Molecular orbitals, potential energy surfaces and symmetry mathematical presentation of molecular symmetry group theory spectroscopy valence theory molecular orbitals Wave functions Hamiltonian: electronic, nuclear wave functions H = Hel + Hvib + Hrot Energy scales: 10-10000 ev, 0.1 ev, 0.01 ev
Molecular orbitals: constructed from atomic orbitals Atomic orbitals have spherical symmetry basis set is spherical harmonics Yl m Molecular orbitals have symmetry determined by molecular geometry basis set is atomic orbitals, ϕ A ϕ B Pauli principle applies Effective bonding energies of atomic orbitals are similar overlap between orbitals strong same symmetry with respect to rotation about molecular bond axis
Molecular bonds: bonding and antibonding Antibonding orbital Node (zero density) between nuclei DESTRUCTIVE INTERFERENCE + Bonding orbital nonzero electron density between nuclei CONSTRUCTIVE INTERFERENCE
Energy level diagrams for homonuclear diatomic molecules Li2 to N2 O2 and F2
Energy level diagram for heteronuclear molecules Atomic orbitals ϕ A ϕ B s-orbitals different atomic energy form σ molecular orbitals
Example: hydrogen molecular ion Calculate orbitals for the hydrogen molecular ion, H2+ Calculate potential energies for bonding and antibonding sigma states Use atomic 1s orbitals as a basis set Effective charge described by ζ (= 1 for hydrogenic case) The LCAO wave function in it s simplest form
Basic strategy and equations Solve the SE using the Hamiltonian for a 3-body system (H1, H2, e-) Two wave functions, two equations use symmetry to simplify solve for energy eigenvalues use atomic units...
Symmetry... For a homonuclear molecule the basis wave functions are identical, and the coefficients in the LCAO wave function are equal. H11=H22 Two solutions: c1 =
Symmetry.. i gerade i ungerade
Calculation of wave functions, potentials and other quantities... A brief MATLAB interlude...
What does it mean?
Polyatomics: Symmetry operations and elements I, identity Cn, n-fold rotation σ Reflection (in a plane) i, inversion (through a center of symmetry) Sn, improper rotation (n-fold rotation in a plane followed by reflection through a plane perpendicular to rot plane)
Rotation Reflection Improper rotation Inversion S = C n. σ"
Symmetry groups Lowest symmetry C1, Cs, Ci Cn: I, n-fold rotation (about principal axis) Cnv: I, n-fold rotation, vertical reflections Cnh: I, n-fold rotation, horizontal reflection Dn: I, n-fold rotation, about principal and perpendicular axis Dnh: Dn and horiz reflection (homonuclear) Sn: I, n-fold improper rotation C2v
Calculus of symmetry elements Symmetry elements generate other elements: C3 2 = C3 x C3 All symmetry elements are generated from others Matrix representation Group theory applies to molecular symmetries
Character tables Symmetry classification of properties of molecules (wave functions, orbitals, states) Degeneracy Symmetry elements Basis for constructing orbitals, vibrational motion, electronic transitions
Example: C2v point group character table (water, SO2, etc) Generating elements Irreducible representation of symmetry species I C 2 σ v σ v A 1 1 1 1 1 A 2 1 1-1 -1 B 1 1-1 1-1 B 2 1-1 1-1
Heteronuclear diatomics Σ+ Σ Π Δ I C... σ v A 1 1 1 1 A 2 1 1-1 E 1 2 2cosφ 0 E 2 2 2cos2 φ$... 0
Molecular bonds: σ symmetry
Molecular bonds: π symmetry
Potential energy function Neutral oxygen molecule
Symmetry adapted Linear Combination AO Example: water C2v symmetry I C2 σ v(xz) σ v(yz) A1 1 1 1 1 A2 1 1-1 -1 B1 1-1 1-1 B2 1-1 -1 1
Construct wave functionsfollowing the rules I C2 σ v(xz) σ v(yz) A B O 2s 1 1 1 1 O 2px 1-1 -1 1 O 2py 1-1 1-1 O 2pz 1 1 1 1 A1 B2 B1 A1 H 1sA 1 H 1sB 1 H 1sB H 1sB 1 H 1sA 1 H 1sA 1/2 1sA + 1sB 1/2 1sA - 1sB A1 B2
Symmetry adapted wave functions Ψ(A1) = c1 O 2s + c2 O 2pz + c3 ϕ (A1) Ψ(B1) = O 2py Ψ(B2) = c4 O 2px + c5 ϕ (B2) where the symmetry adapted molecular orbitals are ϕ (A1) = 1/2 1sA + 1sB ϕ (B2) = 1/2 1sA - 1sB
Molecular orbitals for water
Molecular orbital labels Water molecule: 8 oxygen electrons 2 hydrogen 1s Walsh diagram Ground-state configuration and term bent XH 2 molecules linear O1s 2 1a1 2 1b2 2 2a1 2 1b1 2 1 A1 4a 1 3 g 1b1 1 u 1b2 A1... a1 B1...b1 B2... b2 3a 1 2a 1 1 u 2 g 90 120 150 180 Bond angle 2
Vibrational modes A1 B1 Symmetric antisymmetric
Quantum mechanical treatment of vibrations in molecules The full wave function is a product of electronic and nuclear wave functions (Born-Oppenheimer approx) electrons (r) immediately follow the nuclear motion (R) Break down matrix element to electronic and nuclear transition elements
Quantum mechanical treatment of vibrations in molecules Attractive restoring force (approximate harmonic osc)... solve the Schrödinger equation:! With eigenfunctions based upon Hermite polynomials v = 1 1/2 H 2 v v! 1/2 v (y) exp( y 2 /2) and eigenfrequencies: = 1 2 k µ! 1/2. Table 2.1: Hermite polynomials. v H v (y) v H v (y) 0 1 3 8y 3 12y 1 2y 4 16y 4 48y 2 +12 2 4y 2 2 5 32y 5 160y 3 +120y! 1/2 y = 4 2 µ (r r e ) h
Vibrational progressions E ν = E 0 + ω 0 (ν +1/ 2)+ ω 0 x e (ν +1/ 2) 2 Adiabatic binding energy Harmonic term Anharmonic term
Molecular potential function, vibrations and dissociation
Franck-Condon principle Absorption of radiation-vibronic transitions Electronic transition from state with energy E el and vibrational quantum number v =0 to an excited state E el The electronic state transition is independent of the vibrational transition Electronic state transition: DIPOLE selection rules Vibronic transition, no strict selection rules E el(v ) --> E el(v )
Frank-Condon principle: intuitive picture Franck-Condon region is defined by the ground-state wave function Absorption from the ground state is strongest to the wave function in the excited state that has a maximum in the center of the Franck-Condon region
Franck-Condon principle: quantum mechanical treatment Transition moment for vibrational transitions S = Ψ 0 (x)ψ' o (x)dx general form of vibrational wave functions Hv are Hermite polynomials, and α = 2π/h mhv S 0 0 = e α 4 2 ' ( R e R e ) 2 Δq = 2 S µω gr 1/ 2 ω gr ω fin