Crystal Field Theory

Crystal Field Theory History 1929 1932 1932 1952 1954 1960s Hans Bethe - Crystal Field Theory (CFT) • Developed to interpret color, spectra, magnetism in crystals 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 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 L. E. Orgel - Revival of CFT and development of Ligand Field Theory (LFT) • Slowly replaces VB theory • Explains magnetism and spectra better Y. Tanabe and S. Sugano - Semi-quantitative term splitting diagrams • Used to interpret visible spectra 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 ML6 (Oh) complex first and then extend the approach to other complex geometries. Energies of d orbitals in an Octahedral Complex K Consider a spherical field (R3) equivalent to six electron pairs surrounding a central metal ion, M. • Ligand-metal electron repulsions will perturb the energies of the five degenerate d orbitals, making them rise in energy. E K isolated metal ion metal ion in spherical field Localize the six ligand pairs into the positions of an octahedron (R3 6 Oh). • Five-fold degeneracy among d orbitals will be lifted, in keeping with the direct product listings in the Oh character table. • The dxy, dyz, dxz orbitals constitute a triply degenerate set of t2g symmetry. • The dx2–y2, dz2 orbitals constitute a doubly degenerate set of eg symmetry.1 ! The energies of the t2g orbitals and eg orbitals relative to the perturbed energy of the hypothetical spherical field depend upon their orientation to the six ligand coordination positions. 1 The vector product 2z2 – x2 – y2 indicates the d orbital more commonly labeled z2. d Orbitals in an Octahedral Field x z z y eg y x dx2-y2 d2z2-x2-y2 = dz2 x z z z y x y y t2g x dxy dyz dzx ! The eg orbitals have lobes that point at the ligands and so will ascend in energy. ! The t2g orbitals have lobes that lie between ligands and so will descend in energy. eg +3/5)o = +6Dq )o = 10Dq -2/5)o = -4Dq t2g Energy level of hypothetical spherical field energy R3 Oh Crystal Field Splitting Energy, Δo eg +3/5)o = +6Dq )o = 10Dq -2/5)o = -4Dq t2g Energy level of hypothetical spherical field energy R3 Oh ! The energy gap between t2g and eg levels is designated Δo or 10Dq. ! The energy increase of the eg orbitals and the energy decrease of the t2g 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 = +6Dq while the energy of the t2g set falls by –2/5Δo = –4Dq, resulting in no net energy change for the system. ΔE = E(eg)8 + E(t2g)9 = (2)(+3/5Δo) + (3)(–2/5Δo) = (2)(+6Dq) + (3)(–4Dq) = 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–1 Electron Filling of t2g and eg Orbitals eg o t2g ! Electrons fill t2g and eg orbitals in an aufbau manner, starting with the t2g set, in accord with the Pauli exclusion principle and Hund’s rule. • Spins of successively added electrons are parallel so long as the Pauli exclusion principle allows. • At the point when the set of t2g orbitals is half filled, an additional electron must pair if it is to occupy one of the orbitals of the degenerate set. • But if the mean pairing energy (P) is greater than Δo, a lower energy state will result by putting the electron in the higher eg level. ! For configurations d 1 – d 3 and d 8 – d 10 there is only one ground state configuration. ! For configurations d 4 – d 7 there are two possible filling schemes depending on the magnitudes of P and Δo. • A high spin configuration minimizes pairing by spreading the electrons across both the t2g and eg levels. • A low spin configuration minimizes occupying the higher energy eg level by pairing electrons in the t2g level. ! For a given metal ion, the pairing energy is relatively constant, so the spin state depends upon the magnitude of the field strength, Δo. • Low field strength (Δo < P) results in a high-spin state. • High field strength (Δo > P) results in a low-spin state. High- and Low-Spin Configurations for ML6 Oh eg t2g ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼¿ ¼ ¼ ¼ ¼ ¼¿ ¼¿ d1 d2 d3 d4 high spin d4 low spin d5 high spin d5 low spin ¼ ¼ ¼ ¼ ¼ ¼ ¼¿ ¼¿ ¼¿ eg ¼ ¼ t2g ¼ ¼ ¼¿ ¼¿ ¼¿ ¼¿ ¼ ¼¿ ¼¿ ¼¿ ¼¿ ¼¿ ¼¿ ¼¿ ¼¿ ¼¿ ¼¿ ¼¿ ¼¿ ¼¿ ¼¿ d6 high spin d6 low spin d7 high spin d7 low spin d8 d9 d 10 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 as4 where = number of electrons in t2g orbitals = number of electrons in eg orbitals p = total number of electron pairs P = mean pairing energy Free Ion Examples of CFSE Calculations Oh CFSE Calculation CFSE d3 t2g3 d8 t2g6eg2 [(6)(–0.4) + (2)(+0.6)]Δo + 3P d 7 low t2g6eg1 [(6)(–0.4) + (1)(+0.6)]Δo + 3P –1.8Δo + 3P d 7 high t2g5eg2 [(5)(–0.4) + (2)(+0.6)]Δo + 2P –0.8Δo + 2P (3)(–0.4Δo) –1.2Δo –1.2Δo + 3P ! 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 + 3P = –1.8Δo + 3Δo = 1.2Δo CFSE(d 7 high) = –0.8Δo + 2P = –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 Πc for the Δo term and Πe for the P term in the defining equation. Some sources do not include pairing energy in calculating CFSE. Values of Δo, P, CFSE and Resulting Spin State Complex d n P (cm–1) Δo (cm–1) State CFSE [Cr(H2O)6]2+ d 4 18,800 13,900 high –0.6Δo [Fe(H2O)6]3+ d 5 24,000 13,700 high 0 [Fe(H2O)6]2+ d 6 14,100 10,400 high –0.4Δo + P [Fe(CN)6]4– d6 14,100 33,000 low –2.4Δo + 3P [CoF6]3– d6 16,800 13,000 high –0.4Δo + P [Co(NH3)6]3+ d 6 16,800 23,000 low –2.4Δo +3P [Co(H2O)6]2+ d 7 18,000 9,3000 high –0.8Δo + 2P ! Values of Δo depend on both the metal ion and the ligand. ! Most aquo complexes are high spin, because H2O is a weakfield ligand. ! Almost all Co3+ (d 6) complexes are low spin, including [Co(H2O)6]3+, except [CoF6]3–, which is high spin. ! Second and third row transition metal ions tend to have low spin states. • These ions tend to have larger Δo values. • Larger 4d and 5d orbitals result in smaller P values, owing to lesser electronic repulsions. • 4d and 5d orbitals overlap with ligand orbitals, delocalizing electron density onto the ligands. 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– < C2O42– < H2O < NH3 < en < bipy < phen < CN– . CO • en = ethylenediamine, bipy = 2,2'-bipyradine, phen = o-phenathroline • Ligands up through H2O are weak-field ligands and tend to result in high-spin complexes. • Ligands beyond H2O are strong-field ligands and tend to result in low-spin complexes. Tetrahedral Crystal Field Splitting ! The same considerations of crystal field theory can be applied to ML4 complexes with Td symmetry. • In Td, dxy, dyz, dxz orbitals have t2 symmetry and dx2–y2, dz2 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 t2 orbitals are closer to ligands than are the e orbitals. This can be seen by comparing the orientations of the dx2y2 orbital (e set) and dxy orbital (t2 set) relative to the four ligands. dx2-y2 dxy ! The difference results in an energy split between the two levels by Δt or 10Dq'. Relative to the barycenter defined by the hypothetical spherical field " the e level is lower by –3Δt /5 = –6Dq' " the t2 level is higher by +2Δt /5 = +4Dq' Tetrahedral Crystal Field Splitting - cont. t2 +2/5)t = +4Dq' )t = 10Dq' -3/5)t = -6Dq' e Energy level of hypothetical spherical field energy R3 Td ! In principle, both high and low spin configurations are conceivable for d 3–d 6 ML4 Td complexes. ! With extremely rare exceptions, only high spin configurations are observed. " Δt is much smaller than Δo. " For a given ligand at the same M-L distances, it can be shown that Δt = (4/9)Δo. " Δt << P in ordinary complexes, so high spin is favored. ! The crystal field stabilization energy for tetrahedral complexes is calculated from the following equation: CFSE(Td) = (–0.6ne + 0.4 )Δt + pP 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 slitting 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 ML6 ! A tetragonal distortion to an octahedron results from any change in geometry that preserves a C4 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 D4h. • 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 ML4 complex. Splitting of d Orbital Degeneracies – Oh 6 D4h ! From a correlation table that links the groups Oh and D4h it can be determined that the two eg orbitals of the octahedral field become nondegenerate as a1g and b1g in the D4h tetragonal field. • From the direct product listings in the D4h character table a1g = d2z2-x2-y2 (= dz2) b1g = dx2-y2 ! From the correlation table it can also be shown that the degeneracy among the t2g orbitals in Oh is partially lifted to become b2g and eg in the D4h tetragonal field. • From the direct product listings in the D4h character table b2g = dxy eg = (dxz, dyz) ! Moving the two ligands away from the central metal ion lowers the repulsions between ligand electrons and the metal electrons in those d orbitals that have substantial electron distribution along z. • Thus the energies of the dxz, dyz, and dz2 orbitals are lowered. ! If we assume that the stretch along z is accompanied by a counterbalancing contraction in the xy plane, so as to maintain the overall energy of the system, then the orbitals with substantial electron distribution in the xy plane will experience increased repulsions. • Thus, the dxy and dx2-y2 orbitals rise in energy. Orbital Splitting from a Stretching Tetragonal Distortion Oh D4h b1g (dx2-y2) +*1/2 eg -*1/2 a1g (dz2) )o )o b2g (dxy) +2*2/3 t2g -*2/3 eg (dxz, dyz) increasing stretch along z ! The upper eg orbitals of the perfect octahedron split equally by an amount δ1, with the dx2-y2 orbital (b1g in D4h) rising by +δ1/2 and the dz2 orbital (a1g in D4h) falling by –δ1/2. ! The lower t2g orbitals of the perfect octahedron split by an amount δ2, with the dxy orbital (b2g in D4h) rising by +2δ2/3, and the degenerate dxz and dyz orbitals (eg in D4h) falling by –δ2/3. Magnitudes of the δ1 and δ2 Splittings ! Both the δ1 and δ2 splittings, which are very small compared to Δo, maintain the barycenters defined by the eg and t2g levels of the undistorted octahedron. • The energy gap δ1 is larger than that of δ2, because the dx2-y2 and dz2 orbitals are directed at ligands. • The distortion has the same effect on the energies of both the dx2-y2 and dxy orbitals; i.e. δ1/2 = 2δ2/3. As a result, the energies of both the dx2-y2 and dxy rise in parallel, maintaining a separation equal to Δo of the undistorted octahedral field. L Oh D4h b1g (dx2-y2) +*1/2 eg -*1/2 a1g (dz2) )o )o b2g (dxy) +2*2/3 t2g -*2/3 increasing stretch along z • Note that δ1/2 = 2δ2/3 implies that δ1 = (4/3)δ2. eg (dxz, dyz) Square Planar ML4 Complexes ! If we imagine continuing the stretching of M-L bonds along z, the orbital splittings will become progressively greater, producing successively larger values of δ1 and δ2. ! Eventually the two ligands will be removed, resulting in a square planar ML4 complex. ! At some point before this extreme the a1g (dz2) level may cross and fall below the b2g (dxy) level, resulting in the following splitting scheme.5 b1g (dx2-y2) ~)o eg )o b2g (dxy) a1g (dz2) t2g eg (dxz, dyz) ML6 Oh 5 ML4 D4h The ordering of the lower four d orbitals probably varies among square planar complexes and has been the subject of much debate. ML4 (D4h) vs. ML4 (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 b1g (dx2-y2) level vacant in the ground state. • This is expected, because square planar geometry in firstrow transition metal ions is usually forced by strong field ligands. • Strong field ligands produce a large Δo value. • The energy gap between the b2g (dxy) and b1g (dx2-y2) levels is equivalent to Δo. L A large Δo value favors pairing in the b2g (dxy) level, a low-spin diamagnetic configuration for d 8. ! Tetrahedral d 8 is a high-spin paramagnetic configuration e4t24. L ML4 (D4h) and ML4 (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., [NiCl4]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 [PtCl4]2-.