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Crystal Field Theory History
1929 Hans Bethe - Crystal Field Theory (CFT)
- Developed to interpret color, spectra, magnetism in crystals 1932 J. H. Van Vleck - CFT of Transition Metal Complexes
- Champions CFT to interpret properties of transition metal complexes
- Show unity of CFT, VB, and MO approaches 1932 L. Pauling and J. C. Slater - VB theory
- Apply hybrid orbital concepts to interpret properties of transition metal complexes
- Becomes dominant theory to explain bonding and magnetism until 1950s
- Can't explain colors and visible spectra 1952 L. E. Orgel - Revival of CFT and development of Ligand Field Theory (LFT)
- Slowly replaces VB theory
- Explains magnetism and spectra better 1954 Y. Tanabe and S. Sugano - Semi-quantitative term splitting diagrams
- Used to interpret visible spectra 1960s CFT, LFT, and MO Theories
- Used in conjunction with each other depending on the level of detail required
- MO used for most sophisticated and quantitative interpretations
- LFT used for semi-quantitative interpretations
- CFT used for everyday qualitative interpretations
CFT Principles
! CFT takes an electrostatic approach to the interaction of ligands and metal ions.
- In purest form it makes no allowances for covalent M–L bonding.
! CFT attempts to describe the effects of the Lewis donor ligands and their electrons on the energies of d orbitals of the metal ion.
K We will consider the case of an octahedral ML 6 ( Oh ) complex first and then extend the approach to other complex geometries.
dx2-y2 d2z2-x2-y2 = dz
dxy dyz dzx
eg
t2g
x
x
x
x
x
z
z
z
z
y
z
y y
y y
eg
t2g
)o = 10 Dq -2/5)o = -4 Dq
+3/5)o = +6 Dq
energy R 3 Oh
Energy level of hypothetical spherical field
d Orbitals in an Octahedral Field ! The eg orbitals have lobes that point at the ligands and so will ascend in energy.
! The t 2 g orbitals have lobes that lie between ligands and so will descend in energy.
eg
t2g
)o = 10 Dq
-2/5)o = -4 Dq
+3/5)o = +6 Dq
energy R 3 Oh
Energy level of hypothetical spherical field
Crystal Field Splitting Energy, Δo
! The energy gap between t 2 g and eg levels is designated Δo or 10 Dq.
! The energy increase of the eg orbitals and the energy decrease of the t 2 g orbitals must be balanced relative to the energy of the hypothetical spherical field (sometimes called the barycenter ).
- The energy of the eg set rises by +3/5Δo = +6 Dq while the energy of the t2g set falls by –2/5Δo = –4 Dq , resulting in no net energy change for the system. Δ E = E ( eg ) 8 + E ( t2g ) 9 = (2)(+3/5Δo) + (3)(–2/5Δo) = (2)(+6 Dq) + (3)(–4 Dq ) = 0
! The magnitude of Δo depends upon both the metal ion and the attaching ligands.
! Magnitudes of Δo are typically ~100 – 400 kJ/mol (~8,375 – 33,500 cm–1).^2
(^2) 1 kJ/mol = 83.7 cm–
High- and Low-Spin Configurations for ML 6 Oh
¼ eg ¼ ¼
t2g ¼ ¼ ¼ ¼ ¼ ¼¿ ¼ ¼ ¼ ¼ ¼¿ ¼ ¼¿
d^1 d^2 d^3 d^4 d^4 d^5 d^5 high low high low spin spin spin spin
e^ ¼^ ¼^ ¼^ ¼^ ¼¿ g (^) ¼ ¼ ¼ ¼ ¼¿ ¼¿
t2g ¼ ¼¿ ¼¿ ¼¿ ¼¿ ¼¿ ¼¿ ¼¿ ¼¿ ¼¿ ¼¿ ¼¿ ¼¿ ¼¿
d^6 d^6 d^7 d^7 d^8 d^9 d^10 high low high low spin spin spin spin
Crystal Field Stabilization Energy (CFSE)^3
! Occupancy of electrons in t2g and eg orbitals results in an overall crystal field stabilization energy (CFSE), defined for octahedral complexes as^4
where = number of electrons in t2g orbitals = number of electrons in eg orbitals p = total number of electron pairs P = mean pairing energy
Examples of CFSE Calculations Free Ion Oh CFSE Calculation CFSE d^3 t 2 g^3 (3)(–0.4Δo) –1.2Δo d^8 t 2 g^6 eg^2 [(6)(–0.4) + (2)(+0.6)]Δo + 3 P –1.2Δo + 3 P d^7 low t 2 g^6 eg^1 [(6)(–0.4) + (1)(+0.6)]Δo + 3 P –1.8Δo + 3 P d^7 high t 2 g^5 eg^2 [(5)(–0.4) + (2)(+0.6)]Δo + 2 P –0.8Δo + 2 P
! For d n^ cases that could be high- or low-spin, the configuration that results in the lower CFSE for the Δo of the complex is the spin state that is observed.
- For the hypothetical case Δo = P , neither state would be preferred, as the two CFSEs for d^7 illustrate: CFSE( d^7 low) = –1.8Δo + 3 P = –1.8Δo + 3Δo = 1.2Δo CFSE( d^7 high) = –0.8Δo + 2 P = –0.8Δo + 2Δo = 1.2Δo
- There are no cases for which Δo = P.
(^3) CFSE is also called Ligand Field Stabilization Energy (LFSE). (^4) Meissler & Tarr use Π c for the Δo term and Π e for the P term in the defining equation. Some sources do not include pairing energy in calculating CFSE.
Spectrochemical Series
! For a given metal ion, the magnitude of Δo depends on the ligand and tends to increase according to the following spectrochemical series :
I–^ < Br–^ < Cl–^ < F–^ < OH–^ < C 2 O 4 2–^ < H 2 O < NH 3 < en < bipy < phen < CN–^. CO
- en = ethylenediamine, bipy = 2,2'-bipyradine, phen = o -phenathroline
- Ligands up through H 2 O are weak-field ligands and tend to result in high-spin complexes.
- Ligands beyond H 2 O are strong-field ligands and tend to result in low-spin complexes.
dx2-y^2 dxy
Tetrahedral Crystal Field Splitting
! The same considerations of crystal field theory can be applied to ML 4 complexes with Td symmetry.
- In Td , dxy, dyz, dxz orbitals have t 2 symmetry and dx (^2) – y 2 , dz 2 orbitals have e symmetry.
! Relative energies of the two levels are reversed, compared to the octahedral case. " No d orbitals point directly at ligands. " The t 2 orbitals are closer to ligands than are the e orbitals. This can be seen by comparing the orientations of the dx (^2) - y 2 orbital ( e set) and dxy orbital ( t 2 set) relative to the four ligands.
! The difference results in an energy split between the two levels by Δt or 10 Dq'. Relative to the barycenter defined by the hypothetical spherical field " the e level is lower by –3Δt /5 = –6 Dq' " the t 2 level is higher by +2Δt /5 = +4 Dq'
Crystal Field Splitting for Other Geometries
! We can deduce the CFT splitting of d orbitals in virtually any ligand field by
- Noting the direct product listings in the appropriate character table to determine the ways in which the d orbital degeneracies are lifted
- Carrying out an analysis of the metal-ligand interelectronic repulsions produced by the complex’s geometry.
! Sometimes it is useful to begin with either the octahedral or tetrahedral splitting scheme, and then consider the effects that would result by distorting to the new geometry.
- The results for the perfect and distorted geometries can be correlated through descent in symmetry, using the appropriate correlation tables.
- Can take this approach with distortions produced by ligand substitution or by intermolecular associations, if descent in symmetry involves a group-subgroup relationship.
Crystal Field for Tetragonally Distorted ML 6
! A tetragonal distortion to an octahedron results from any change in geometry that preserves a C 4 axis.
- Tetragonal distortion occurs whenever two trans related ligands are differentiated from the remaining four.
! A useful tetragonal distortion to consider involves equally stretching two trans related ligands, thereby causing a descent in symmetry Oh 6 D 4 h.
- The stretching occurs along the z axis, leaving the four positions in the xy plane equivalent to each other.
- Ultimately, such a stretching leads to removal of the two ligands, leaving a square planar ML 4 complex.
+* 1 /
+2* 2 /
)o )o
eg
t2g
Oh D4h
increasing stretch along z
eg (dxz, dyz)
b2g (dxy)
a1g (dz 2 )
b1g (dx2-y2)
Orbital Splitting from a Stretching Tetragonal Distortion
! The upper eg orbitals of the perfect octahedron split equally by an amount δ 1 , with the dx (^2) - y 2 orbital ( b 1 g in D 4 h ) rising by +δ 1 /2 and the dz 2 orbital ( a 1 g in D 4 h ) falling by –δ 1 /2.
! The lower t2g orbitals of the perfect octahedron split by an amount δ 2 , with the dxy orbital ( b 2 g in D 4 h ) rising by +2δ 2 /3, and the degenerate dxz and dyz orbitals ( eg in D 4 h ) falling by
+* 1 /
+2* 2 /
)o )o
eg
t2g
Oh D4h
increasing stretch along z
eg (dxz, dyz)
b2g (dxy)
a1g (dz 2 )
b1g (dx2-y2)
Magnitudes of the δ 1 and δ 2 Splittings
! Both the δ 1 and δ 2 splittings, which are very small compared to Δo, maintain the barycenters defined by the eg and t 2 g levels of the undistorted octahedron.
- The energy gap δ 1 is larger than that of δ 2 , because the dx (^2) - y 2 and dz 2 orbitals are directed at ligands.
- The distortion has the same effect on the energies of both the dx (^2) - y 2 and dxy orbitals; i.e. δ 1 /2 = 2δ 2 /3.
L As a result, the energies of both the dx (^2) - y 2 and dxy rise in parallel, maintaining a separation equal to Δo of the undistorted octahedral field.
- Note that δ 1 /2 = 2δ 2 /3 implies that δ 1 = (4/3)δ 2.
ML 4 ( D 4 h ) vs. ML 4 ( Td )
! Most square planar complexes are d^8 and less often d^9.
! In virtually all d^8 cases a low spin configuration is observed, leaving the upper b 1 g ( dx (^2) - y 2 ) level vacant in the ground state.
- This is expected, because square planar geometry in first- row transition metal ions is usually forced by strong field ligands.
- Strong field ligands produce a large Δo value.
- The energy gap between the b 2 g ( dxy ) and b 1 g ( dx (^2) - y 2 ) levels is equivalent to Δo.
L A large Δo value favors pairing in the b 2 g ( dxy ) level, a low-spin diamagnetic configuration for d^8.
! Tetrahedral d^8 is a high-spin paramagnetic configuration e^4 t 24.
L ML 4 ( D 4 h ) and ML 4 ( Td ) can be distinguished by magnetic susceptibility measurements.
! Ni2+^ ion tends to form square planar, diamagnetic complexes with strong-field ligands (e.g., [Ni(CN) 4 ]2-), but tends to form tetrahedral, paramagnetic complexes with the weaker-field ligands (e.g., [NiCl 4 ]2–).
! With second and third row transition metal ions the Δo energies are inherently larger, and square planar geometry can occur even with relatively weak field ligands; e.g., square planar [PtCl 4 ]2-.
