Chapter 10 Atomic Structure and Atomic Spectra

Objectives: Objectives: Apply quantum mechanics to describe electronic structure of atoms Apply quantum mechanics to describe electronic structure of atoms Obtain experimental information from atomic spectra Obtain experimental information from atomic spectra Set up Schrödinger equation and separate Set up Schrödinger equation and separate wavefunction into radial and angular parts Use hydrogenic atomic orbitals to describe Use hydrogenic atomic orbitals to describe structures of many-electron atoms Use term symbols to describe atomic spectra Use term symbols to describe atomic spectra

Fig 10.1 Emission spectrum of atomic hydrogen

Conservation of quantized energy when a photon is emitted.

n f = 1 n i = 2 n f = 1 n i = 3 n f = 2 n i = 3 Energy levels of the hydrogen atom where: 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 meme mNmN

Schrödinger equation Separation of internal motion Separate motion of e - and n from motion of atom as a whole Use reduced mass, Result: where R(r) are the radial wavefunctions Structure of Hydrogenic Atoms

Fig 10.2 Effective potential energy of an electron in the H atom Shapes of radial wavefunctions dependent upon V eff V eff consists of coulombic and centrifugal terms: When l = 0, V eff is purely coulombic and attractive When l ≠ 0, the centrifugal term provides a positive repulsive contribution

Hydrogenic radial wavefunctions L n,l (p) is an associated Laguerre polynomial R = (N n,l ) (polynomial in r) (decaying exponential in r)

Fig 10.4 Radial wavefunctions of first few states of hydrogenic atoms, with atomic # Z

Interpretation of the Radial Wavefunction 1)The exponential ensures that R(r) → 0 at large r 2)The ρ l 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

Potential energy between an electron and proton in a hydrogen atom aoao One-electron wavefunction = an atomic orbital