(2πi/6)

z, Rz

x 2 + y 2 , z2

(x, y)(Rx , Ry )

(yz, xz)

(x 2 − y 2 , xy)

= exp(2πi/7)

z, Rz

x 2 + y 2 , z2

(x, y)(Rx , Ry )

(yz, xz)

(x 2 − y 2 , xy)

¯3

¯

Cˆ 83

[x(x 2 − 3y 2 ),

y(3x 2 − y 2 )]

3

Cˆ 85

Cˆ 87

= exp(2πi/8)

1

1

1

1 z, Rz

1 −1 −1 −1

−i −¯ −

¯

(x, y)(Rx , Ry )

i − −¯

1 −1

1 −1

−1

−1

(x 2 − y 2 , xy)

2

−i

i

−i

i

¯

x 2 + y 2 , z2

(yz, xz)

i

−i

−i

i

(x 2 − y 2 , xy)

¯

−¯

[x(x 2 − 3y 2 ),

y(3x 2 − y 2 )]

194

A Character Tables

The Dihedral Groups Dn (n = 2, 3, 4, 5, 6)

D2

Cˆ 2z

y

Cˆ 2

Cˆ 2x

A

B1

B2

B3

1

1

1

1

1

1

−1

−1

1

−1

1

−1

1

−1

−1

1

D3

2Cˆ 3

3Cˆ 2

A1

A2

E

1

1

2

1

1

−1

1

−1

0

Cˆ 2 (= Cˆ 42 )

z, Rz

y, Ry

x, Rx

x 2 + y 2 , z2

z, Rz

(x, y)(Rx , Ry )

2Cˆ 2

2Cˆ 2

1

−1

1

−1

0

1

−1

−1

1

0

D4

2Cˆ 4

A1

A2

B1

B2

E

1

1

1

1

2

1

1

−1

−1

0

D5

2Cˆ 5

2Cˆ 52

A1

A2

E1

E2

1

1

2

2

1

1

2 cos(2π/5)

2 cos(4π/5)

1

1

2 cos(4π/5)

2 cos(2π/5)

D6

2Cˆ 6

2Cˆ 3

Cˆ 2

3Cˆ 2

3Cˆ 2

A1

A2

B1

B2

E1

E2

1

1

1

1

2

2

1

1

−1

−1

1

−1

1

1

1

1

−1

−1

1

1

−1

−1

−2

2

1

−1

1

−1

0

0

1

−1

−1

1

0

0

1

1

1

1

−2

x 2 , y 2 , z2

xy

xz

yz

(xz, yz)(x 2 − y 2 , xy)

x 2 + y 2 , z2

z, Rz

(x, y)(Rx , Ry )

x2 − y2

xy

(xz, yz)

5Cˆ 2

1

−1

0

0

x 2 + y 2 , z2

z, Rz

(x, y)(Rx , Ry )

(xz, yz)

(x 2 − y 2 , xy)

x 2 + y 2 , z2

z, Rz

(x, y)(Rx , Ry )

x(x 2 − 3y 2 )

y(3x 2 − y 2 )

(xz, yz)

(x 2 − y 2 , xy)

A.1 Finite Point Groups

195

The Conical Groups Cnv (n = 2, 3, 4, 5, 6)

C2v

Cˆ 2z

σˆ vxz

σˆ v

A1

A2

B1

B2

1

1

1

1

1

1

−1

−1

1

−1

1

−1

1

−1

−1

1

C3v

2Cˆ 3

3σˆ v

A1

A2

E

1

1

2

1

1

−1

1

−1

0

yz

z

Rz

(x, y)(Rx , Ry )

C4v

2Cˆ 4

Cˆ 2

2σˆ v

2σˆ d

A1

A2

B1

B2

E

1

1

1

1

2

1

1

−1

−1

0

1

1

1

1

−2

1

−1

1

−1

0

1

−1

−1

1

0

C5v

2Cˆ 5

2Cˆ 52

5σˆ v

A1

A2

E1

E2

1

1

2

2

1

1

2 cos(2π/5)

2 cos(4π/5)

1

1

2 cos(4π/5)

2 cos(2π/5)

1

−1

0

0

C6v

2Cˆ 6

2Cˆ 3

Cˆ 2

3σˆ v

3σˆ d

A1

A2

B1

B2

E1

E2

1

1

1

1

2

2

1

1

−1

−1

1

−1

1

1

1

1

−1

−1

1

1

−1

−1

−2

2

1

−1

1

−1

0

0

1

−1

−1

1

0

0

x 2 , y 2 , z2

xy

xz

yz

z

Rz

x, Ry

y, Rx

x 2 + y 2 , z2

(x 2 − y 2 , xy)(xz, yz)

z

Rz

(x, y)(Rx , Ry )

x 2 + y 2 , z2

x2 − y2

xy

(xz, yz)

z

Rz

(x, y)(Rx , Ry )

x 2 + y 2 , z2

z

Rz

x 2 + y 2 , z2

(x, y)(Rx , Ry )

(xz, yz)

(x 2 − y 2 , xy)

x(x 2 − 3y 2 )

y(3x 2 − y 2 )

(xz, yz)

(x 2 − y 2 , xy)

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7 Application: Spin Hamiltonian for the Octahedral Quartet State

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