Crystal (ligand) field theory Literature Title page • C.E. Housecroft, A.G. Sharpe: Inorganic chemistry 1 Crystal (ligand) field theory • Sphere symmetry Title page • jj-coupling, LS- coupling 2 Schrödinger equation Crystal field 2 ˆ 2m (r) V (r) (r) E(r) Schrödinger equation m: electron mass kinetickinetická energy E. potenciálnpotential energyí E. o: permitivity of vacuum 2 2 2 : wave functions x2 y2 z2 e: electron charge Hydrogen atom: Vˆ e2 4 or E: energy ħ: Planck’s constant R: radial function in spherical coordinates: n,l,m Rn,l (r)Yl,m (,) Y: angular function 2 2 2 r2 2 2 ˆ ˆ ˆ ˆ n: principal quantum number Hn,l,m Enn,l,m H T V l: orbital quantum number ˆ2 2 L Yl,m l(l 1) Yl,m determine the orbital angular momentum ˆ l = 0 ... n-1 LzYl,m mlYl,m ml: magnetic quantum number projection of the angular momentum into z-axis m l = -l … l 3 Radial function Crystal field 8 7 2 2 4r ·Rnl(r) 6 5 4 1s Bohr’s radius a0 = 0.52918 Å 2s 3 The distance of the electron (maximum 2p of the orbital 1s density) from nuclei in 3s 2 hydrogen atom 3p 3d 1 2 4 0 a0 2 0 me 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 r / a0 Radial function – the dependence on Z Crystal field The radius of maximal electron density Effective nuclear charge a0* = a0 / Z* Z* = Z - σ σ = screening constant, a sum over all electrons The electrons are merged into groups () (1s)(2s,2p)(3s,3p)(3d)(4s,4p)(4d)(4f)(5s,5p)(5d)(5f)... Slater’s rules: electron on the right does not contribute to σ Electrons inside the group screen 0.35 (1s only 0.30) n–1 (s,p) screen 0.85 n–2 and lower screen 1.00 If the electron is in d or f, all electron on the left screen 1.0 An example for Fe (26): 5 Atomic orbitals – spherical harmonic functions (angular parts) Crystal field Solution of Schrödinger equation Linear combination: (complex function): . (real function): . ml Yl 0 s Y0 0 pz Y1 p 1 Y 1 Y 1 x 2 1 1 p i Y 1 Y 1 y 2 1 1 d Y 0 z 2 2 d 1 Y 2 Y 2 x 2 y 2 2 2 2 d i Y 2 Y 2 xy 2 2 2 d 1 Y 1 Y 1 xz 2 2 2 d i Y 1 Y 1 yz 2 2 2 33 Y3 f ... 6 Atomic orbitals Crystal field Orbital and spin angular momentun Crystal field m: electron mass o: permitivity of vacuum : eigenfunctions sˆ2 2s(s 1) s 1 2 e: electron charge ˆ E: energy Lˆ Y m Y sz ms z l,m l l,m 2 ħ: Planck’s constant R: radial function Y: angular function n: principal quantum number l: orbital quantum number determine the orbital angular momentum l = 0 ... n-1 ˆ ˆ ˆ ˆ ml: magnetic quantum number Hn,l,m Enn,l,m H T V projection of the angular ˆ2 2 L Yl,m l(l 1) Yl,m momentum into z-axis m l = -l … l s: spin angular momentum ms: projection into z-axis ms = -1/2 … 1/2 8 Dirac’s equation Crystal field 2 2 2 퐸 = 푐 푝Ԧ + 푚0푐 Relativistic formula for the total energy of a free particle 1 0 2 2 2 2 where 훽 = 훼Ԧ = 0 휎Ԧ 퐸 = 푐 −ℏ ∆ + 푚0푐 = −푖ℏ푐훼Ԧ훻 + 훽푚0푐 0 −1 휎Ԧ 0 0 1 0 −푖 1 0 Pauli matrixes: 휎 = 휎 = 휎 = 푥 1 0 푦 푖 0 푧 0 −1 s: spin quantum number | s | s(s 1) determines angular momentum of electron 1 s 2 ms: projection into z-axis ms = -s … s sz ms 1 1 ms 2 , 2 9 Spin-orbit coupling Magnetism Spin orbit interaction is known in atomic physics as Hso SL relativistic correction of the electron energies (푠Ԧ, 푝Ԧ, and 퐿 mean vector operators of spin, momentum and angular momentum) > 0 : less than ½ occupied orbital, ℏ 푉푆퐿 = 2 푠Ԧ ∙ ∇푉 × 푝Ԧ/푚0 J=|L-S|, direction of L and S opposite 4푚0푐 ℏ 1 푑푉 < 0 : more than ½ occupied orbital, 푉 = 퐿 ∙ 푠Ԧ 푆퐿,푠푝ℎ 4푚 푐2 푟 푑푟 J=L+S, direction of L and S identical 0 3d < 4d < 5d M2+ < M3+< … Approximate solution of Dirac’s equation: ˆ ˆ ˆ ˆ Vm : Relativistic mass correction HD HSchr Vm VD Vso VD : Contact interaction (Darwin’s term), significant for s orbitals only. Vso : Spin-orbit coupling Gyromagnetic ratio Magnetism Gyromagnetic ratio is a ration of m: electron mass e: electron charge magnetic momentum and angular momentum l l ħ: Planck’s constant l: orbital angular momentum B e Gyromagnetic ratio for s: spin angular momentum l 1 2m orbital momentum B: Bohr’s magneton r: radius of electron circuit e 2 B Gyromagnetic ratio for v: velocity of electron s m spin momentum : time of electron circulation I: current e : magnetic moment B 2m Derivation for orbital angular momentum: . 2r / v l m r v I e / ev / 2r e 2 l IS (ev / 2r)(r ) 2me e v r 11 2 Energy of orbitals in sphere symmetry – hydrogen atom Crystal field One-electron scheme Hydrogen atom – 1 electron: (energy only depends on the Wave length of the transitions principal quantum number n) between energy levels is determined 3s —— 3p —— 3d —— by Rydberg’s formula: 2s —— 2p —— 1s —— 12 Energy of orbitals in sphere symmetry Crystal field One-electron scheme (energy only depends J: total angular momentum on the principal quantum number n) J = |L+S| … |L-S| 3s —— 3p —— 3d —— MJ: -J, ... , J 2s —— 2p —— j = |l+s| … |l-s| 1s —— mj = -j, ... , j Many-electrons scheme J=|L+S| … |L-S| j=|l+s| … |l-s| Coulombic interaction is predominant spin-orbit interaction is predominant 3d kvantová čísla Vazebná energie 3d —— 5/2 n l j l = ±1 j = 0, ±1 pro Cu (eV) 3p —— 3p 3d 3d 3 2 5/2 L1 3/2 3/2 1,6 3s —— 3 2 3/2 3s 3p M 3p 3 1 3/2 K 1/2 1/2 1 73,6 3 1 1/2 K3 2p —— 3s 3 0 1/2 119,8 2s —— 2p3/2 2p 2 1 3/2 K1 931,1 L 2s1/2 2p1/2 2 1 1/2 K2 951,0 1s —— 2s 2 0 1/2 1096,1 K 1s 1 0 1/2 8978,9 1s1/2 LS coupling J ~ L + S = l + s Valence electrons, spectroskopic j-j coupling J ~ j = (s+l) symbols Core electrons 13 LS and jj coupling Crystal field one electron electron state Orbital momentum of atom LS coupling 퐿 = σ 푚푙(occupied orbitals) so<Coulomb 푀퐿 = −퐿, … , 퐿 Spin momentum Orbital momentum 푆 = σ 푚 (occupied orbitals) Total momentum 푙 = 0, 1, 2, 3, … 푠 푀푆 = −푆, … , 푆 퐽 = 퐿 − 푆 , … , 퐿 + 푆 푚푙 = 0 푀 = −퐽, … , 퐽 Spin momentum 퐽 1 푠 = Τ2 2S+1 1 LJ 푚푠 = ± Τ2 Total momentum 푗 = 푙 − 푠 , … , 푙 + 푠 jj coupling 푚푗 = −푗, … , 푗 so>Coulomb 14 jj-coupling Crystal field m j l s m l s + 3/2 l +1 s + 1/2 + 1/2 + 1/2 0 - 1/2 - 1/2 - 1/2 -1 - 3/2 mj = -3/2, -1/2, +1/2, +3/2 j = 3/2 p3/2 mj = -1/2, +1/2 j = 1/2 p1/2 15 jj-coupling Crystal field quantum number Binding energy spin-orbit interaction n l j l = ±1 j = 0, ±1 for Cu (eV) is predominant 3d5/2 3d 3 2 5/2 L1 1,6 3p 3d 3/2 3/2 3 2 3/2 3s1/2 3p1/2 M 3p 3 1 3/2 K 1 73,6 3 1 1/2 2p3/2 K3 2s1/2 2p1/2 3s 3 0 1/2 119,8 1s1/2 2p 2 1 3/2 K1 931,1 L j-j coupling J ~ j = 2 1 1/2 K2 951,0 (s+l) 2s 2 0 1/2 1096,1 Core electrons K 1s 1 0 1/2 8978,9 16 Orbital and spin momentum, state of the atom Crystal field one electron orbital momentum spin momentum s s s 1 sz ms l ll 1 lz ml the whole atom orbital momentum spin momentum S M L LL 1 Lz M L S SS 1 z S M m M L ml S s electronic state of the atom multiplicity = (2S+1)(2L+1) 2S+1 L L: S, P, D, F, G, H, I, … 17 Weak coupling - LS-coupling Crystal field Russel-Saunders scheme: L li S si (LS-coupling) i i spin-orbit coupling J JJ 1 J z M J J L S J = L+S, L+S-1, ... ,L-S MJ = -J, ..., 0 , ... , J x>y x<y 2J+1 values ground state x – number of electrons y – number of orbitals 2S+1 values for S<L 2L+1 values for S>L state of the atom 2S+1 multiplicity = (2J+1) LJ 18 Spectroscopic symbols Crystal field 1 p ml: -1 0 1 l = 1 L = 1 P s = ½ S = ½ 2S+1=2 J = 1+½, 1+½-1 (= |1-½| ) 2 2 P3/2, P1/2 Multiplicity of the state 2P: (2L+1)(2S+1) = 32 = 6 (2J+1) = (23/2+1)+(21/2+1) = 4+2 = 6 19 Spectroscopic symbols – ground state Crystal field LL occupation of orbitals starts from maximal ml.