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Atomic Structure and Atomic Spectra Chapter 10 Objectives: Apply quantum mechanics to describe electronic structure of atoms Obtain experimental information from atomic spectra Set up Schrödinger equation and separate wavefunction into radial and angular parts Use hydrogenic atomic orbitals to describe structures of many-electron atoms Use term symbols to describe atomic spectra Fig 10.1 Emission spectrum of atomic hydrogen Conservation of quantized energy when a photon is emitted. Energy levels of the hydrogen atom ni = 3 ni = 3 ni = 2 nf = 2 1 1 R H 2 2 nf ni where: R H 109 677 cm 1 R H 2.18 x 10 18 J nnf f==11 Rydberg constant Structure of Hydrogenic Atoms • Schrödinger equation • Separation of internal motion • Separate motion of e- and nucleus from motion of atom as a whole Coordinates for discussing separation of relative motion of two particles Center-of-mass me mN Structure of Hydrogenic Atoms • Schrödinger equation • Separation of internal motion • Separate motion of e- and n from motion of atom as a whole • Use reduced mass, • Result: mN me 1 μ mNme me (r, , ) R(r)Y(, ) • where R(r) are the radial wavefunctions Fig 10.2 Effective potential energy of an electron in the H atom • Shapes of radial wavefunctions dependent upon Veff • Veff consists of coulombic and centrifugal terms: Veff Ze 2 l(l 1) 2 4 πε or 2μr 2 •When l = 0, Veff is purely coulombic and attractive • When l ≠ 0, the centrifugal term provides a positive repulsive contribution Hydrogenic radial wavefunctions R = (Nn,l) (polynomial in r) (decaying exponential in r) l R n,l (r) Nn,l L n,l ()e 2 n n Ln,l(p) is an associated Laguerre polynomial 2Zr ao Fig 10.4 Radial wavefunctions of first few states of hydrogenic atoms, with atomic # Z Interpretation of the Radial Wavefunction l R n,l (r) Nn,l L n,l ()e 2 n n 1) The exponential ensures that R(r) → 0 at large r 2) The ρl ensures that R(r) → 0 at the nucleus 3) The associated Laguerre polynomial oscillates from positive to negative and accounts for the radial nodes 1s 2s 3s 2p 3d 3d Potential energy between an electron and proton in a hydrogen atom +- + - + - ao One-electron wavefunction = an atomic orbital