Transcript Chapter 10 - Lecture 1

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 2n
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 2n
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