1

3. Electrons, states, energy levels

2

Single electron spatial quantum numbersn: principal (radial) quantum number

l: orbital angular momentum (new azimuthal) quantum numberml: magnetic quantum number

For one electron atoms, wavefunctions characterized by even l have even parity:

Pi |nlml> = (-1)l |nlml>

H&I Ch. 2 http://www.chemtube3d.com/orbitals-f.htm

http://media-2.web.britannica.com/eb-media/12/7512-004-91DF97CB.gif

http://winter.group.shef.ac.uk/orbitron/AOs/4f/index.html

3

Some basic principles

Eigenvalue equation2 2

2[ ( )] ( ) ( )

2

dV x x E x

m dx

operator H Hamiltonian

energy

wave function

matrix elements: *A | A |b a a bdx

expectation value

* |b a a b abdx 1 if a = b; 0 if a ≠ b

* H |a a a a a adx E E diagonal matrix energies are state energies

orthonormality

4

Perturbation theory

describes how state a is perturbed by another state b, giving the corrected state a’. States must be of same symmetry, and mixing is inversely proportional to energy difference.

describes how the Hamiltonian describing state a is changed by a perturbation. Example: a Hamiltonian H0 has the perturbation V which changes the energies Ei and wavefunctions ni.

H = H0 + V

http://en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics)

5

Single electron spin quantum numbers: electron spin ½

ms: magnetic quantum number, ±1/2No two electrons in an atom can have the same values of these four quantum numbers

Ce3+ 4f1 - a configuration shows electron structure

Pr3+ 4f2 Electrons are indistinguishablethese 2 are the same stateno Kramers degeneracy for even electron systems

odd electron systems have Kramers degeneracy: 2 different states with same energy, in absence of magnetic field.

E

E

6

Spin orbit coupling: j = l + s

E

|nlsmlms> or |nlsjmj>

For light atoms, l and s are good quantum numbers

= an unique value can specify a state

For lanthanides, only the total angular momentum is (nearly) a good quantum number

|nls j=l+1/2 mj>

|nls j=l-1/2 mj>

[(2l+1)/2] ζ

e.g. l=3, s=1/2 splits into

j = 5/2 and j = 7/2.

7

Many electron systems

Coulomb interaction (e-e repulsion) splits 4fN configuration into LS multiplet terms, whilst spin-orbit coupling (SOC) splits terms into J multiplets.

2S+1LJ : total degeneracy 2J+1 (or J+1/2, Kramers systems)

Example: 3P splits into 3P0, 3P1, 3P2 since S and L can vector couple in different ways.

http://www.ias.ac.in/j_archive/currsci/51/19/934-936/viewpage.html

8

J multiplets To find the multiplet terms for a given 4fN

configuration: consider microstates

|l ml s ms>

easy for 4f1 Ce3+ l = 3; s = ½; j = 3/2 or 5/2 gives 2F7/2 and 2F5/2.

more complicated for many electron systems, e.g. 4f2 Pr3+

ml1 = -3…+3; ms1 = ±½

ml2 = -3…+3; ms2 = ±½(no 4 quantum numbers can be the same for 1 and 2)

http://www.astro.sunysb.edu/fwalter/AST341/qn.html

http://books.google.com/books?id=VLIUnG9YimMC&pg=PA31&lpg=PA31&dq=multiplet+terms+of+configuration&source=bl&ots=_4O6IYjVfa&sig=pkilI6xDB6ttJunosC9fLlvX7W8&hl=zh-TW&ei=hY_7S5rXCYHk7AObhJki&sa=X&oi=book_result&ct=result&resnum=5&ved=0CCcQ6AEwBDgU#v=onepage&q=multiplet%20terms%20of%20configuration&f=false

9

ml ML MS Term -3 -2 -1 0 1 2 3 + - -6 0 1I + - -5 0 +- -4 0 + - -3 0 +- -2 0 + - -1 0 +- 0 0 + - 1 0 +- 2 0 + - 3 0 +- 4 0 + - 5 0 +- 6 0 -+ 0 0 1S -+ 2 0 1D - + 1 0 - + 0 0 - + -1 0 -+ -2 0 4 0 1G 3 0 2 0 1 0 0 0 -1 0 -2 0 -3 0 -4 0

4f2 Pr3+

has 91 microstates

(14×13)/(1×2)

can you fill in

3F

3P

3H

ms written as + or -

Microstates of Pr3+

10

Clebsch-Gordon coefficients

When two angular momentum states are coupled, the new states can be expressed as the coupled representation:

1 2

1 1 2 2

1 2 1 1 2 2 1 1 2 2| ( ) | |j j

m j m j

j j j m j m JJM j m j m M

The Clebsch-Gordon coefficient is related to a 3-j symbol:

1 2 1 21 1 2 2

1 2

| ( 1) 2 1j j M j j Jj m j m JM J

m m M

3-j symbols have special properties, including:

invariant to even permutation of columns

m1+m2=M

j1+j2+J integer

triangle rule: (j1-j2)≤ J ≤ (j1+j2)

http://en.wikipedia.org/wiki/Table_of_Clebsch-Gordan_coefficients

11

Example of Clebsch-Gordon coefficients

or abbreviated:

Example for j1 =1/2; j2 = 1/2

H&I, page 134

12

Electronic ground state

Hund’s rule tells us the ground state multiplet.

1. Terms with maximum spin multiplicity (2S+1) has lowest energy.

2. Then choose the one with largest L.

3. For less than, or half-filled shells: choose the one with smallest J. For more than half-filled, choose largest J.

Ce

Yb

Pr

Tm

Nd Er

Pm Ho

Sm Dy

Eu Tb

Gd

4fN 1 2 3 4 5 6 7

SL 1 7 17 47 73 119 119

SLJ 2 13 41 107 198 295 327

Γ 14 91 364 1001

2002

3003

3432

http://en.wikipedia.org/wiki/List_of_Hund's_rules

e.g. Multiplets for Pr3+ and Tm3+ are 1S0, 3P0,1,2， 1I6, 1D2, 1G4, 3F4,3,2, 3H4,5,6

13

Crystal field

The environment of Ln3+ is different in a crystal than in the free ion since the symmetry is lower.

The crystal field (CF) = all surrounding charges, multipoles, etc.

The descending symmetry causes splitting of degenerate levels.

The SOC interaction is of greater magnitude than the CF interaction for Ln3+ 4fN configurations.

14

Crystal field levels

The split J levels are described by irreducible representations (irreps) of the site symmetry point group of Ln3+.

We can determine the symmetry irreps of a given J if we know the site symmetry, but we do not know the energy level ordering without detailed calculation.

Ch. 3 HI

15

Character table and irreps

http://www.webqc.org/symmetry.php

http://en.wikipedia.org/wiki/List_of_character_tables_for_chemically_important_3D_point_groups

http://www.cryst.ehu.es/rep/point.html

http://www.webqc.org/symmetrypointgroup-td.html

16

Character table for Oh molecular point group

Oh (m3m) E 8C3 6C2 6C4 3C2 i 6S4 8S6 3h 6d

1g A1g 1 1 1 1 1 1 1 1 1 1 x2+y2+z2 2g A2g 1 1 -1 -1 1 1 -1 1 1 -1 3g Eg 2 -1 0 0 2 2 0 -1 2 0 (2z2-x2-y2, x2-y2) 4g T1g 3 0 -1 1 -1 3 1 0 -1 -1 (Rx,Ry, Rz) 3g T2g 3 0 1 -1 -1 3 -1 0 -1 1 xz, yz, xy 1u A1u 1 1 1 1 1 -1 -1 -1 -1 -1 2u A2u 1 1 -1 -1 1 -1 1 -1 -1 1 3u Eu 2 -1 0 0 2 -2 0 1 -2 0 4u T1u 3 0 -1 1 -1 -3 -1 0 1 1 (x, y, z) 5u T2u 3 0 1 -1 -1 -3 1 0 1 -1

Multiplication Table for O and O* molecular point group (Oh = O Ci)

O A1 A2 E T1 T2 E (6) E (7) U (8)

A1 (1) A1 A2 E T1 T2 E E U A2 (2) A2 A1 E T2 T1 E E U E (3) E E A1+A2+E T1+T2 T1+T2 U U E+E+U T1 (4) T1 T2 T1+T2 A1+E+T1+T2 A2+E+T1+T2 E+U E+U E+E+2U T2 (5) T2 T1 T1+T2 A2+E+T1+T2 A1+E+T1+T2 E+U E+U E+E+2U E (6) E E U E+ U E+ U A1+T1 A2+T2 E+T1+T2 E (7) E E U E+ U E+ U A2+T2 A1+T1 E+T1+T2

U (8) U U E+E+U E+E+2U E+E+2U E+T1+T2

E+T1+T2

A1+A2+E+2T1+2T2

17

χ(φ) = [sin(J+1/2)φ]/[sin(φ/2])

e.g. for C2, φ = π;

For E,

χ(φ)

= lim φ→0[sin(J+1/2)φ]/[sin(φ/2])

=(J+1/2)φ]/[(φ/2])=(2J+1)

Crystal field irreps from J-values

18

Theoret. Chim. Acta 74 (1988) 219 http://www.springerlink.com/content/n02ml8613m5xl82h/fulltext.pdf

19

Bases for irreps in Oh

Γ1

J=0: |0>J=4: (24)-1/2 [(14)1/2 |0> + (5)1/2 (|4>+|-4>)]J=6: (1/4)[(2)1/2 |0>- (7)1/2(|4>+|-4>)]

Γ2

J=3: (2)-1/2(|2>-|-2>)J=6: (32)-1/2[(5)1/2(|6>+|-6>)- (11)1/2(|2>+|-2>)]etc.

20

Calculation of energy levels of Ln3+

1. ab initio methods accurate to hundreds thousands of cm-1

2. Semi-empirical methods accurate to 5-50 cm-1.

A. Reid, Eur. J. Inorg. Chem. doi:10.1002/ejic.20100182

Calc A Calc B

B.

21

Calculation of energy levels of Ln3+

Need only to consider 4fN electrons since electrons in closed shells form spherical charge distribution which do not result in energy splitting - only a shift.

(H – Ei)Ψi = 0 (1)

3. Diagonalize matrix using trial parameter values to get agreement between calculated and observed energy levels. Wavefunctions generated can be tested by calculations of spectral intensities, etc. Wybourne book

1. Choose all possible arrangements of 4fN as bases.

2. Calculate the Hamiltonian matrix.

22

Basics of energy level calculations

1 1 2 2( ) ( ) ..k kk

x a x a a

abbreviate 1 1 11| H | H

Eigenvalue equation is 11 12 1 1

21 22 2 2

H H

H H

a aE

a a

or simply: 11 12 1

21 22 2

H H0

H H

E a

E a

(just consider two terms)

then solve for the 2 energies E1, E2

Eigenvectors a1, a2 must be orthonormal

For a Ln3+ system:

23

Electronic Hamiltonian and states of 4fN systems

The electronic states of rare earth ions in crystals are N-body localized states, since the N electrons of 4fN are coupled strongly, and move around the corresponding ion core without extending far away. The semiempirical calculations for the 4fN

energy level systems employ a parametrized hamiltonian H under the appropriate site symmetry for Ln3+:

H = HAT + HCF + HADD (2)

where HAT comprises the atomic hamiltonian, which includes all interactions which are spherically symmetric; HCF is the operator comprising the nonspherically symmetric CF; HADD contains other interactions.

Crosswhite J. Opt. Soc. Am. B 1 (1984) 246

24

Pr3+ energy levels

Free ion: Coulomb repulsion

Spin-orbit coupling

Further smaller splittings due to crystal field

H&I p. 398

major interactions for Ln3+

are Coulomb and SOC

25

SOC and CFTransition metal d-electron systems: CF > SOCLanthanide 4fN: SOC > CFH = HCoul(ff) + Hso(f) + Hcf(f)+....Most important interactions aree-e repulsion > SOC > CFi.e. weak CF

Do not judge by parameter values:

But by energy splittings, e.g. Ce3+ 4f1 levels in Oh symmetryn.b. for Ce3+, 4f1, no Coulomb interaction 2F5/2

2F7/2

7

8

7

8

6

0 cm-1

570 cm-1

2160 cm-1

2661 cm-1

3048 cm-13048

2661

2160

570

0

Energy (cm-1)

Label states 2S+1LJ

(7/2) ×ζ4f

26

The atomic Hamiltonian:

k ifk

kAVAT RGFE )(γG)(βG1)αL(LlsfH 72ii

t p ms k js k j

s k j

T P M (3)

First term EAV (containing F0) adjusts the configuration barycentre energy with respect to

other configurations. Usually chosen to put ground state at E = 0.

Slater parameters Fk represent the electron-electron repulsion interactions and are two-

electron radial integrals, where the fk represent the angular operator part of the interaction.

Spin orbit magnetic coupling constant, f, controls the interaction and the mixing of states

from different SL terms with the same J manifold. These two parameters are the most

important in determining the atomic energies.

Note that the terms f

N

if Neie 44 )( involving the kinetic energy of the electrons and their

nuclear attraction do not give rise to a splitting of the 4fN levels and are contained in EAV.

27

Other interactions

The two-body configuration interaction parameters α, β, γ parametrize the second-order Coulomb interactions with higher configurations of the same parity.

For fN and f14-N, N>2 the three body parameters Ts (s = 2,3,4,6,7,8) are employed to represent Coulomb interactions with configurations that differ by only one electron from fN.

With the inclusion of these parameters, the free ion energy levels can usually be fitted to within 100 cm-1.

The magnetic parameters Mj (j = 0,2,4) describe the spin-spin and spin-other orbit interactions between electrons, and the electrostatically correlated spin-orbit interaction Pk (k = 2,4,6) allows for the effect of additional configurations upon the spin-orbit interaction. Usually the ratios M0:M2:M4 and those of P2:P4:P6 are constrained to minimize the number of parameters, which otherwise already total 20.

28

Crystal field parameters CFP

The HCF operator represents the nonspherically symmetric components of

the one-electron CF interactions, i.e. the perturbation of the Ln3+ 4fN electron

system by all the other ions. The general form of the CF hamiltonian HCF is

given by:

'0 0

; 0 1

H C ( ) ( (C ( ) ( 1) C ( )) (C ( ) ( 1) C ( )))k

k k k k q k k k q kCF q q q q q q

i k q

B i B i i jB i i

(4)

where the kqB are parameters and the )(C ik

q are tensor operators (rank k, with q =

k, k-1,..., -k) related to the spherical harmonics, and the sum on i is over all

electrons of the 4fN configuration.

Garcia, Handbook on Phys. Chem. Rare Earths 21 (1995) 263

Gőrller-Walrand, Handbook on Phys. Chem. Rare Earths 23 (1996) 121

29

Crystal field parameters CFP

• Since the hamiltonian is a totally symmetric operator, only those values of k whose angular momentum irreps transform as the totally symmetric representation of the molecular point group are included.

• In C1 symmetry: 27 CFP• In higher symmetries, the angular interactions cancel. • For atoms, spherical symmetry, no CFP.• For LnCl63- systems, Oh symmetry, 2 CFP.

30

CFP in higher symmetryFor Oh symmetry, values of k whose angular momentum irreps contain Γ1g give nonzero k

qB , i.e. k = 4 and 6. The nonzero CF parameters for the Oh group

reduce to .,,, 64

60

44

40 BBBB The unit tensor normalized CFP )(k

qB (used by

Richardson) are related to the spherical tensor CFP kqB (used by Wybourne)

by:

000

33)7(|C| )()( k

BffBB kq

kkq

kq (5)

so that )4(0B = 1.128 4

0B , and )6(0B = -1.277 6

0B . However, the kB4 parameters are

related to kB0 , and in the Wybourne spherical tensorial notation herein, and in the

cubic environment, 44B = 4

0B (5/14)1/2 and 64B = 6

0B (7/2)1/2. Thus only 2

additional parameters are required to model the CF splittings of J terms by the octahedral CF, and this presents a more severe test for theory than for low symmetry systems.

31

Theoretical analysis More convincing for LnX6

3- systems because fewer independent crystal field

parameters are involved.

Pr3+ in Cs2NaLnCl6 (Oh): 40B and 6

0B (J. Chem. Phys. 114 (2001) 10860)

Pr3+ in YPO4 (D3d): 20B , 4

0B , 44B , 6

0B , 64B

(J. Alloys. Compds. 323-324 (2001) 783)

Pr3+ in La2O3 (C3v): 20B , 4

0B , 43B , 6

0B , 63B , 6

6B (J. Lumin. 85 (1999) 59)

Pr3+ in Pr(trensal) (C3): 20B , 4

0B , 43B , 6

0B , 63B , 6

6B (Inorg. Chem. 41 (2002) 5024)

32

4f1 and 4f13 systems

These are simple because only one electron (or one hole) so no Coulomb repulsion.

33

Crystal field analysis of 4f13

Inorg. Chem. 16 (1976) 1694

647

648

647

648

646

2

7

3

442

7

3

22

1218

162

2014

bb

bb

bb

bb

bb

606

404

64

64

6

8

7

429

533

1

8410

568

2

3

4

35

4

3

54

3

4

1

2

13

30

Bb

Bb

bb

bb

J. Chem. Phys. 94 (1991) 942

2F5/2

2F7/2

6

87

87 Different notations employed

4 60 0, , ,aveE B B

we use:

34

Crystal field analysis of 4f13

2

1

2

7|

12

5

2

7

2

7|

12

56

(Equation 2)

(Equation 1)

2

3

2

7|

2

3

2

5

2

7|

2

1

2

5

2

5|

6

5

2

3

2

5|

6

1

8

2

3

2

7|

2

1

2

5

2

7|

2

3

2

3

2

5|

6

5

2

5

2

5|

6

1

7

JMJtcoefficien |

Griffith, The Theory of Transition –Metal Ions; Cambridge University Press, 1961

35

2

3

429

20

11

2 60

40 BB

2

3

312

25

77

25

143

205

77

4

5143

205

77

42

21

2

60

40

60

40

60

40

40

BBBB

BBB

2

3

143

20

77

18

3143

120

377

203143

120

377

202

21

4

60

40

60

40

60

40

40

BBBB

BBB

Γ7:

Γ8:

Γ6:

Energy matrices of 4f13

Crystal field analysis of 4f13

Take advantage of symmetry factorization

36

0

200

400

600

800

9000

9500

10000

10500

250

10718

10248

572

En

erg

y (c

m-1)

0

exp.

6076 10691

5662 10277

-4021 594

-4382 233

-4615 0

cal.

Cs2NaYbCl

6

Comparison of experimental and calculated energy levels of Cs2NaYbCl6.

The published parameters of , and were used40B 6

0B

J Alloys Compds 215 (1994) 349

37

Anticipate the effect of crystal field parameters on the energy gaps.

2

3

312

25

77

25

143

205

77

4

5143

205

77

42

21

2

60

40

60

40

60

40

40

BBBB

BBB

2

3

143

20

77

18

3143

120

377

203143

120

377

202

21

4

60

40

60

40

60

40

40

BBBB

BBB

Γ7:

Γ8:

The effect of on Γ7 should be more prominent than that on Γ8 .

Crystal field analysis of 4f13

40B

The effect of on Γ8 might be more prominent than that on Γ7 .60B

38

-200 -100 0 100 200

0.0

0.2

0.4

0.6

0.8

1000 1500 2000

0.0

0.2

0.4

0.6

0.8

10.2

10.4

10.6

10.8

10.2

10.4

10.6

10.8

2F7/2

6

2F7/2

6

2F7/2

7

2F7/2

7

2F7/2

8

2F7/2

8

2F5/2

8

2F5/2

8

Ene

rgy

(10

3cm

-1)

2F5/2

7

B4

0

2F5/2

7

B6

0

Energy against crystal parameters of Cs2NaYbCl6, calculated by f-shell program.

Crystal field analysis of 4f13

39

Fitting programs

• Prof. M. Reid: f shell programs, from Prof. F.S. Richardson’s group.

• Profs. Edvardsson, Åberg:

http://cpc.cs.qub.ac.uk/summaries/ADMZ

• Dr. Michèle Faucher: ATOME

40

Determination of site symmetry molecular point group of Ln3+

Need published crystal structure.1. Use International Tables of Crystallography: Vol 4A, Space

Group symmetry: Hermann-Mauguin notation.

41

2. Use Appl. Spectrosc. 25 (1971) 155: Schoenflies notation.

http://neon.otago.ac.nz/chemlect/chem203/symmetrylectures/molecularsymmetry.pdf

number of equivalent atoms in Bravais cell

number of different kinds of site with this symmetry

42

What use are the parameters from energy level calculations?

If useful:

1A. Energy level fitting accurately reproduce the experimental data set.

1B Predict missing or unexplored energy levels.

1C. Energy level dataset representative (i.e. extending over a wide range) and fairly complete.

1D. Wave functions resulting from the parametrization should be capable of accurately predicting other properties such as g-factors and spectral intensities.

Duan, J. Phys. Chem. A 114 (2010) 6055

43

What use?

2.Parameters expected to show some type of systematic variation for materials comprising a series of closely-related elements.

3. Parameters should be related to other physical quantities in a systematic manner. Also the parameters should show explicable trends over various crystal hosts for a particular ion.

44

Critical test: Cs2NaLnCl6

1. Ln3+ in octahedral symmetry: only 2 CFP

2. Representivity (100 × Nexp/Ntotal) of the dataset does vary considerably for different Ln3+, being 100% for Ce3+, Yb3+; over 90% for Pr3+, Tm3+, but much less for the more extensive 4fN configurations, such as 3% for Gd3+.

3. Standard deviations of most fits are around 20 cm-1.

45

Results: Prediction of energy levels No. Mult. Irrep Ecalc

1 5I 4 T2 0 2 5I 4 E 245 3 5I 4 T1 247 4 5I 4 A1 257 5 5I 5 T1 1578 6 5I 5 E 1607 7 5I 5 T2 1633 8 5I 5 T1 1707 9 5I 6 A1 3151

10 5I 6 T1 3173 11 5I 6 T2 3197 12 5I 6 A2 3226 13 5I 6 T2 3278 14 5I 6 E 3282 15 5I 7 T1 4804 16 5I 7 E 4843 17 5I 7 T2 4860 18 5I 7 A2 4884 19 5I 7 T1 4981 20 5I 7 T2 4988 21 5I 8 E 6480 22 5I 8 T2 6480 23 5I 8 T2 6535 24 5I 8 T1 6591 25 5I 8 A1 6793 26 5I 8 T1 6828 27 5I 8 E 6840 28 5F 1 T1 12241 29 5F 2 E 12544 30 5F 2 T2 12782 31 5F 3 T1 13462 32 5F 3 T2 13502 33 5F 3 A2 13723 34 5S 2 E 14038 35 5S 2 T2 14044 36 5F 4 A1 14391 37 5F 4 T1 14446 38 5F 4 T2 14464 39 5F 4 E 14486 40 5F 5 T1 15626 41 5F 5 E 15674 42 5F 5 T2 15824 43 5F 5 T1 15908 44 3K2 6 E 16814 45 3K2 6 T2 16821 46 3K2 6 A2 16844 47 3K2 6 A1 16847 48 3K2 6 T1 16860 49 3K2 6 T2 16867 50 5G 2 E 17221

No. Mult. Irrep Ecalc

51 5G 2 T2 17339 52 3H4 4 T2 17519 53 5G 3 T1 17809 54 3H4 4 E 17827 55 5G 3 A2 17837 56 5G 3 T1 18016 57 3K2 7 T2 18021 58 5G 4 A1 18049 59 3K2 7 A2 18078 60 5G 3 T1 18128 61 3K2 7 T2 18149 62 3K2 7 T2 18180 63 3K2 7 T1 18195 64 3K2 7 E 18195 65 3K2 8 E 19603 66 3K2 8 T2 19640 67 3K2 8 T1 19644 68 3K2 8 T2 19653 69 3K2 8 E 19697 70 3K2 8 A1 19701 71 3K2 8 T1 19732 72 3K2 8 E 19751 73 5G 5 T1 19762 74 5G 4 T2 19839 75 5G 5 T1 19885 76 5G 4 T2 19896 77 5G 4 A1 20176 78 5G 4 T1 20178 79 5G 4 E 20180 80 3G2 3 A2 21440 81 3G2 3 T2 21540 82 3G2 3 T1 21541 83 5G 5 E 21777 84 5G 6 T2 21827 85 5G 5 T1 21919 86 5G 6 E 21994 87 5G 6 T2 22020 88 5G 5 T1 22023 89 5G 6 A2 22044 90 5G 6 A1 22332 91 5G 6 T1 22350 92 5G 6 T2 22362 93 3D1 2 E 22738 94 3D1 2 T2 22973 95 3L 7 T1 23190 96 3L 7 T2 23220 97 3L 7 A2 23536 98 3L 7 T2 23606 99 3L 7 E 23645

100 3L 7 T1 23754

No. Mult. Irrep Ecalc

101 3P2 1 T1 23934 102 3H4 6 A1 24103 103 3H4 6 T1 24157 104 3H4 6 T2 24171 105 3G2 4 E 24334 106 3H4 6 T2 24345 107 3L 8 A1 24350 108 3H4 6 A2 24351 109 3L 8 T1 24388 110 3H4 6 T2 24429 111 3L 8 E 24433 112 3G2 4 E 24543 113 3L 8 A1 24544 114 3G2 4 T1 24544 115 3L 8 T2 24679 116 3L 8 T1 24683 117 3L 8 T2 24738 118 3L 8 E 24754 119 3L 9 T1 25394 120 3L 9 T2 25421 121 3L 9 E 25436 122 3L 9 T1 25446 123 3D1 3 T2 25495 124 3D1 3 A2 25504 125 3D1 3 T1 25531 126 3P2 0 A1 25576 127 3L 9 A1 25600 128 3L 9 T1 25641 129 3L 9 T2 25669 130 3L 9 A2 25695 131 3M 8 E 25776 132 3M 8 T1 25794 133 3M 8 A1 25845 134 3M 8 T2 25938 135 3M 8 T1 25944 136 3M 8 E 25957 137 3M 8 T2 25995 138 3P2 2 E 26163 139 3P2 2 T2 26238 140 3G2 5 E 26797 141 3F4 4 T2 26809 142 3G2 5 T1 26820 143 3G2 5 T1 26879 144 3G2 5 T2 26930 145 3F4 4 A1 27054 146 3F4 4 T1 27057 147 3G2 5 E 27083 148 3P2 1 T1 27708 149 3P2 2 T2 28382 150 3I1 5 E 28491

421 Energy levels of Pm3+ in Cs2NaPmCl6

Using interpolated

parameters

46

Prediction of luminescent levels

Ln3+ 2S+1L J IR Ecalc Eexpt Egap Lum

Ce 2F7/2 E″ 2160 2160 1590 ?

Pr 3H5 T1 2280 2300 1614 ?

Pr 3H6 E 4376 4386 1678 ?

Pr 3F3 *T2 6583 6616 1330 ?

Pr 1G4 A1 9728 9847 2484 √

Pr 1D2 T2 16661 16666 6235 √

Pr 3P0 A1 20604 20625 3418 √

Pr 1S0 A1 46380 23967 ×

Nd 4I11/2 U 1893 1921 1562 ?

Nd 4I13/2 E″ 3828 3861 1695 ?

Nd 4I15/2 U 5771 5797 1664 ?

Nd 4F3/2 U 11309 11335 5129 √

Nd 4G7/2 E″ 18620 18640 1377 √

Nd 2P1/2 E′ 23005 23043 1289 ?

Nd 2P3/2 U 25933 2300 ?

Nd 4D3/2 U 27602 27617 1669 √

Nd 2F5/2 U 37898 37838 3866 ?

Nd 2F7/2 E′ 39253 1257 ?

Nd 2G9/2 U 46884 7495 ?

Pm 5I5 T1 1578 1321 ?

Pm 5I6 A1 3151 1445 ?

Pm 5I7 T1 4804 1523 ?

Pm 5I8 E 6480 1491 ?

Pm 5F1 T1 12241 5401 ?

Pm 3K8 E 19603 1408 ?

Pm 3G3 A2 21440 1260 ?

Pm 3F4 A1 35230 1541 ?

Pm 3G5 T2 41592 1224 ?

Pm 1G4 T2 46895 1565 ?

Pm 1L8 T2 49215 1688 ?

Pm 1I6 A2 55016 1590 ?

Pm 1G4 A1 56607 1235 ?

Sm 6H13/2 U 5028 5005 1310 ?

Sm 6F11/2 E′ 10458 10461 1254 ?

Sm 4G5/2 U 17747 17742 7120 √

Eu 5D0 A1 17209 17208 11938 √

Eu 5D1 T1 18961 18961 1752 √

Eu 5D2 T2 21393 21385 2432 √

Eu 5D3 T1 24281 24261 2810 √

Eu 5H3 T1 30563 1839 ?

Gd 6P7/2 E″ 31940 31954 31940 √

Gd 6I7/2 E″ 35624 35623 2485 √

Gd 6D9/2 E′ 39233 39219 2736 ?

Gd 6G7/2 E″ 48957 8226 ?

Tb 7F5 T1 2071 2083 1703 ?

Tb 5D4 A1 20457 20470 14620 √

Tb 5D3 A2 26200 26219 5612 √

Tb 5H6 A1 32785 32732 1333 ?

Dy 6H13 U 3587 3606 3185 ?

Dy 6H11/2 E′ 5917 5929 2191 ?

Dy 6H9/2 U 7692 7713 1661 ?

Dy 6F5/2 E″ 12396 12392 1302 √

Dy 4F9/2 U 20964 20957 7182 √

Dy 4F5/2 E″ 40051 1233 ?

Ho 5I7 T1 5088 5116 4805 √

Ho 5I6 E 8602 8620 3346 √

Ho 5I5 T1 11183 11198 2434 √

Ho 5I4 A1 13225 13232 1939 √

Ho 5F5 T1 15317 15353 1869 √

Ho 5S2 *T2 18370 18365 2809 √

Ho 5F3 A2 20384 20420 1849 √

Ho 5G5 T1 23794 23779 1529 ?

Ho 5G4 A1 25673 25719 1745 √

Ho 3D3 T1 32928 2129 ?

Ho 5D4 A1 41166 41163 1228 √

Ho 1D2 T2 44749 1913 ?

Er 4I13/2 E′ 6490 6492 6188 √

Er 4I11/2 E′ 10165 10166 3469 √

Er 4I9/2 U 12344 12357 2079 √

Er 4F9/2 U 15172 15152 2651 √

Er 4S3/2 U 18272 18265 2923 √

Er 4F7/2 E′ 20388 20374 1238 √

Er 4F5/2 U 22079 22056 1622 ?

Er 4F9/2 U 24369 24425 1926 √

Er 4G11/2 U 26131 26098 1613 √

Er 2P3/2 U 31369 31367 3387 ?

Er 2K13/2 E′ 32613 32613 1244 √

Er 2H9/2 U 36252 36224 1598 √

Er 4D5/2 E″ 38168 38164 1799 ?

Er 2I11/2 E″ 40674 40668 1720 √

Er 4D1/2 E′ 46600 3191 ?

Er 2H11/2 U 50514 1935 ?

Er 2F7/2 E′ 53846 2964 ?

Tm 3F4 T2 5577 5547 5103 √

Tm 3H5 T1 8275 8240 2293 √

Tm 3H4 T2 12572 12538 3968 √

Tm 3F3 *A2 14381 14428 1445 √

Tm 1G4 T2 20952 20852 5799 √

Tm 1D2 T2 27697 27653 6171 √

Tm 1I6 E 34332 34117 6579 √

Tm 3P2 E 37543 37462 1609 √

Yb 2F5/2 U 10276 10248 9680 √

Predicted 4fN luminescent levels (using the energy gap law Egap > 4 Ephonon ~ 1200 cm-1) with energies up to 57000 cm-1. Here Ecalc, Eexpt, Egap,

2S+1L J, and IR are the calculated energy, experimental energy, calculated energy gap below the level, the multiplet term, and the irreducible representation, respectively, of the predicted luminescent level. The column Lum depicts whether luminescence has been experimentally observed (√), whether luminescence does not occur (×), or if the investigation has not been made conclusively. Energy units are cm-1.

47

Prediction of spectral intensities

Discussed later for Gd3+ TPA

48

Systematic variation F2 = (61573±610) +(3223.3±79.4)N (9)

F4 = (46213±830) + (2054±108)N (10)

• F6 = (25631±1620)+(3594.3±18)N

-(121.6±36.4) N2 (11)

Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm0

20

40

60

80

100

120

2 4 6 8 10 12 14

Fk (

103 c

m-1)

Ln3+ in Cs2NaLnCl

6

F2

F4

F6

N

Central Field approximation: F4/F2 and F6/F2 being stable for Ln3+, at 0.70 and 0.54

F4/F2 = (0.7437±0.0138) - (0.00316±0.00180)N (12) F6/F2 = 0.4413 + (0.02261 ± 0.0069)N - (0.0014 ± 0.00048)N2 (13)Comparison with the free-ion values: for Pr3+ Fk are 7.5±1.2% smaller in the Cs2NaPrCl6

crystal. This Nephelauxetic Effect has been ascribed to various causes, including the reduced repulsion between 4f electrons due to interpenetration of ligand electrons.

Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm0.3

0.4

0.5

0.6

0.7

0.8

F6/F2

Rat

io b

etw

een

Fk

Ln3+ in Cs2NaLnCl

6

F4/F2

49

Variation of spin-orbit coupling

Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb0

1000

2000

3000

2 4 6 8 10 12

cm

-1)

Ln3+ in Cs2NaLnCl

6

ζ4f = (539.4±10.2) + (87.82±3.33)N + (7.095±0.233)N2 (12)

50

Variation of crystal field parameters

Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb0

150

300

1600

2000

Cry

stal

-fie

ld p

ram

eter

(cm

-1)

Ln3+ in Cs2NaLnCl

6

B4

B6

B4 = (2176.2±29.2) - (56.7±3.6)N (13)

B6 = (285.5±24.6) - (12.6±3.0)N (14)

The CFP for the point charge model vary according to rk/[R(Ln-Cl)](k+1), where rk (k = 4,6) are the radial integrals <4f|r4|4f> and <4f|r6|4f>, respectively, R(Ln-Cl) is the metal-ligand distance. This model therefore predicts that the ratio B4(Yb3+)/B4(Ce3+) ~ 0.35 and B6(Yb3+)/B6(Ce3+) ~0.24, far from the optimized crystal-field parameter ratios of 0.67 and 0.33 herein. This indicates that main contribution to crystal-field interaction is not the point charge of the ligands.

51

Available values of ζ4f(F) for

Ln = Eu, Tb, Er, Tm, Yb agree with the values of ζ4f(Cl) within the uncertainties of determination.

Parameter trends over various crystal hosts

Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb0

500

1000

1500

2000

2500

3000

Ln3+

Chloride Fluoride

f (cm

-1)

52

Parameter trends over various crystal hosts Kawabe has presented the variation of ζ4f against

Z4 using the experimental values for aquo ions

from Carnall's work. From the resulting linear

plot the value of the 4f screening constant was

evaluated to be 32.0. An analogous plot from the

Cs2NaLnCl6 data, of [ζ4f ]0.25 against Z is

presented in Fig. S6 (N =12; R = 0.9995). The

derived equation is:

[ζ4f]0.25 = 0.193(Z - 31.9) = 0.193 Z*, (16)

where Z* is the effective nuclear charge, which gives a similar value (31.9) for the screening constant.

58 60 62 64 66 68 704

6

8

f (cm

-1)]

0.25

Z

Upper 95% Confidence Limit Lower 95% Confidence Limit

0.25 = -6.154+-0.124+ (0.19307+-0.00193)Z

0.25= 0.193(Z-32)

R = 0.9995 N=12

5 6 76

7

8

9

10 Upper 95% Confidence Limit Lower 95% Confidence Limit

F2 (

104 cm

-1)

(4f (cm-1))0.25

Y = (-18603+-2253)+(16560+-364)X

R= 0.99783; N=11

Kawabe, Geochem. J. 26 (1992) 309.

Carnall et al. J. Chem. Phys. 90 (1989) 3443.

53

Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb0

250

500

2000

3000

4000

B6

Cry

stal

fie

ld p

aram

eter

(cm

-1)

Ln3+ in Cs2NaLnF

6

B4

Comparison of CFP for Cs2NaLnX6

(X = Cl, F) systems

B4(F)/B4(Cl) = 1.77±0.27 (17)B6(F)/B6(Cl) = 1.81±0.78 (18)

54

Multiple parameter sets

For cases where many crystal field parameters (CFP) are involved (orthorhombic/monoclinic crystal symmetry), the resulting parameter values are not unique and depend upon the choice of coordinate system for the Hamiltonian.

This can result in misleading comparisons of CFP between systems where different conventions were employed.

Rudowicz has given some guidelines but it may be more appropriate to employ crystal field strengths in such comparisons, where ambiguities do not arise.

Physica B 279 (2000) 302

55

Comparison of CFP with other

systems

The comparison of the energy level datafits with those for other systems should take into account the fact that only two CF parameters have been employed for M2ALnX6, whereas the number (Np) is considerably greater for other systems where Ln3+ is doped. As an example, the CFP from the Cs2NaErCl6 datafit are compared with those for Er3+ in several lattices using the CF strength, Sk, which is a spherical parameter independent of the crystal symmetry:

Sk =

kkq

kqB

kk

,

2/12

2

])(000

33)12/(7[ (19)

System Site

symmetry

Np S4 S6

YAG D2 9 337 207

YVO4 D2d 184 92

LaCl3 C3h 4 38 76

LaF3 C2v 9 123 153

CsCdBr3 C3v 6 238 31

Cs3Lu2Br9 C3v 6 251 44

Cs3Lu2Cl9 C3v 6 257 48

Cs2NaErCl6 Oh 2 287 60

caution! there are many definitions of Sk

56

Standardization of CFP• In order to compare CFP from different systems with different

geometry, Rudowicz proposed a standardization method, based on the rhombicity ratio being in the “standard” range (0,√6) Rudowicz Physica B. 291 (2000) 327

• Burdick has shown that enforcing a standardization based exclusively upon rank 2 terms will result in the dominant rank 4 and 6 terms having different, incommensurate parameter values, even if their parameters started out (prior to standardization) being nearly identical.

• Burdick Spectr. Lett. 43 (2010) in press• More meaningful to compare CF strengths of various ranks.

57

Problems with analyses: experiment

1. Presence of electron-phonon couplings confuses some energy levels

e.g. Tm3+ in Cs2NaTmCl6 ground state

0

56108

146

261

370

394

} 1+ν5; a5

1

4

2

b5

3

58

Problems with analyses: theory

1. CFP show a multiplet term dependence

2. Some anomalous “rogue” multiplets not well-fitted

e.g. Ho3+ 3K8; Pr3+ 1D2,1G4; Nd3+ 2H(2)11/2

N level label Cs2NaPrCl6 expt. 4f 2 calc. |Error| cm-1 cm-1 cm-1 24 1G4 1 9847 9766 81 25 1G4 4 9895 9852 43 26 1G4 3 9910 9901 9 27 1G4 5 10327 10441 114 28 1D2 5 16666 16705 39 29 1D2 3 17254 17209 45

59

What are the parameters HADD?In the conventional CF analyses of lanthanide ion systems, the CF splittings of certain multiplet terms were poorly modeled:e.g. Ho3+ 3K8; Pr3+ 1D2; Nd3+ 2H(2)11/2.

The SLJ-dependence of the CF parameters, and their irregular behaviour on crossing the lanthanide series prompted the introduction of further phenomenological parameters into the CF hamiltonian.

CF parameters which accurately model low-lying multiplets give poor fits for higher-energy multiplet terms. The one-electron CF model assumes that the CF potential experienced is independent of the properties of the remaining electrons, despite their strong electrostatic correlation.

Berry et al. J Lumin 66&67 (1996) 272

60

The one-electron CF model assumes that the CF potential experienced is independent of the properties of the remaining electrons.

Correlation CF corrections to the hamiltonian have been proposed by Newman and Judd, which take into account the different interactions with the ligand field of multiplet terms with different orbital angular momenta (orbitally-correlated CF, OCCF) as well as the different interactions of multiplet terms with different spin (spin-correlated CF, SCCF).

The two-electron correlation terms introduce up to 637 parameters for systems with C1 site symmetry, which reduce down to 41 parameters for Oh symmetry.

Which parameters to choose?

How can we improve the energy level parametrization?

61

Correlation CF• Correlation CF corrections to the hamiltonian have been proposed by

Newman and Judd, which take into account the different interactions with the ligand field of multiplet terms with different orbital angular momenta (orbitally-correlated CF) as well as the different interactions of multiplet terms with different spin (spin-correlated CF: SCCF).

• The SCCF has received most attention. It is argued that because spin-parallel electrons are subject to an attractive exchange force they are expected to occupy orbitals with a more compact radial distribution than spin-antiparallel electrons. Thus, spin-anti-parallel (minority spin) electrons should be subject to stronger CF interactions.

• Most of the earlier CF analyses for M2ALnX6 systems were carried out on limited datasets pertaining to maximum multiplicity states, so that the importance of this hypothesis could not be evaluated fairly.

Garcia, Faucher, Handbook on the Physics and Chemistry of Rare Earths 21 (1995) 263

Newman et al. J Phys C: Solid State Phys 15 (1982) 3113

62

4fN-15d and CT energy levels

Energies of lowestf → d and π → f(○ expt.; ● calc.)transitions for LnCl6

3-

(aq)and lowestf → d transitions for LnCl6

3- crystals (+ and I)

Ionova et al., New J. Chem. 19 (1995) 677

Tanner, Topics Curr. Chem. 241 (2004) 1

63

Calculation of 4fN-15d energy levels 4fN-15d multi-electron energy levels are calculated using an

extension of the standard phenomenological crystal-field Hamiltonian H(4f) :

H= H(4f) + H(5d)+ Hint(4f,5d)

where H(4f), H(5d) and Hint(4f,5d) describe the interactions felt by or between the 4f electrons;

felt by the 5d electron;

and the interaction between 4f and 5d electrons, respectively,

64

Additional terms in Hamiltonian( )

5d 5d 5d

int exc2,4 1,3,5

H(5d) (d) (d);

H (4f,5d) (fd) (fd) (fd) (fd).

k kq q

kq

k kk k

k k

B

E F G

s l C

f g

Italic and bold letters represent parameters and operators, respectively, and (d) and (fd) are used to show that the operators are interactions for the 5d electron and interactions between 4f and 5d electrons, respectively.

For H(5d), the two terms are the spin-orbit and the strong crystal-field interactions felt by the 5d electron. For the Hint(4f,5d) parameter, Eexc describes the separation between the barycenters of the 4fN and 4fN-15d levels; and the second and third terms are the direct and exchange Coulomb interaction between 4f and 5d electrons.

65

CF for d electrons

The crystal-field interactions felt by 4f and 5d electrons for a

lanthanide ion occupying an octahedral site can be simplified to be

written as:

(4) (4) (4) (6) (6) (6)cf 4 0 4 4 6 0 4 4

5 7

14 2H B C C C B C C C

. (3)

In the case of the 5d electron, only the first term needs to be included

and the parameter is denoted as B4(d) in the following, to be

distinguished from the one for 4f electrons, B4.

66

Parameters for 4fN-15d• Although both the 4fN configuration and the 4fN-1 core of the 4fN-

15d configuration contain H(4f), the parameter values for these may be slightly different due to the contraction of 4f orbitals in the 4fN-15d configuration.

• For the 4fN-15d core, the same parameters as for the 4fN configuration were used, except for Fk (k = 2, 4, 6) and 4f, which are enlarged from those values for the 4fN configuration by the ab initio ratio 1.06.

• The parameters Fk (fd) (k = 2,4) and Gk (fd) (k = 1, 3, 5) describe the Coulomb interaction between the 4fN-1 core and the d electron.

• The values used were scaled from the HF values by a factor of ηfd = 0.55

Taylor, Carter, J. Inorg. Nucl. Chem. 24 (1962) 387