chapter 10 atomic structure and atomic spectra. objectives: objectives: apply quantum mechanics to...
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Chapter 10
Atomic Structure and Atomic Atomic Structure and Atomic SpectraSpectra
Objectives:Objectives:
Apply quantum mechanics to describe Apply quantum mechanics to describe electronic structure of atomselectronic structure of atoms
Obtain experimental information from atomic Obtain experimental information from atomic spectraspectra
Set up Schrödinger equation and separateSet up Schrödinger equation and separate
wavefunction into radial and angular partswavefunction into radial and angular parts
Use hydrogenic atomic orbitals to describe Use hydrogenic atomic orbitals to describe
structures of many-electron atomsstructures of many-electron atoms
Use term symbols to describe atomic spectraUse term symbols to describe atomic spectra
Fig 10.1 Emission spectrum of atomic hydrogen
Conservation of quantized energy when a photon is emitted.
nf = 1
ni = 2
nf = 1
ni = 3
nf = 2
ni = 3
Energy levels of the hydrogen atom
2i
2f
H n1
n1
R
where:
J 10 x 2.18R
cm 677 109R18
H
1H
Rydberg constant
Structure of Hydrogenic AtomsStructure 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
• 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 AtomsStructure of Hydrogenic Atoms
),(Y)r(R),,r(
eeN
eN
m
1
mm
mm
μ
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:
•When l = 0, Veff is purely coulombic and attractive
• When l ≠ 0, the centrifugal term provides a positive repulsive contribution
2
2
o
2
effr2
)1l(l
r4
ZeV
μπε
Hydrogenic radial wavefunctionsHydrogenic radial wavefunctions
n2e)(Ln
N)r(R l,n
l
l,nl,n
oaZr2
LLn,ln,l(p) is an (p) is an associatedassociated
Laguerre polynomialLaguerre polynomial
R = (NR = (Nn,ln,l) (polynomial in r) (decaying exponential in r)) (polynomial in r) (decaying exponential in r)
Fig 10.4 Radial wavefunctions of first few states
of hydrogenic atoms, with atomic # Z
n2e)(Ln
N)r(R l,n
l
l,nl,n
Interpretation of the Radial Wavefunction
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 oscillatesfrom positive to negative and accounts for theradial 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