ZZ_0_As_Example - Molecular Orbitals ­part I ...

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Unformatted text preview: Molecular Orbitals ­part I Diatomic molecules: bonding proper8es Once energies (E) are known, we may determine wavefunc8ons (MO wavefunc8ons) The Born ­Oppenheimer approxima8on •  The Born ­Oppenheimer (BO) approxima8ons consists in assuming that for a broad class of molecular and solid systems, the separa8on between nuclear and ionic mo8on is approximately correct. The approxima8on rests on the fact that the nuclei (with mass Mα) are much more massive than the electrons (with mass me ) and thus in most cases they are nearly fixed with respect to electronic mo8on. Hence we can write: M. Born, R. Oppenheimer, Ann. Phys. (Leipzig) 84, 457 (1927). The Hartree ­Fock approxima8on Approxima+on for the many ­body electronic wavefunc+on which is expressed as an an+symmetrized product of N orthonormal single par+cle orbitals, each wri@en as a product of a spa+al orbital and a Prof. D.R. spin func+on: Hartree, Cambridge, UK, 1897 ­ 1958 Prof. V.A.Fock, St.Petersburg, Russia 1898 ­1974 Hydrogen Fluoride dimer Non Bonding p Bonding σ 2s F orbitals not shown OH: Oxygen 2p closer to hydrogen 1s. The highest MO of OH are also non bonded; occupaCon is different than in HF NaEalene, not aniline, may be described by the Hueckel approximaCon Walsh diagrams: BeH2 and H2O Rota%onal axes and mirror planes of the water molecule: C2 principal axis C2 σv mirror plane C2 σv mirror plane The water molecule has only one rotaConal axis, its C2 axis, which is also its principal axis. It has two mirror planes that contain the principal axis, which are therefore σv planes. It has no σh mirror plane, and no center of symmetry. •  Show that if an Hamiltonian H is separable into two terms, one involving only coordinates q1 and the other involving only coordinates q2 [H(q1,q2) = H1 (q1) + H2(q2)], then the eigenfuncCons of H are products of the eigenfuncCons of H1 and H2 and the eigenvalues of H are the sums of eigenvalues of H1 and H2. •  Can the Hamiltonian of a molecular system composed of nuclei and electrons be wriFen in a fully separable form (i.e. as a sum of a nuclear and electronic part)? •  Explain what the Born ­Oppenheimer approximaCon is and consider the molecule, in the following group, for which you think the BO approximaCon is most appropriate: HCl, HI and KCl ( jusCfy your choice). The infrared spectrum of such a molecule has an intense line at 278 cm ­1. Calculate the force constant and the vibraConal frequency of the molecule. Use the harmonic approximaCon. KCl (heaviest): From the allowed energies of the HO and knowing the reduced mass of the molecule one may compute the force constant: ν= (1/2π) sqrt(k/µ) •  Compute the average value of the square of the posiCon operator in one dimension for the ground state of a harmonic oscillator. Use the expression to evaluate the root mean square amplitude of vibraCon in the molecule chosen in 3. <x2> = h/[4π sqrt (µk)] •  Explain what the following types of spectroscopic techniques are and which physical quanCty they may be used to probe: Infrared, Raman, NMR, UV ­visible, laser spectroscopy. •  Explain what the Fermi golden rule is and which approximaCons are involved in its derivaCon. Write an expression for the rate of transiCon between an iniCal and an excited state of a molecule in the presence of electromagneCc radiaCon (consider electric field only for simplicity). Discuss selecCon rules. ...
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