Transcript Electrons, states, energy levels

3. Electrons, states, energy levels
1
Single electron spatial quantum numbers
n: principal (radial) quantum number
l: orbital angular momentum (new azimuthal) quantum number
ml: magnetic quantum number
For one electron atoms, wavefunctions characterized by even l
have even parity:
Pi |nlml> = (-1)l |nlml>
http://winter.group.shef.ac.uk/orbitron/AOs/4f/index.html
http://media-2.web.britannica.com/eb-media/12/7512-004-91DF97CB.gif
H&I Ch. 2
2
http://www.chemtube3d.com/orbitals-f.htm
Some basic principles
Eigenvalue equation
2
wave
d2
[
 V ( x)] ( x)  E  ( x)
2
function
2m dx
operator H
Hamiltonian
matrix elements:
orthonormality
expectation value
energy
*

 b A a dx   a | A | b 
*

 b  a dx   a | b   ab
1 if a = b; 0 if a ≠ b
*

 a H a dx  Ea   a |  a  Ea
diagonal matrix energies are state energies
3
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)
4
Single electron spin quantum number
s: electron spin ½
ms: magnetic quantum number, ±1/2
No two electrons in an atom can have the same values of these four quantum numbers
Ce3+ 4f1 - a configuration shows electron structure
E
Pr3+ 4f2
E
odd electron systems have
Kramers degeneracy: 2 different
states with same energy, in
absence of magnetic field.
Electrons are indistinguishable
these 2 are the same state
no Kramers degeneracy for even
electron systems
5
Spin orbit coupling: j = l + s
|nls j=l+1/2 mj>
|nlsmlms> or |nlsjmj>
[(2l+1)/2] ζ
E
For light atoms, l and s are
good quantum numbers
= an unique value can specify
a state
|nls j=l-1/2 mj>
e.g. l=3, s=1/2 splits into
j = 5/2 and j = 7/2.
For lanthanides, only the total
angular momentum is (nearly) a
good quantum number
6
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+1L :
J
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
7
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
8
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
Microstates of
Pr3+
-3
++
-2
++
-1
++
ml
0
++
1
++
ms written as +
or -+
4f2 Pr3+
has 91 microstates
(14×13)/(1×2)
can you fill in
3F
3P
3H
-+
+
-+
+
+
2
++
ML
MS
Term
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
0
0
0
0
0
0
0
0
0
0
0
0
0
1
I
0
0
1
S
2
1
0
-1
-2
0
0
0
0
0
1
D
4
3
2
1
0
-1
-2
-3
-4
0
0
0
0
0
0
0
0
0
1
G
3
+-
9
Clebsch-Gordon coefficients
When two angular momentum states are coupled, the new states
can be expressed as the coupled representation:
| ( j1 j2 ) JM 
j1
j2
 
m1  j1 m2  j2
| j1m1 j2 m2   j1m1 j2 m2 | JM 
The Clebsch-Gordon coefficient is related to a 3-j symbol:
 j j J 
 j1m1 j2 m2 | JM  (1) j1  j2  M 2 J  1  1 2

 m1 m2  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
10
Example of Clebsch-Gordon coefficients
or abbreviated:
Example for j1 =1/2; j2 = 1/2
H&I, page 134
11
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 halffilled 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
100
1
200
2
300
3
343
2
e.g. Multiplets for Pr3+ and Tm3+ are 1S0, 3P0,1,2， 1I6, 1D2, 1G4, 3F4,3,2, 3H4,5,6
http://en.wikipedia.org/wiki/List_of_Hund's_rules
12
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.
13
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
14
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
15
Character table for Oh molecular point group
Oh (m3m)
E
8C3
6C2
6C4
3C2
i
6S4
8S6
3h
6d
1g
2g
3g
4g
3g
1u
2u
3u
4u
5u
1
1
2
3
3
1
1
2
3
3
1
1
-1
0
0
1
1
-1
0
0
1
-1
0
-1
1
1
-1
0
-1
1
1
-1
0
1
-1
1
-1
0
1
-1
1
1
2
-1
-1
1
1
2
-1
-1
1
1
2
3
3
-1
-1
-2
-3
-3
1
-1
0
1
-1
-1
1
0
-1
1
1
1
-1
0
0
-1
-1
1
0
0
1
1
2
-1
-1
-1
-1
-2
1
1
1
-1
0
-1
1
-1
1
0
1
-1
A1g
A2g
Eg
T1g
T2g
A1u
A2u
Eu
T1u
T2u
x2+y2+z2
(2z2-x2-y2, x2-y2)
(Rx,Ry, Rz)
xz, yz, xy
(x, y, z)
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)
A2 (2)
E (3)
T1 (4)
T2 (5)
E (6)
E (7)
A1
A2
E
T1
T2
E
E
A2
A1
E
T2
T1
E
E
E
E
A1+A2+E
T1+T2
T1+T2
U
U
T1
T2
T1+T2
A1+E+T1+T2
A2+E+T1+T2
E+ U
E+ U
T2
T1
T1+T2
A2+E+T1+T2
A1+E+T1+T2
E+ U
E+ U
U (8)
U
U
E+E+U
E+E+2U
E+E+2U
E
E
U
E+U
E+U
A1+T1
A2+T2
E+T1+
T2
E
E
U
E+U
E+U
A2+T2
A1+T1
E+T1+
T2
U
U
E+E+U
E+E+2U
E+E+2U
E+T1+T2
E+T1+T2
A1+A2+E+
2T1+2T2
16
Crystal field irreps from J-values
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)
17
Theoret. Chim. Acta 74 (1988) 219 http://www.springerlink.com/content/n02ml8613m5xl82h/fulltext.pdf
18
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.
19
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.
Calc A
Calc B
Reid, Eur. J. Inorg. Chem. doi:10.1002/ejic.20100182
B.
20
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.
1. Choose all possible arrangements of
4fN as bases.
2. Calculate the Hamiltonian matrix.
Wybourne book
21
Basics of energy level calculations
For a Ln3+ system:  ( x)   akk ( x)  a11  a22  ..
k
abbreviate  1 | H | 1  H11
Eigenvalue equation is
or simply:
 H11  E
 H
21

 H11
H
 21
(just consider two terms)
H12  a1 
 a1 
 E 


H 22  a2 
 a2 
H12  a1 
0


H 22  E  a2 
then solve for the 2
energies E1, E2
Eigenvectors a1, a2 must be orthonormal
22
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
23
Pr3+ energy levels
major interactions for Ln3+
are Coulomb and SOC
H&I p. 398
Free ion: Coulomb
repulsion
Spin-orbit coupling
Further smaller
splittings due to
crystal field
24
SOC and CF
Transition metal d-electron systems:
CF > SOC
Lanthanide 4fN: SOC > CF
H = HCoul(ff) + Hso(f) + Hcf(f)+....
Most important interactions are
e-e repulsion > SOC > CF
i.e. weak CF
Label states 2S+1LJ
Energy (cm-1)
Do not judge by parameter values:
6
2
F7/2
8
7
But by energy splittings,
e.g. Ce3+ 4f1 levels in Oh symmetry
n.b. for Ce3+, 4f1, no Coulomb interaction
2
F5/2
-1
3048 cm
3048
2661
2661 cm-1
2160 cm-1
2160
(7/2)
×ζ4f
8
570 cm
570
7
0 cm-1
-1
0
25
The atomic Hamiltonian:
H AT  E AV   F k f k   f s i  l i  αL(L 1)  βG(G2 )  γG( R7 )
k
i
T s t s   P k p k   M j m j
s
k
(3)
j
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 twoelectron radial integrals, where the f k 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.
N
Note that the terms
e
4f
(i)  Ne4 f involving the kinetic energy of the electrons and their
i
nuclear attraction do not give rise to a splitting of the 4f N levels and are contained in EAV.
26
Other interactions
The two-body configuration interaction parameters α, β, γ parametrize the secondorder 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.
27
Crystal field parameters CFP
The HCF operator represents the nonspherically symmetric components of
the one-electron CF interactions, i.e. the perturbation of the Ln 3+ 4fN electron
system by all the other ions. The general form of the CF hamiltonian HCF is
given by:
k
 k k

   B0 C0 (i)   ( Bqk (Ck q (i)  (1) q Ckq (i))  jBq' k (Ck q (i)  (1) q Ckq (i))) 
i ;k 0 
q 1


HCF
(4)
where the Bqk are parameters and the Ckq (i) 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
28
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.
29
CFP in higher symmetry
For Oh symmetry, values of k whose angular momentum irreps contain Γ1g
give nonzero Bqk , i.e. k = 4 and 6. The nonzero CF parameters for the Oh group
reduce to B04 , B44 , B06 , B46 . The unit tensor normalized CFP Bq(k ) (used by
Richardson) are related to the spherical tensor CFP Bqk (used by Wybourne)
by:
 3 k 3
(k )
k
(k )
k

(5)
Bq  Bq f | C | f  Bq (7)
0
0
0


so that B0( 4) = 1.128 B04 , and B0(6) = -1.277 B06 . However, the B4k parameters are
related to B0k , and in the Wybourne spherical tensorial notation herein, and in the
cubic environment, B44 = B04 (5/14)1/2 and B46 =  B06 (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.
30
Theoretical analysis
More convincing for LnX63- systems because fewer independent crystal field
parameters are involved.
Pr3+ in Cs2NaLnCl6 (Oh): B04 and B06
(J. Chem. Phys. 114 (2001) 10860)
Pr3+ in YPO4 (D3d): B02 , B04 , B44 , B06 , B46
(J. Alloys. Compds. 323-324 (2001) 783)
Pr3+ in La2O3 (C3v): B02 , B04 , B34 , B06 , B36 , B66
(J. Lumin. 85 (1999) 59)
Pr3+ in Pr(trensal) (C3): B02 , B04 , B34 , B06 , B36 , B66
(Inorg. Chem. 41 (2002) 5024)
31
4f1 and 4f13 systems
These are simple because only one electron (or
one hole) so no Coulomb repulsion.
32
Crystal field analysis of 4f13
7
8
2F
5/2
Different notations employed
we use:
Eave ,  , B04 , B06
7
8
2F
7/2
6
6  14b4  20b6
8  2b4  16b6
7  18b4  12b6
22
7
b4  b6
3
2
44
7
7   b4  b6
3
2
8 
Inorg. Chem. 16 (1976) 1694
3
7
1
3   
2
1
3
 
5
4
4
8
3
3
5     
4
4
3
6
   
2
  8b4  56b6
0
  10b4  84b6
1 4
b4 
B0
33
5 6
b6  
B0
429
J. Chem. Phys. 94 (1991) 942
33
Crystal field analysis of 4f13
6
8
7
5 7 7
5 7 1
|  
|
 coefficient | J M J 
12 2 2
12 2 2
1 5 3
5 5
5

|

|  
6 2 2
6 2
2
1 7
5
3 7 3
|  
|

2 2
2
2 2 2
1 5
|
6 2
3 7
|
2 2
5
5 5 3

|  
2
6 2 2
5 1 7 3
 |  
2 2 2 2
(Equation 1)
(Equation 2)
Griffith, The Theory of Transition –Metal Ions; Cambridge University Press, 1961
34
Crystal field analysis of 4f13
Γ6
 2 4 20 6 3 
 11 B0  429 B0  2  
Γ8:
2 4


B0  2

21
4
20

5B04 
5B06
143
 77
:
Γ7:
4 4

B0  2

21

 20 B04  120 B06
 77 3
143 3
4
20

5B04 
5B06 
77
143
2 4 25 6 3 
 B0 
B0   
77
312
2 

B06 
77 3
143 3 
18 4 20 6 3 
B0 
B0  
77
143
2 
20
B04 
120
Energy matrices of 4f13
Take advantage of symmetry factorization
35
The published parameters of B04, B06 and  were used
cal.
exp.
6076
10691
 10248
5662
10277
 572
-4021
594
 250
-4382
233
 0
-4615
0
 10718
10500
-1
Energy (cm )
10000
9500
9000
800
600
400
200
0
Cs2NaYbCl6
Comparison of experimental and calculated energy levels of Cs2NaYbCl6.
J Alloys Compds 215 (1994) 349
36
Crystal field analysis of 4f13
Anticipate the effect of crystal field parameters on the energy gaps.
Γ8:
Γ7:
2 4

 B0  2

21
4
20

5B04 
5B06
143
 77
4
20

4
5B0 
5B06 
77
143
2 4 25 6 3 
 B0 
B0   
77
312
2 
4 4

B0  2

21

 20 B04  120 B06
 77 3
143 3

B06 
77 3
143 3 
18 4 20 6 3 
B0 
B0  
77
143
2 
20
B04 
120
The effect of
B04 on Γ7 should be more prominent than that on Γ8 .
The effect of
B06 on Γ8 might be more prominent than that on Γ7 .
37
Crystal field analysis of 4f13
2
F5/27
F5/27 10.8
10.6
10.6
2
10.4
F5/28 10.4
10.2
10.2
2
F5/28
3
-1
Energy (10 cm )
2
10.8
2
F7/27
0.8
0.8
2
F7/27
0.6
0.6
2
2
F7/28 0.4
0.4
F7/28
0.2
0.2
2
F7/26
0.0
-200
-100
0
100
6
B0
0.0
200 1000
2
F7/26
1500
2000
4
B0
Energy against crystal parameters of Cs2NaYbCl6, calculated by f-shell program.
38
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
39
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.
40
2. Use Appl. Spectrosc. 25 (1971) 155: Schoenflies notation.
number of equivalent atoms
in Bravais cell
number of different kinds of
site with this symmetry
41
http://neon.otago.ac.nz/chemlect/chem203/symmetrylectures/molecularsymmetry.pdf
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
42
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.
43
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.
44
Results: Prediction of energy levels
421 Energy
levels of Pm3+
in Cs2NaPmCl6
Using interpolated
parameters
No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
Mult.
5I 4
5I 4
5I 4
5I 4
5I 5
5I 5
5I 5
5I 5
5I 6
5I 6
5I 6
5I 6
5I 6
5I 6
5I 7
5I 7
5I 7
5I 7
5I 7
5I 7
5I 8
5I 8
5I 8
5I 8
5I 8
5I 8
5I 8
5F 1
5F 2
5F 2
5F 3
5F 3
5F 3
5S 2
5S 2
5F 4
5F 4
5F 4
5F 4
5F 5
5F 5
5F 5
5F 5
3K2 6
3K2 6
3K2 6
3K2 6
3K2 6
3K2 6
5G 2
Irrep
T2
E
T1
A1
T1
E
T2
T1
A1
T1
T2
A2
T2
E
T1
E
T2
A2
T1
T2
E
T2
T2
T1
A1
T1
E
T1
E
T2
T1
T2
A2
E
T2
A1
T1
T2
E
T1
E
T2
T1
E
T2
A2
A1
T1
T2
E
Ecalc
0
245
247
257
1578
1607
1633
1707
3151
3173
3197
3226
3278
3282
4804
4843
4860
4884
4981
4988
6480
6480
6535
6591
6793
6828
6840
12241
12544
12782
13462
13502
13723
14038
14044
14391
14446
14464
14486
15626
15674
15824
15908
16814
16821
16844
16847
16860
16867
17221
No.
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
Mult.
5G 2
3H4 4
5G 3
3H4 4
5G 3
5G 3
3K2 7
5G 4
3K2 7
5G 3
3K2 7
3K2 7
3K2 7
3K2 7
3K2 8
3K2 8
3K2 8
3K2 8
3K2 8
3K2 8
3K2 8
3K2 8
5G 5
5G 4
5G 5
5G 4
5G 4
5G 4
5G 4
3G2 3
3G2 3
3G2 3
5G 5
5G 6
5G 5
5G 6
5G 6
5G 5
5G 6
5G 6
5G 6
5G 6
3D1 2
3D1 2
3L 7
3L 7
3L 7
3L 7
3L 7
3L 7
Irrep
T2
T2
T1
E
A2
T1
T2
A1
A2
T1
T2
T2
T1
E
E
T2
T1
T2
E
A1
T1
E
T1
T2
T1
T2
A1
T1
E
A2
T2
T1
E
T2
T1
E
T2
T1
A2
A1
T1
T2
E
T2
T1
T2
A2
T2
E
T1
Ecalc
17339
17519
17809
17827
17837
18016
18021
18049
18078
18128
18149
18180
18195
18195
19603
19640
19644
19653
19697
19701
19732
19751
19762
19839
19885
19896
20176
20178
20180
21440
21540
21541
21777
21827
21919
21994
22020
22023
22044
22332
22350
22362
22738
22973
23190
23220
23536
23606
23645
23754
No.
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
Mult.
3P2 1
3H4 6
3H4 6
3H4 6
3G2 4
3H4 6
3L 8
3H4 6
3L 8
3H4 6
3L 8
3G2 4
3L 8
3G2 4
3L 8
3L 8
3L 8
3L 8
3L 9
3L 9
3L 9
3L 9
3D1 3
3D1 3
3D1 3
3P2 0
3L 9
3L 9
3L 9
3L 9
3M 8
3M 8
3M 8
3M 8
3M 8
3M 8
3M 8
3P2 2
3P2 2
3G2 5
3F4 4
3G2 5
3G2 5
3G2 5
3F4 4
3F4 4
3G2 5
3P2 1
3P2 2
3I1 5
Irrep
T1
A1
T1
T2
E
T2
A1
A2
T1
T2
E
E
A1
T1
T2
T1
T2
E
T1
T2
E
T1
T2
A2
T1
A1
A1
T1
T2
A2
E
T1
A1
T2
T1
E
T2
E
T2
E
T2
T1
T1
T2
A1
T1
E
T1
T2
E
Ecalc
23934
24103
24157
24171
24334
24345
24350
24351
24388
24429
24433
24543
24544
24544
24679
24683
24738
24754
25394
25421
25436
25446
25495
25504
25531
25576
25600
25641
25669
25695
25776
25794
25845
25938
25944
25957
25995
26163
26238
26797
26809
26820
26879
26930
27054
27057
27083
27708
28382
28491
45
Prediction of luminescent levels
Ln3+
Ce
Pr
Pr
Pr
Pr
Pr
Pr
Pr
Nd
Nd
Nd
Nd
Nd
Nd
Nd
Nd
Nd
Nd
Nd
Pm
Pm
Pm
Pm
Pm
Pm
Pm
Pm
Pm
Pm
Pm
Pm
Pm
Sm
Sm
Sm
Eu
Eu
Eu
Eu
Eu
Gd
Gd
Gd
Gd
Tb
Tb
Tb
Tb
Dy
2S+1
L J IR Ecalc Eexpt
Egap Lum
2
?
F7/2 E″ 2160 2160 1590
3
?
H5 T1 2280 2300 1614
3
?
H6
E 4376 4386 1678
3
?
F3 *T2 6583 6616 1330
1
√
G4 A1 9728 9847 2484
1
√
D2 T2 16661 16666 6235
3
√
P0 A1 20604 20625 3418
1
×
S0 A1 46380
23967
4
?
I11/2
U 1893 1921 1562
4
?
I13/2 E″ 3828 3861 1695
4
?
I15/2
U 5771 5797 1664
4
√
F3/2
U 11309 11335 5129
4
√
G7/2 E″ 18620 18640 1377
2
?
P1/2 E′ 23005 23043 1289
2
?
P3/2
U 25933
2300
4
√
D3/2
U 27602 27617 1669
2
?
F5/2
U 37898 37838 3866
2
?
F7/2 E′ 39253
1257
2
?
G9/2
U 46884
7495
5
?
I5 T1 1578
1321
5
?
I6 A1 3151
1445
5
?
I7 T1 4804
1523
5
?
I8
E 6480
1491
5
?
F1 T1 12241
5401
3
?
K8
E 19603
1408
3
?
G3 A2 21440
1260
3
?
F4 A1 35230
1541
3
?
G5 T2 41592
1224
1
?
G4 T2 46895
1565
1
?
L8 T2 49215
1688
1
?
I6 A2 55016
1590
1
?
G4 A1 56607
1235
6
?
H13/2
U 5028 5005 1310
6
?
F11/2 E′ 10458 10461 1254
4
√
G5/2
U 17747 17742 7120
5
√
D0 A1 17209 17208 11938
5
√
D1 T1 18961 18961 1752
5
√
D2 T2 21393 21385 2432
5
√
D3 T1 24281 24261 2810
5
?
H3 T1 30563
1839
6
√
P7/2 E″ 31940 31954 31940
6
√
I7/2 E″ 35624 35623 2485
6
?
D9/2 E′ 39233 39219 2736
6
?
G7/2 E″ 48957
8226
7
?
F5 T1 2071 2083 1703
5
√
D4 A1 20457 20470 14620
5
√
D3 A2 26200 26219 5612
5
?
H6 A1 32785 32732 1333
6
?
H13
U 3587 3606 3185
Dy
Dy
Dy
Dy
Dy
Ho
Ho
Ho
Ho
Ho
Ho
Ho
Ho
Ho
Ho
Ho
Ho
Er
Er
Er
Er
Er
Er
Er
Er
Er
Er
Er
Er
Er
Er
Er
Er
Er
Tm
Tm
Tm
Tm
Tm
Tm
Tm
Tm
Yb
6
H11/2
E′ 5917 5929 2191 ?
6
H9/2
U 7692 7713 1661 ?
6
F5/2 E″ 12396 12392 1302 √
4
F9/2
U 20964 20957 7182 √
4
F5/2 E″ 40051
1233 ?
5
I7 T1 5088 5116 4805 √
5
I6
E 8602 8620 3346 √
5
I5 T1 11183 11198 2434 √
5
I4 A1 13225 13232 1939 √
5
F5 T1 15317 15353 1869 √
5
S2 *T2 18370 18365 2809 √
5
F3 A2 20384 20420 1849 √
5
G5 T1 23794 23779 1529 ?
5
G4 A1 25673 25719 1745 √
3
D3 T1 32928
2129 ?
5
D4 A1 41166 41163 1228 √
1
D2 T2 44749
1913 ?
4
I13/2
E′ 6490 6492 6188 √
4
I11/2
E′ 10165 10166 3469 √
4
I9/2
U 12344 12357 2079 √
4
F9/2
U 15172 15152 2651 √
4
S3/2
U 18272 18265 2923 √
4
F7/2
E′ 20388 20374 1238 √
4
F5/2
U 22079 22056 1622 ?
4
F9/2
U 24369 24425 1926 √
4
G11/2
U 26131 26098 1613 √
2
P3/2
U 31369 31367 3387 ?
2
K13/2
E′ 32613 32613 1244 √
2
H9/2
U 36252 36224 1598 √
4
D5/2 E″ 38168 38164 1799 ?
2
I11/2 E″ 40674 40668 1720 √
4
D1/2
E′ 46600
3191 ?
2
H11/2
U 50514
1935 ?
2
F7/2
E′ 53846
2964 ?
3
F4 T2 5577 5547 5103 √
3
H5 T1 8275 8240 2293 √
3
H4 T2 12572 12538 3968 √
3
F3 *A2 14381 14428 1445 √
1
G4 T2 20952 20852 5799 √
1
D2 T2 27697 27653 6171 √
1
I6
E 34332 34117 6579 √
3
P2
E 37543 37462 1609 √
2
F5/2
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.
46
Prediction of spectral intensities
Discussed later for Gd3+ TPA
47
Systematic variation
2
(11)
6
8
10
12
14
2
F
80
4
F
-1
3
60
6
F
k
-(121.6±36.4) N2
4
100
F (10 cm )
F2 = (61573±610) +(3223.3±79.4)N (9)
F4 = (46213±830) + (2054±108)N (10)
• F6 = (25631±1620)+(3594.3±18)N
N
120
40
20
F4/F2
Central Field approximation:
and
being stable for Ln3+, at 0.70 and 0.54
F6/F2
0
Pr
Nd Pm Sm Eu Gd Tb
Dy Ho
Er Tm
3+
Ln in Cs2NaLnCl6
0.8
4
2
F /F
k
0.7
Ratio between F
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.
0.6
6
2
F /F
0.5
0.4
0.3
Pr
Nd Pm Sm
Eu
3+
Gd
Tb
Dy
Ln in Cs2NaLnCl6
Ho
Er
Tm
48
Variation of spin-orbit coupling
2
4
6
8
10
12
3000
-1
 cm )
2000
1000
0
Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb
3+
Ln in Cs2NaLnCl6
ζ4f = (539.4±10.2) + (87.82±3.33)N + (7.095±0.233)N2 (12)
49
Variation of
crystal field
parameters
B6 = (285.5±24.6) - (12.6±3.0)N
-1
Crystal-field prameter (cm )
B4 = (2176.2±29.2) - (56.7±3.6)N
2000
(13)
(14)
1600
B4
300
150
B6
0
Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb
3+
Ln in Cs2NaLnCl6
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
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.
50
Parameter trends over various crystal
hosts
Available values
of ζ4f(F) for
Ln = Eu, Tb, Er, Tm, Yb
3000
2500
Chloride
Fluoride
2000
-1
f (cm )
agree with the
values of ζ4f(Cl)
within the
uncertainties of
determination.
1500
1000
500
0
Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb
Ln
3+
51
Parameter trends over various crystal hosts
8
Kawabe has presented the variation of ζ4f against

0.25
Z4 using the experimental values for aquo ions

0.25
plot the value of the 4f screening constant was
= 0.193(Z-32)
R = 0.9995 N=12
0.25
-1
f (cm )]
from Carnall's work. From the resulting linear
= -6.154+-0.124
+ (0.19307+-0.00193)Z
6
Upper 95% Confidence Limit
Lower 95% Confidence Limit
evaluated to be 32.0. An analogous plot from the
4
Cs2NaLnCl6 data, of [ζ4f ]0.25 against Z is
58
60
62
64
66
68
70
Z
presented in Fig. S6 (N =12; R = 0.9995). The
10
derived equation is:
Upper 95% Confidence Limit
Lower 95% Confidence Limit
(16)
-1
= 0.193(Z - 31.9) = 0.193 Z*,
where Z* is the effective nuclear charge, which
gives a similar value (31.9) for the screening
constant.
Kawabe, Geochem. J. 26 (1992) 309.
Carnall et al. J. Chem. Phys. 90 (1989) 3443.
8
Y = (-18603+-2253)
+(16560+-364)X
2
4
[ζ4f]
F (10 cm )
9
0.25
7
R= 0.99783; N=11
6
5
6
7
-1
(4f (cm ))
0.25
52
Comparison of CFP for Cs2NaLnX6
(X = Cl, F) systems
(17)
(18)
-1
B4(F)/B4(Cl) = 1.77±0.27
B6(F)/B6(Cl) = 1.81±0.78
Crystal field parameter (cm )
4000
3000
B4
2000
500
250
B6
0
Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb
3+
Ln in Cs2NaLnF6
53
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
54
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:
3 3 k 

S = [7 /(2k  1)
0
0
0


2
k
System
 (B
q   k ,k
Site
k
q
) 2 ]1 / 2
Np
(19)
S4
S6
symmetry
YAG
D2
YVO4
D2d
LaCl3
C3h
LaF3
9
337
207
184
92
4
38
76
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
Comparison of
CFP with other
systems
55
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.
56
Problems with analyses:
experiment
1. Presence of electron-phonon couplings confuses some
energy levels
e.g. Tm3+ in Cs2NaTmCl6 ground state
394
3
370
b5
261
2
146
108
56
0
} 1+ν5; a5
4
1
57
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
24
25
26
27
28
29
level
1
G4
1
G4
1
G4
1
G4
1
D2
1
D2
label
1
4
3
5
5
3
Cs2NaPrCl6
expt.
cm-1
9847
9895
9910
10327
16666
17254
4f2 calc.
cm-1
9766
9852
9901
10441
16705
17209
|Error|
cm-1
81
43
9
114
39
45
58
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
59
How can we improve the energy level parametrization?
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.
60
Which parameters to choose?
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 spinparallel 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
61
4fN-15d and CT energy levels
Energies of lowest
f → d and π → f
(○ expt.; ● calc.)
transitions for LnCl63(aq)
and lowest
f → d transitions for
LnCl63- crystals
(+ and I)
Ionova et al., New J. Chem. 19 (1995) 677
Tanner, Topics Curr. Chem. 241 (2004) 1
62
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,
63
Additional terms in Hamiltonian
H(5d)   5d s5d  l 5d   Bqk (d)C(qk ) (d);
kq
H int (4f,5d)  Eexc 
F
k  2,4
k
(fd)f k (fd) 

k 1,3,5
G k (fd)g k (fd).
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.
64
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)

 (6)

5
7 (6)
(4)
(4)
(6)
H cf  B4 C0 
C4  C4   B6 C0 
C4  C4  .


14
2




(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.
65
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
66
Taylor, Carter, J. Inorg. Nucl. Chem. 24 (1962) 387