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Sophia Klokishner Institute of Applied Physics of the Academy of Sciences of Moldova Academy str.5, Kishinev, MD 2028,Moldova [email protected] OUTLINE 1.General introduction 2.Crystal field effects in transition metal complexes 3.Intensity of the optical line 4.Electron-vibrational interaction. Semiclassical adiabatic approximation. Franck-Condon principle- classical representation 5.Absorption optical d-d bands in the semiclassical approximation: shape and temperature dependence 6.Charge transfer bands NATURE OF THE RADIATION Oscillating electric and magnetic fields: radiation, traveling in z-direction consists of electric and magnetic fields perpendicular to each other and to the z-direction. The wave travels in the z-direction with the velocity of the c=3· 10 10 cm sec -1

Y oscillating magnetic field

Z direction of the light extension X oscillating electric field (polarization) Approximate , and for selected regions of the IMPORTANT VALUES AND SPECTROSCOPIC UNITS

Plank’s constant Speed of light h = 6.623x10-27 erg sec c = 3x1010 cm sec-1

ν - of the radiation, ν (sec-1)=c(cm sec-1)/ λ (cm), wavenumber (cm-1)=1/ λ (cm), λ -

1 cm = 108 Å (angstroms) E- of the : E=h ν =hc/ λ

Units for the energy h = h/2 π , Ω =2 π ν 1 ev=8067 cm-1 The following sequence represents decreasing energy: > visible > > > radiofrequency

Radiofrequency → Nuclear Magnetic Resonance (NMR)

Microwave → Electron Spin (or Paramagnetic) Resonance (ESR or EPR)

Far IR (Infrared), Near IR → IR

Visible, Near UV (Ultraviolet) → Electronic Spectroscopy

X ray → Photoelectron Spectroscopy

γ ray → Mössbauer Spectroscopy MAIN FEATURES OF THE ATOMIC SPECTRA

Narrow lines of absorption and emission : Each line in the atomic spectrum corresponds to a certain transition between two atomic levels 1 and 2 : 2( )

Absorption Emission 1( )

RESONACE CONDITION ( ENERGY CONSERVATION ):

2 − 1 = hEE ν = h Ω Ω theis frequency theof

absorbed emittedor photon

h = h 2/ π Ω = 2 πν Two- and three- level schemes Intensity, arbitrary units

0 Ω −=Ω ()EE 12 / h Two-level system gives rise to the only optical line, a more complicated energy pattern results in a more complicated structure of the absorption or luminescence spectrum. Three-level scheme

3

I13 2 I12 two transitions Intensity, arbitrary units 1

I12, I13-intensities of the lines Ω Ω12 Ω13 in the medium (ensemble): (each having, let say, three levels): 1 2 3 Due to interaction with the medium (thermostat or bath) electrons ( red balls) jump “up” (absorption of heat) and “down” (emission of heat) traveling among the levels 1, 2 and 3. These jumps are very fast, so one can speak about the distribution of the electrons over the levels in the thermodynamic equilibrium of the ensemble.

N1 -mean number of the molecules with the energy E1

N2 -mean number of the molecules with the energy E2

N3 -mean number of the molecules with the energy E3

N= N1+N2+N3 - total number of the molecules.

P1 = N1 / N -probability to find a molecule with the populated level 1 (i.e. with the energy E1), etc… The main question: what are these probabilities? Probability to find a molecule with the populated level “ i ”

(i.e. a molecule with the energy Ei ) E 1 − i p = e kBT i Z

kB - Boltzman constant, T - temperature Z – partition function E − i Z = ∑e kBT i (summation over all levels)

Probability pi depends on the energy Ei and on the temperature T BOLTZMAN DISTRUBUTION- ILLUSTRATION FOR TWO LEVELS E 2 E − 1 += eZ kT

1 0 1 ,0 2 == EEE E − 1 e kT p1 = E , p 2 = E − − 1 + e kT 1 + e kT

T ⇒ p1 = p 2 = 0,1,0 ()only level "1" is populated 1 T , pp ==∞→ 21 2 equally are "2" and "1" (levels "1" and "2" are equally populated) POPULATION OF THE ENERGY LEVELS IN THE THERMODYNAMIC EQUILIBTIUM Energy E

1 − Ei kT E 5 ()Tp = e i Z E 4 E 3 T ≠ 0 E 2 E 1 0 population 1 p 1)Population exponentially decreases with the increase of the energy; level “ i” is populated significantly if kT ≥ E i ; 2)The ground level is always (at any T ) the most populated level . TRANSITIONS FROM THE EXCITED STATES Three-level scheme: energies E 1 =0, E 2 , E 3 we assume also that kT << E 3 , so that only the levels 1 and 2 are populated 3 intensit y (13)

T 1 (13) (23) (23) 2 fre q uenc y 1 intensity (13) (23) Intensities: T 2 p 1 (T ) I 13 and p 2 (T ) I 23 frequency T 2 > T 1 n – the main number ( n = 1, 2, 3…) l – quantum number of the orbital angular ( l = 0, 1, 2, …, n-1)

ml – magnetic quantum number ( ml = – l, – l+1,…,l – 1,l) ms – spin projection quantum number (spin s = 1/2, ms= ± 1/2) p – parity of the , p = (-1)l spectroscopic notations: l = 0, 1, 2, 3, 4, 5,… s, p, d, f, g, h, … “even” and “odd” states: s – even (l=0), p – odd (l=1), d –even (l=2), etc. ELECTRONIC CONFIGURATIONS OF 3d ELEMENTS The closed d-shell contains 10 electrons The filling up of the 3d shell takes place in the iron group [(1s)2 (2s)2(2p)6 (3s)2 (3p)6] –closed electronic shell Instead of a regular sequence of configurations of the type 3dn (4s)2 with increasing n, configurations of the type of 3dn+2 4s are found

Sc Ti V Cr Mn Fe Co Ni Cu

3d14s2 3d24s2 3d34s2 3d54s1 3d54s2 3d64s2 3d74s2 3d84s2 3d104s1

Ions of transition metals of iron group have unfilled d-shells: n=3, l=2 TYPICAL OXIDATION DEGREES AND dn CONFIGURATIONS d1(Ti3+), d2 (V3+), d3 (Cr3+,V2+), d4(Mn3+,Cr2+), d5(Fe3+,Mn2+),

d7(Co2+), d8(Ni2+), d9(Cu2+) BAND STRUCTURE OF : dielectrics, semiconductors Atomic levels are split into bands when form a crystal Conduction band is empty

forbidden band gap, Δ

Valent band is filled, each level is occupied by a pair of electrons at T=0 BAND STRUCTURE OF SOLIDS - levels of transition metal ions

‰ Levels of of transition metal ions fall into the levels of d-ions forbidden band gap ‰ These levels are populated by d-electrons THE MAIN CONSEQUENCES: levels of transition metal ions

1. Additional bands of light 3 absorption and emission related to the transitions between the levels of the d- ions

1 2 2. Photoconductivity-conductivity induced by light absorption – transitions from d-levels to the conduction band. Electron jumps from the d-level to the conduction band

3. Caption of the electrons from the conduction band-decrease of conductivity CRYSTAL FIELD Crystal field theory treats a complex as if it consists of a positive metal ion surrounded by negatively charged ions (ligands) PHYSICAL PICTURE Shapes of the electronic clouds: Ψ(r) – wave-function |Ψ(r)|2 – spatial distribution of the electronic density (shape of the electronic “cloud”) Energy in the crystal field =Interaction of the electronic cloud with the charges of the ligands CRYSTAL FIELD

The crystal field theory assumes that the ligands are point charges or point dipoles and the d-orbitals are entirely non-bonding

The energy of interaction V of an electron of the

metal ion M with the negatively charged ligands Li L5 6 2 L3 Ze V = ∑ r L r 4 i =1 − Rr i r M L2 Ze - ligand charge, e - the electron charge, r r - the electron coordinate

L1 r - the position –vector of the ligand L L5 Ri - i M – transition metal ion

Li –ligand SchrödingerSchrödinger EquationEquation forfor thethe stationarystationary statesstates Hˆ Ψ = EΨ Hˆ -the Hamiltonian of the system (operator of the energy including kinetic and potential energy of the system of electrons) Ψ (r) -the wave function By solving SchrödingerSchrödinger EquationEquation one can find the energy levels En and the wave-functions corresponding to these energy levels 2 Ψ ()r -spatial distribution of the probability (electronic density) SPLITTING OF THE d-LEVEL INTO A TRIPLET AND A DOUBLET IN THE CUBIC CRYSTAL FIELD Octahedral transition metal Transition metal ions –d orbitals complex l =2→m =-2 -1,0, 1,2 Point symmetry group : ______cubic,Oh Five orbitals with the same energy

L5 The d-level is 5-fold degenerate !!! L 3 Five d –functions : L 4 ϕ3dlm(r, θ,ϕ )=R3d(r)Ylm( θ,ϕ)

M L2 r R3d(r) – radial wave function,

Ylm( θ,ϕ) –spherical harmonic Angular parts of d-functions L1 L5 Y2,-2, Y2,-1,Y2,0, Y2,1,Y2,2 (2) M – transition metal ion D =T2+E - crystal field L –ligand i (triplet + doublet) Wavefunctions for the T2 state 22/1 ϕyz = 1,2 + −1,2 ϕϑϕϑ3d rRYYi = π 3d rRryz )()/()4/15()()),(),()(2/( ≡ϕξ 22/1 ϕxy −= 2,2 − −2,2 ϕϑϕϑ3d rRYYi = π 3d rRrxy )()/()4/15()()),(),()(2/( ≡ϕς 22/1 ϕzx −= 1,2 − −1,2 ϕϑϕϑ3d rRYY = π 3d rRrzx )()/()4/15()()),(),()(2/1( ≡ ϕη

In a regular octahedral complex(point group Oh) ,where there is no distinction between the x,y,z coordinates, the zx , yz and xy orbitals lying between these coordinates are undistinguishable

and of equal energy: the T2 –state is triply degenerate dyz d z z y xy dzx

y x x

x z y

Shapes of three d-orbitals of t2 type Wavefunctions for the E state 2222/1 22 = ϕϑϕrRY = π − rRrrz )()/)3(()4/15()(),( ≡ϕ 3 −rz 30,2 d 3 ud 2222/1 ϕ 22 = + ϕϑϕϑrRYY = π − rRryx )()/()6/15()()),(),()(2/1( ≡ϕ −yx 2,2 −2,2 3d 3d v The z2 and x2-y2 orbitals lie along the x,y,z axes. Since these axes are indistinguishable,it follows that these two orbitals are also indistinguishable, have equal energy and therefore form the basis for the two dimensional representation E

Shapes of two orbitals of e-type SPLITTING OF THE d-LEVEL INTO A TRIPLET AND A DOUBLET IN THE OCTAHEDRAL CRYSTAL FIELD

Cubic crystal field parameter Dq Eg 42 1 rZe d Dq = 10Dq 5 6 R 0

Ze −charge of the ligands T2g R0 − metal-ligand distance in the octahedral surrounding Octahedral surrounding 10 Dq − crystal field splitting of the one-electron d-level − mean value of r4 for M the d-electron

R0 d1-electron → Ti3+, V4+, Mo5+,Zr3+ Qualitative interpretation of the obtained result

d-orbitals of t2 type

dyz d z z y xy dzx

y x x x z y ™The two sets of orbitals are d- orbitals of e-type distinguishable because the orbitals dyz,dxy,dzx lie between 2 2 the axes, and the orbitals dx -y , 2 dz along them

™ An electron in the e set points directly towards the charge of the ligand and therefore suffers greater destabilization than an

electron in the t2 set which points between the ligands d-level splitting in a crystal field

T2g 42 Eg rZe Dq)( oct = 5 T2 6 R 0 4(Dq)cub 6(Dq)oct 1 d 4(Dq) 42 tet 2 rZe Dq)( tet −= 5 4(Dq) 6(Dq)tet 27 R0 oct 6(Dq)cub 4 rZe 42 T E 2g Dq)( cub −= 5 Eg 27 R0

(Dq)oct=−(9/4)(Dq)tet

= −(9/8)(Dq)cub Octahedral Tetrahedral Cubic ligand ligand ligand surrounding surrounding surrounding ABSORPTION SPECTRA OF d1 IONS 2S+1 Γ 2S+1-spin multiplicity, Γ -irreducible representation tet oct

Octahedral complex Tetrahedral complex 2 2 Eg T2g Intensity, a.u. 1 Ω 1 d d 10(Dq) oct 10(Dq)tet Ω2 Ω1

2 Eg peak positions of the 2 absorption bands T2g

(Dq)oct > (Dq)tet→Ω1>Ω2

transition metal ion

ligand 3+ OCTAHEDRAL Ti(H2O)6 COMPLEX TETRAHEDRAL VCl4 COMPLEX Ti3+(d1) V4+(d1)

-1 The band at 20100 cm-1 is a crystal The band at 9000 cm is 2 2 assigned to the 2E→2T field transition T2g → Eg 2 transition ‰The tetrahedral TiCl 3- complex has 6 Octahedral V4+ complexes an absorption band at 13000 cm-1 have bands in the range of 3+ ‰Few tetrahedral complexes of Ti 14000-18000 cm-1 are known The absorption d-d bands of tetrahedral complexes are red shifted in comparison with those arising from octahedral ones Typical (experimental ) values of Dq in transition metal

complexes with H2O ligands Ion Ti3+ V3+ Cr3+ Mn3+ Fe3+ Co3+ (d1) (d2) (d3) (d4) (d5) (d6)

Dq,cm-1 2030 1860 1720 2100 1350 1920

Ion Cr2+ Mn2+ Fe2+ Co2+ Ni2+ Cu2+ (d4) (d5) (d6) (d7) (d8) (d9) Dq,cm-1 1390 1230 1030 840 820 1220

ƒCrystal field splitting 10Dq in octahedral complexes is of the order of 10,000-20,000 cm-1 ƒEmpirical rule: irrespective of the ligand and metal ions, 10Dq in the systems with divalent metal ions is around 10,000 cm-1 and in those with trivalent metal ions around 20,000 cm-1 (1 eV =8066 cm-1) SPECTROCHEMICAL SERIES

9 The change of the ligand elements gives the variation of 10Dq which is of the order of 1000 cm-1 9When the metal element is fixed and the ligand is varried, the magnitudes of 10Dq may be arranged in the following order: I

9When the ligand is fixed and the metal ion is varied, the magnitudes of 10Dq may be arranged in the following order: Mn2+

group Oh-cubic y -metal ion x -ligand

Tetragonallyx distorted octahedral complexes point symmetry group D4h: metal-ligand distances are equal in the equatorial plane (x,y), but different from those for axial ligands (z-axis)

Elongated Compressed along the C 4 along the C4 axis axis Octahedral complex elongated along the C4 axis 2- 2 E2g B1g (x y ) d 2 d 2 z x -y 2 A (z2) d1 1g

B2g (xy) T2g Perfect octahedron, E (zy,zx) x g Oh point group Elongated along the C4 d d dxy z zy z zx y axis, D4h point group

The optical band y x x splits into three bands

‰ The z-axis is distinquishable from the x and y axes ‰ Due to the elongation of the octahedron along the z-axis the z2 orbital suffers less repulsion from the ligands than the x2-y2 orbital and its energy becomes lower than that of the x2-y2 orbital ‰The energy of the xy orbital is higher than that of the zy and zx orbitals because it suffers more repulsion from the ligands UV/Vis/NIR spectra of vanado-molybdophosphate

heteropoly acid H4PVMo11O40 •y H2O treated in He

‰Treatment in He 663 K from RT to 663 K leads to the strong 2 467 K increase of the 422 K apparent absorption 371 K ‰ The NIR absorption 1 bands of the crystal water disappear

Kubelka-Munk units (a.u.) 326 K completely 0 ‰At higher 500 1000 1500 2000 2500 temperatures a Vis wavelength (nm) broad absorption band, and additionally a NIR absorption band can be clearly detected 2 1 3 −Mo, V −O 3 4 R1=1.68Å 5 5 O1, O2, O3, O4 –bridging R=1.93Å (O ) 6 4 2 b 12 O5 –terminal (Ot ) R =2.32Å 2 O −oxygen in the 1 6 11 phosphorus environment 9 6 (O ) 8 10 p

7 According to structural data in heteropoly compounds of Keggin type Idealized structure of a the site symmetry of Mo and V Keggin molecule ions is close to C 1. V , 2-12− Mo 4v

At 300K the loss of crystal water begins after a certain time of gas flux. At this initial stage there is no reduction, the HPA is partially hydrated, the protons are not localized and reside on the bridging water + moieties H5O2 . Thus, initially the spectra should originate from 5+ 4+ d-d transitions in the unimpaired Mo (O)6 and V (O)6 units d-d transitions in the unimpaired MoO6 and VO6 units a (3z2) 1 5 3 2 2 2 2 e( x -y ,3z -r ) R1=1.68Å 2 2 b1 (x -y ) R=1.93Å R1

C4V ™The orbitals x2-y2 and z2 lie along the x-,y- and z-axes, respectively, whereas the orbitals xy,zy,zx lie between the axes, therefore, the interaction between the orbitals x2-y2 and z2 and the ligands is stronger, and they are higher in energy ™ The z2 orbital has the highest energy because it overlaps with the orbital of the ligand 5 having the shortest distance R1 from the metal ion. Under the condition R1

MoO6 and VO6 V4+ Mo5+

b2→e b2→b1 b2→e b2→b1 770 625 710 440 d-d transitions in protonated units At temperatures from 326 to 422 K the crystal water loss is accompanied by the localization of acidic protons the most preferable sites of which are the bridging Ot oxygens 2 Proton a1 (3z ) Ob yz R =1.68Å added 1 e( x2-y2,3z2-r2) 2 2 R=1.93Å b1 (x -y ) 2 2 2 4+ 5+ ϕ=α(x -y )+β(z ) Ob V , Mo Ob R =2.32Å 1 e(zy,zx) zx 2 d t2 (xy,zy,zx) Ob O Op b2 (xy) xy C4V ™ If a proton resides on the bridging oxygen the charge of Cs this ligand becomes less Peak positions (in nm) of the d-d transitions negative, the repulsion between the xy orbital and the ligand is V4+ Mo5+ weaker, and the energy of this Intact b2→e b2→b1 b2→e b2→b1 orbital diminishes species 770 625 710 440 ™ The energy of the xz orbital Protonated xy→zx xy→ϕ xy→zy xy→zx xy→zy xy→ϕ also decreases ,but it is higher species 743 720 520 700 505 495 than that of xy, because the orbital xz lies in the xz-plane , Protonated reduced species give rise to new d-d transitons and R1

Questions to be discussed ---- − calculated 1 – contribution from the d-d 1. Which d-d transitions are transitions allowed ? 2 − contribution from the charge 2. What interaction is transfer band responsible for the shape of 3 − total calculated spectrum 4 − experimental data the d-d bands ? 3. What is the origin of charge transfer bands? Which d-d transitions are allowed ? Probabilities of transitions in

Two d- levels with energies E1 and E2 , corresponding wave functions Ψ1 and Ψ2

Ψ2 E2

Light absorption Light emission

E Ψ1 1 The intensity of the optical line – probability of the transition 1↔2 . The transition 1↔2 is caused by the interaction of the quantum system with light. This interaction is mainly the interaction of the dipole moment of the quantum system with the electric component of the radiation (electric dipole transitions) Probabilities of transitions Operator of the dipole moment for an electron d = er r is the coordinate of the electron Intensity of the is proportional to the squared matrix element of the dipole moment : * = ∫ 221 ( )()1 rdrd dτΨΨ Dirac’s notation for the matrix element d 12 Intensity of the optical line 1↔2 2 d 12

The electronic transition is assumed to be allowed if the matrix element of the transition <2|d|1> is non-zero What interaction is responsible for the shape of the d-d bands ? MainMain observationsobservations ¾ The optical d-d bands are very broad ¾ The widths of the d-d bands are of the order of 103-6•103 cm-1 ¾ The d-d bands in crystals are considerably broader than the lines in the atomic spectra ¾The interaction of d-electrons with the crystal or molecular vibrations is responsible for the width of the d-d bands Electron-vibrational interaction

™The transiton metal center is considered as a complex formed by the central ion and the adjacent ions of the lattice (ligands) ™The displacements of the ligands appear due to the modulation of the crystal field by lattice vibrations ™The interaction of the d-ion electrons with the ligand displacements represents the electron -vibrational interaction . Actually, the electron- vibrational interaction is the change in the energy of the cluster due to the displacements of the ligands

Increase in energy Decrease in energy Full –symmetric vibration of the octahedral metal z complex 3 4 Octahedral ML6 complex: 5 coordinate system and enumeration of ligands 1 2 y x R0-equilibrium metal-ligand (M-L) distance 6 Xi, Yi,Zi − ligand displacements

Q –full symmetric (“breathing”) collective coordinate

of the ML6 complex R>R0 R=R0 R

r Potential energy kx 2 spring xU )( = 2 k =M ω 2 –force constant, M –mass, 0 M ω- frequency of vibration Force acting on the mass M 2 X F(x) ∂ ()xU ∂ kx ()xF −= −= , ()xF −= kx ∂ ∂xx 2 Vibronic parameters

2 Two d- levels with energies E1 and E2 ‰The value ρ (r)=| ψ(r)| and wave functions Ψ and Ψ 1 2 is the probability to find the electron in the proximity of the Ψ2 E2 point with the coordinate r

Light ‰ eρ(r) is the density of the absorption electronic charge(“cloud”) in E the point r (e –charge of electron) Ψ1 1 Electron-vibrational interaction ‰ electronic densities are different in the states ψ and ψ r 1 2 = )( QrvH 2 ev ρ1(r)=| ψ1(r)| 2 ρ2 (r)=| ψ2(r)|

™Vibronic parameters v1 and v2 represent the mean values of the operator v(r) in the states Ψ1 and Ψ2 2 2 v1 =∫ dr v(r)| ψ1(r)| , v2 =∫ dr v(r)| ψ2(r)| ™Vibronic parameters do depend on the electronic state and are different for states 1 and 2 HAMILTONIAN OF THE COUPLED ELECTRON-VIBRATIONAL SYSTEM

ω ⎛ ∂ 2 ⎞ ˆ = + υ + h ⎜ 2 − ⎟ e ()rHH ()r Q ⎜ Q 2 ⎟ 2 ⎝ ∂ Q ⎠

electronic vibronic vibrational energy interaction energy

potential energy kinetic energy ω ω ∂ 2 h 2 − h Q 2 2 2 ∂ Q

electrons move fast ligands move slowly

M ion >> m electron At the first step one can neglect the kinetic energy of ligands (heavy ions) TWO STEPS OF THE ADIABATIC APPROXIMATION

1) Neglect of the kinetic energy of ligands (heavy ions) Initial (exact) Hamiltonian ω ∂ 2 rr )()( h QQrvrHH 2 −++= )( e 2 ∂Q 2

Kinetic Electronic Vibronic Potential energy energy interaction energy of ligands of ligands New (approximate) Hamiltonian ω rr )()( ++= h QQrvrHH 2 e 2 2) Averaging of the vibronic interaction over all possible positions of the electron New concept : Adiabatic potential What does it mean ?

ADIABATIC POTENTIALS U1(Q) AND U2(Q) FOR A TWO LEVEL SCHEME

() υ hω QQEQU 2 ⇒++= state "1" 1 11 2 ω () υ h QQEQU 2 ⇒++= state "2" 2 22 2 E1,E2−energies of an electron in a static crystal field (full energy of an electron)

υ2 , υ1 –vibronic parameters for states 1 and 2 υ1Q , υ2Q –averaged energies of the interaction of the electron with the vibration Q ω h Q 2 −potential energy of vibrations 2 ADIABATIC POTENTIAL CAN BE ASSOCIATED WITH THE FULL ENERGY OF THE SYSTEM , WHEN THE KINETIC ENERGY OF LIGANDS IS NEGLECTED ADIABATIC POTENTIALS U1(Q) AND U2(Q) FOR A TWO LEVEL SCHEME

() υ hω QQEQU 2 ⇒++= state "1" 1 11 2 ω () υ h QQEQU 2 ⇒++= state "2" 2 22 2

ƒAdiabatic potential=Full energy of the system (electron+vibration) in the adiabatic approximation

ƒThe full energy of the system = the energy of the electron in the crystal 2 field (E1,E2) + the potential energy of the vibration(hω Q /2) + the energy of the interaction between the electron and ligands (υ1Q , υ2Q ) averaged over all instant positions of the electron

ƒThe full energy of the system depends on the vibrational coordinate Q ADIABATIC POTENTIALS U1(Q) AND U2(Q) FOR A TWO LEVEL SCHEME

U(Q) 2()QU

U2(Q)-excited state adiabatic potential

E h Ω= mm 1()QU

U1(Q)-ground state adiabatic potential

0 Q0 Q ELECTRON-VIBRATIONAL TRANSITIONS- FRANCK-CONDON PRINCIPLE

‰ This principle represents the main tool for understanding the optical absorption and emission in electron vibrational systems and the basis for calculation of the shape of the optical bands

REMINDER ELECTRONS ⇒ “fast” subsystem, LIGANDS ⇒”slow” subsystem

“CLASSICAL” FRANCK-CONDON PRINCIPLE

‰ In the event of light absorption the electronic state (“fast” subsystem) changes so rapidly, that during the time of electronic transition in course of the photon absorption the ligands (“slow” subsystem) have no time to move (to change their positions)

‰ The condition implied by the adiabatic approximation should be fulfilled: the kinetic energy of the ligands is zero ORIGIN OF THE OPTICAL d-d BAND

U(Q) U2(Q) •Instead of one transition we see a series of transitions at different Q Ωm that form the absorption d-d band Ω4 Ω3 • Each transition from the lower Ω5 adiabatic sheet to the upper one Ω 2 corresponds to a certain frequency Ω6 Ω 1 •In correspondence with the Franck- Condon principle only vertical

U1(Q) transitions are allowed •Non-vertical transitions are 0 Q forbidden because in this case the ligands change their positions Ω6>Ω5>Ω4>Ωm >Ω3>Ω2>Ω1 during the event of the light absorption frequencies Adiabatic potentials (already found):

() υ hω QQEQU 2 ⇒++= state "1" 1 11 2 ω () υ h QQEQU 2 ⇒++= state "2" 2 22 2 SHAPE-FUNCTION OF THE ABSORPTION BAND

3ω ⎛ 3ω Ω−Ω )( 2 ⎞ )( =Ω dF 2 h exp⎜− h m ⎟ 21 2 ⎜ 2 ⎟ π B − vvTk 12 )(2 ⎝ π B − vvTk 12 )(2 ⎠ ν − vv )( EE +−=Ω 211 * h m 12 = 221 1 rdrd )()( dτψψ hω ∫ ™The absorption band is described by a “GAUSSIAN “ curve

™ The band has a maximum at Ω= Ωm , the band can be observed if d21 ≠0 ™ The shape of the band and the position of the maximum

depend on the vibronic parameters υ1, υ2 , temperature T and the energy gap E2 –E1 between the electronic levels in the crystal field Optical absorption band ⇒symmetric “bell-

shaped” curve with the maximum at Ω=Ωm F(Ω ) –height of the band 2 m Intensity of absorption, F(Ω)/d21 in the maximum 1 3 2 h ω 0.8 m )( =Ω dF 21 2 half-width BTk −ννπ12 )(2 0.6 half-height 0.4 Half-height

2 3 0.2 Ωm )( dF 21 h ω = 2 BTk −ννπ12 )(222 -2 -1 1 2 Ω m Ω/ω Half-width 2 Frequency of the absorbed light −ν 12 )( kTv δ =Ω 2ln22 3 h ω 3ω ⎛ 3ω Ω−Ω )( 2 ⎞ )( =Ω dF 2 h exp⎜− h m ⎟ 21 2 ⎜ 2 ⎟ δ ∼Ω T π B − vvTk 12 )(2 ⎝ π B − vvTk 12 )(2 ⎠ F m ∼Ω /1)( T EXAMPLE

Structural data: Malachite contains two types of Cu species

z Malachite Cu1 1 5 R1=2.51 Ǻ, R2=2.64 Ǻ,

R3=R5=1.91 Ǻ, R4=R6=2.02 Ǻ 6 4 y 3 Malachite Cu2 2 x R1=2.28 Ǻ, R2=1.92 Ǻ,

R3=R5=2.08 Ǻ, R4=R6=2.37 Ǻ Cu O Malachite Cu2(CO3)(OH)2 (Germany, FHI, Nov. 2006)

1 Cu1: 740 nm Cu2: 840 nm, 1100 nm 0.8 Cu1+Cu2: 788 nm, 1100 nm trary units i 0.6 Experimental data 0.4 The red, blue and Absorption, arb green curves 0.2 are calculated

500 1000 1500 2000 2500 Wavelength, nm What is the origin of charge transfer bands? CHARGE TRANSFER BANDS

‰ Charge transfer bands arise as a consequence of the absorption of light which occurs when an electron is transferred from an orbital primarily on the ligand to one primarily on the metal, or vice versa ‰ Charge transfer or electron transfer processes are termed “ligand to metal charge transfer” (LMCT) and “metal to ligand” (inverse) charge transfer (MLCT), respectively . Such a transfer occurs frequently in transition metal complexes, but only if this absorption occurs in the visible range it will give rise to a color visible to the human eye ‰ Charge transfer transitions are commonly of higher energies than the crystal field transitions and generally lie at the blue end of the visible region, or in the ultraviolet region

‰ If the metal is easily oxidisable and the ligand readily reducible ,or vise versa , then charge transfer transitions may occur at quite low energies EXAMPLE : METAL HEXAHALIDES

Three main regions are observed in the charge transfer electronic spectra of these species: ™nd3,nd4,nd5 complexes exhibit narrow LMCT bands between 15,000 and 30,000 cm-1

™All hexahaloanions of the 4d and 5d elements exhibit broad strong absorption in the region 25,000-45,000 cm-1

™In certain cases it is possible to observe a very strong band lying above 44,000 cm-1 Simplified approach to the explanation of charge transfer spectra in hexahalides ¾A single metal-halogen bond is considered ¾The lone pairs of the halogen form the basis for the σ and π bonds to the d-ion ν2 eg π ν1 t2g d p π zy y d ν4 σ π ν METAL 3 2 dz p σ z L---Mt σ ν1 π 2g

Ligand ν2 πL---Meg M.O. ν3 σL ---Mt2g A molecular orbital diagram illustrating the potential ligand to metal charge transfer ν4 σL ---Meg transitions ASSIGNMENT OF THE LMCT TRANSITIONS IN HEXAHALIDES

ν2 • The first set of eg ν comparatively low energy 1 bands observed in the t2g hexahalides may be π ascribed to the ν d ν4 1 transitions. They are π ν narrow because they METAL 3 involve the transitions σ between orbitals almost non-bonding σ Ligand •The ν2 transitions are M.O. expected to be broad because the upper level is antibonding • The very high energy, and very intense, bands

are assigned to ν4 INTERVALENCE OPTICAL ABSORPTION Let us consider a cluster containing a single d-electron over the filled ion shells ⇒d0-d1 (Mo5+-Mo6+cluster) •The moving electron can be with equal 1 0 0 1 d d d d probability on each of the cluster ions •the wave functions ϕ and ϕ abab a b corresponding to the states of the electron ϕ ϕb a localized on ions a and b are not the stationary states of the system

1 ψ + += ϕϕba )( 2 2p 1 ψ − −= ϕϕba )( 2 The energy spectrum of such a system consists of two tunnel electronic levels with energies ±p 1 and wavefunctions ψ ± ±= ϕϕba )( 2 Mo5+ 6+ Mo Mo5+-Mo6+ -cluster

Qa, Qb are the totally symmetric (breathing) modes of the two q>0 a b cluster moieties The breathing mode best of all “pushes” out or “draws” in the excess electron when the nearest surrounding of the q=0 a b metal ion contracts or expands

1/2 q=(Qa-Qb)/(2) is the antisymmetric mode of the whole cluster consisting q<0 a b of two coupled octahedral Mo6+ Mo5+ metal centers

1/2 q=(Qa-Qb)/(2) (light induced electron transfer)

1.2 U(q) ab U1(q) 1 Dimeric cluster Mo5+ – Mo6+ 0.8 a, b- Mo ions 0.6 U2(q) 0.4 2p 0.2 absoption relaxation q the transition -2 -1 1 2 -0.2 a(Mo6+) b(Mo5+) a(Mo5+) b(Mo6+) a(Mo6+)b(Mo5+) a(Mo5+)b(Mo6+) Is induced by light

ADIABATIC POTENTIALS p – transfer parameter, ω – vibrational ω h 2 222 frequency, υ – vibronic constant, Q -full 2,1 )( +±= υ qpqqU c 2 symmetric breathing mode of the local 1 −= QQq )( surrounding of ion c (c=a,b) 2 ba 2 2 2 T υ ⎛ υ ⎞ 4 Tk υ 4 +=Ω ⎜ ⎟ − B T h max 3 ω ⎝ 22 ω ⎠ hhh ω T2 T 1 The energy of the maximum of the charge transfer band Absorption, a. u. Absorption, a. decreases with temperature

Ω T1< T2 < T3 < T4 h hΩmax1 > hΩmax2 > hΩmax3 > hΩmax4 RED SHIFT λ max1 < λ max2 < λ max3 < λ max4 CHARGE TRANSFER BAND IN THE HETEROPOLY . ACID H4PVMo11O40 y H2O AT 422 K ‰The TG experiments evidence complete crystal water loss at 422K ‰ After crystal water loss all four acidic protons are attached to the polyanion ‰ EPR data show that in the untreated HPA the Keggin unit represents a one electron-reduced structure. The electron can be delocalized over twelve metal sites (11Mo and V) of the Keggin molecule CHARGE TRANSFER BAND IN THE HETEROPOLY

ACID H4PVMo11O40 .y H2O AT 422 K ‰The charge transfer band was shown to be formed by intervalent optical transitions in intact and protonated 5+ 6+ 5+ 6+ species (Mo Hb)(Mo Hb)O11, (Mo Hb)(Mo )O11,

5+ 5+ 5+ 6+ (Mo Hb)(V )O11, Mo Mo O11 4+ 6+ 4+ 6+ 4+ 6+ (V Hb)(Mo Hb)O11, (V Hb)(Mo )O11, V Mo O11

protonated species intact species

proton added Absorption band of the H4PVMo11O40 compound at 422 K

4

2 3

1

---- − calculated Contributions from different intervalent 1 – contribution from the d-d transitions in binuclear species transitions to the charge transfer band 2 − contribution from the charge 1.(MoH )(MoH )O -13.2%; transfer band b b 11 2. (MoHb)MoO11-33%; 3 − total calculated spectrum 3. (MoHb)VO11-15%;4.Mo2O11-18.2%; 4 − experimental data 5. (VHb)(MoHb)O11-1.6%; 6. (VHb)MoO11-4%; 7.VMoO11-15%; 8-total charge transfer band