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3.5 Matrix Elements and Selection Rules

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3.5 Matrix Elements and Selection Rules 3.5 Matrix elements and selection rules The direct (outer) product of two irreducible representations A and B of a group G, ⎛ a11b11 a11b12 a12b11 a12b12⎞ ⎜ ⎟ ⎛ a11B a12B⎞ a11b21 a11b22 a12b21 A ⊗ B = C = ⎜ ⎟ = ⎜ ⎟ ⎝ a21B a22B⎠ ⎜ a21b11 ⎟ ⎜ ⎟ ⎝ ⎠ gives us the chance to find out the representation for which the product of two functions forms a basis. This representation will in general be reducible. € Theorem: The character of the direct product representation matrix is equal to the product of the characters of the separate irr. rep. matrices. Note that we are dealing with the representations within one group, since we deal with a system with well-defined symmetry. Thus “products” of functions such as <Ψb|Ψa>, where each function forms the basis for an irreducible representation Γa and Γb of the group, form a basis for an (often reducible) representation Γa ⊗ Γb. Using the reduction formula, we can decompose the reducible representation of the products into the irreducible ones of the group. An expression of the form <Ψf|O|Ψi>, where O is some operator which connects the initial with the final state, i.e. a transition matrix element is an integral over all space. If the product function does not contain a function that forms a basis for the totally symmetric irreducible representation, i.e. if that function is not of even parity under all symmetry operations of the group, then by necessity that integral will be zero. <Ψf|O|Ψi> 0 ≠ <Ψf|O|Ψi> r 0 = <Ψf|O|Ψi> Theorems about direct product representation Theorem: Matrix elements of an operator H which is invariant under all operations of a group are identically zero for functions which belong to different irreducible representations Why? “H invariant under all operations of a group” means it belongs to the totally symmetric irreducible representation. Then H ⊗ Ψa belongs to the irreducible representation to which Ψa belongs. Thus the functions in <Ψf | H | Ψi> transform as tot symm Γf ⊗ Γ ⊗ Γi = Γf ⊗ Γi and functions which belong to different irreducible representations of a group are orthogonal. Generalization to operators which are not invariant with respect to all operations of the group: Similar theorems apply, e.g. Mfi =<Ψf |O| Ψi> . Find out for which irreducible representation the functions and the operator form a basis, multiply and see whether the product forms a basis for the totally symmetric irreducible representation as tot symm Γf ⊗ ΓO ⊗ Γi = Γ ? Theorem: A matrix element <Ψf |O| Ψi> must be identically zero if the direct product representation Γf ⊗ ΓO ⊗ Γi does not contain the totally symmetric irreducible representation. Selection rules: a worked example Consider an optical dipole transition matrix element such as used in absorption or emission spectroscopies ∂ω 2π Fermi’s golden rule = ψ f Hʹψi δ(E f − Ei − hω) t ∂ h The operator for the interaction between the system and the electromagnetic field is e r r r r r Hʹ = (A ⋅ pv€ + p ⋅ A ) often, p ⋅ A neglected mc ∂ω 2π v v 2 = ψ f A ⋅ p ψi δ(E f − Ei − hω) ∂t h 2 2π 2 v = A ψ f p ψi δ(E f − Ei − hω) h approximately. € the momentum operator p transforms as a spatial coordinate: 2 ∂ω 2π ∂ = Ax ψ f ψi ∂t h ∂x 2 ∂ω 2π ∂ = Ay ψ f ψi ∂t h ∂y 2 ∂ω 2π ∂ = A ψ ψ ∂t z f ∂z i h Now split up the transition matrix element to search for possible polarization€ selection rules (s-polarized light: E vector along y, z p-pol light: E vector along x and z) plane of incidence x y example: optical dipole transitions in a Ni atom in a fourfold site on an Cu (100) surface; C4v symmetry applies electric dipole transitions: final state operator initial state 2 2 dx - y z px,y irr.rep: B1 A1 E B1⨂A1⨂E = E transition not allowed dxz,yz z px,y red irr.rep: E A1 E E⨂A1⨂E = Γ = A1 + A2 + B1 + B2 contains A1 - transition is allowed ! Magnetic dipole transitions magnetic dipole moment operator µM = r × p = mr × (dr/dt) axial vector, i.e vector product, or antisymmetric second rank tensor* ! R = A × B, that is Rx = AyBz - AzBy etc. Such axial vectors transform as the rotations R given in the character tables. Then the same procedure can be applied as in the case of the electric dipole operator. *Transforms as a polar vector under rotations but is invariant under inversion (two sign changes in cross product) Basics of photoelectron spectroscopy kinetic energy, Ekin Ekin vacuum level Φ hν Fermi level valence band adsorbate core level • photon polarization - linear, circular • photon angle of incidence • electron emission direction bulk core • electron spin levels • sample state (T,M) 122 initial state is a molecular orbital, or a Bloch wave of an electron in the solid. final state is an electron travelling in the plane of detection - must be symmetric with respect to the travelling plane (normal emission - totally symmetric!) electric dipole transitions: final state operator initial state (polarization of light) z x px irr.rep: A1 B1 B1 allowed x py not allowed x irr.rep: A1 B1 B2 x dz2 not allowed irr.rep: A1 B1 A1 y study effect of symmetry lowering by surface allowed electric dipole transitions: final state operator initial state A1 E (s,p) A1(p) A1 A1g A1 E (s,p) A1(p) E1 E1u coordinates A1 E (s,p) A1(p) A1 A2u x -> E y -> E z -> A1 A1 E (s,p) A1(p) E1 E1g No crossing rule Not a selection rule, but a rule governing interactions between wavefunctions of specific symmetries. Consider eigenfunctions of a Hamiltonian H0 with different eigenvalues Ψ1 and Ψ2, which transform as the same irreducible representation. Assume there is now a perturbation V that has the same symmetry as the Hamiltonian H0. The integrals are then eigenvalues H11 = ψ1 H0 + V ψ1 2 H22 = ψ2 H0 + V ψ2 H11 + H12 1 (H11 − H12 ) 2 E = ± + 4H 1,2 2 2 12 H12 = ψ2 H0 + V ψ1 = H21 interaction term cannot have the same energy if H12 not zero € H H H E With 12 << 11 − 22 € H 2 this leads to E = H + 12 1 11 (H H ) 11 − 22 minimum separation |H12| 2 H12 € E1 = H11 − (H11 − H22 ) V0 So two energy levels which stem from wavefunctions that form bases of the same irreducible representations cannot cross when they interact. € 4. Vibrations in molecules Why study vibrations in molecules? Because they provide information about the molecular environment of an atom, the bonding in the molecule, and indeed the molecular structure itself. Vibrational spectroscopy is a standard technique in many fields of science: chemistry, biology, physics,... Monochro- Two techniques: mator sample detector Infrared (transmission, reflection) spectroscopy IR beam Raman spectroscopy Monochro- Visible mator or UV detector beam sample Which energy range are we speaking of? Most important molecular vibrations are in the range 80 meV - 500 meV 100 - 4000 cm-1 (1 eV = 8067 cm-1) Physical mechanisms a) IR spectroscopy Many molecules have a permanent dipole moment (charge transfer between atoms in the molecule µ = er) If the molecule has a natural vibration frequency, it can interact with incident light by means of the dipole - in fact by the dynamic dipole moment, i.e. change in dipole moment as the atoms in the molecule move -> vibrations which change the dipole moment are IR active modes Raman spectroscopy The Raman effect: Incident light with frequency ω is modulated by the vibrations of atoms in the medium changing the susceptibility. Two side bands produced: The Raman effect is based on an induced dipole moment in a molecule. The induced dipole moment is proportional to the applied external field µind = α E α polarizability tensor Classical explanation is based on a kind of frequency modulation -> change in polarizability during excitation, “side bands” occur at lower and higher frequency (ω0 ± ωi , “Stokes” and “anti-Stokes” line) of the incident radiation IR and Raman spectrum of benzene 4.1 Number and symmetry of normal modes Complex internal motion of atoms in a molecule can be described by a superposition of a small number of simple vibrations -> “normal modes” with fixed frequency A1 A2 Number of normal modes in a molecule: N atoms, with 3 degrees of freedom each Molecule as such has 3 translational and 3 E E rotational degrees of freedom (2 for a linear molecule) So a molecule with N atoms has 3N - 6 vibrational E E degrees of freedom (3N - 5 for a linear molecule). 2- Example: CO3 carbonate ion describe the resultant motion of an atom in a vibration by a superposition of the movement of each atom along three basic vectors, i.e. a Cartesian system on each atom. z1 An important theorem governing the z4 z3 y 1 shape of molecular vibrations, or x1 better vibrational wave functions: y4 y3 z2 Theorem: Each of the normal modes x4 x3 of a molecule transforms as (forms the basis for) an irreducible y2 representation of the molecule. x2 This means that when we consider the movement of all atoms in a complex vibration, i.e. we consider a transformation of the type The transformation matrix Γij is a reducible representation of the movement of the atom in the vibration; a superposition of several normal modes We only deal with the characters of the matrices as usual, so we are only interested in those coordinates that transform into + or - themselves and thus contribute to the character. Example: H2O The identity operation transforms all coordinates into z themselves -> character χ(E) = 9 Operation C2: atoms H1 and H2 interchange - no contribution.
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