Biophysical Chemistry
Chemistry 24a
Winter Term 200910
Instructor:
Sunney I. Chan
Lecture 5
January 13, 2010
Atomic Structure and Chemical
Bonding
The Hydrogen Atom
The Hamiltonian
The Hamiltonian (
Ĥ
) is the sum of the kinetic
energies and potential energies associated
with an atomic and molecular system:
Ĥ
isolated molecule
(p
1
, r
1
; p
2
, r
2
;
.,
p
i
, r
i
;
..
)
=
Σ
i
p
i
2
/2m
i
+
V(r
1
; r
2
;.. r
i
;…)
kinetic energy
potential energy
where p
i
, r
i
and m
i
refer to the momentum,
coordinates, and the mass of the i
th
particle
(electrons and nuclei), respectively.
Typically for a molecule
V(r
1
; r
2
;.. r
i
;…)
=
Σ
N
Σ
k
(z
N
e
2
/r
Nk
)
(coulomb attraction between
nuclei and electrons)
+
Σ
k’
≠
k
(e
2
/r
kk’
)
(electrostatic repulsion between
electrons)
+
Σ
N’
≠
N
z
N
z
N’
e
2
/R
NN’
(nuclearnuclear repulsion)
where
Σ
k
denotes sum over all electrons;
and
Σ
N
denotes sum over all nuclei in the
molecule.
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For an atom
V(r
1
; r
2
;.. r
i
;…)
=
Σ
k
(z
N
e
2
/r
Nk
)
(coulomb attraction between
nucleus and electrons)
+
Σ
k’
≠
k
(e
2
/r
kk’
)
(electrostatic repulsion
between electrons)
where
Σ
k
denotes sum over all the
electrons in the atom.
For the hydrogen atom
V(r) =
 e
2
/
r
(coulomb attraction between proton and electron)
So,
Ĥ
atom
=

ħ
2
/2
μ
▽
2

e
2
/
r
(when kinetic energy is expressed relative to the center of
mass of the twoparticle system)
μ
=
M
p
m
e
/(M
p
+ m
e
)
≈
m
e
(since M
p
>> m
e
)
(reduced mass
)
where M
p
and m
e
are the masses of
the proton and electron, respectively.
For the hydrogen atom,
Schr
ő
edinger equation for the problem:
Ĥ
atom
Φ
n
(
r
,
θ
,
φ
) =
 (
ħ
2
/2m
e
)
▽
2
Φ
n
(
r
,
θ
,
φ
)
 (e
2
/
r
)
Φ
n
(
r
,
θ
,
φ
)
=
E
n
Φ
n
(
r
,
θ
,
φ
)
where
r
is the magnitude of the vector r,
namely, just the distance of the electron
from the proton.
Φ
n
is the wavefunction associated with
the energy
E
n
.
Polar coordinates
r
r
x
y
z
θ
φ
z =
r
cos
θ
x =
r
sin
θ
cos
φ
y =
r
sin
θ
sin
φ
Problem has spherical symmetry!