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