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8 The wavefunction of the electron
The potential energy in the SE is centrosymmetric and the
equation is separable into radial and angular components.
 (r ,  ,  )  R ( r )Y ( ,  )
We will end up with three quantum numbers. One coming from the
radial dependence R denoted n, and two from the spherical
component Y, denoted l and ml. The quantum numbers l and ml
are both related to the angular momentum of the electron.
The SE separates into two equations, one for R and the other for Y.
The SE equation for R has an effective potential Veff:

 2 d 2 R 2 dR
(

)  Veff R  ER
2 dr 2 r dr
Veff  
e2
l (l  1) 2

4 0 r
2 r 2
8.2 The radial wavefunction of the electron
The equation

2
2
 d R 2 dR
(

)  Veff R  ER
2 dr 2 r dr
8.1 Effective potential of an electron
in the hydrogen atom
When the electron has zero orbital
angular momentum, the effective
potential energy is the Coulombic
potential energy. When the electron
has nonzero orbital angular
momentum, the centrifugal effect
gives rise to a positive contribution,
which is very largely close to the
nucleus. We can expect l=0 and
l  0 wavefunctions to be very
different near the nucleus.
8.3 What do orbitals ‘look like’?-The radial solutions
The radial wavefunctions R(r) depend on the values of both n
and l (but not ml).
can be solved analytically.
Acceptable solutions can be found only for integral values of a
quantum number n, with n=1,2,3.... and l= 0,1,2,...,n-1.
Therefore, the solutions R(r) depend on the two quantum numbers,
n and l:
Rn,l(r):
R1,0(r), R2,0(r), R2,1(r) ...
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8.3.1 Radial distribution functions
A constant volume electronsensitive detector (the small
cube) gives its greatest reading
at the nucleus for functions with
l=0, and a smaller reading
elsewhere. The same reading is
obtained anywhere on a circle of
given radius: the s (l=0) orbital is
spherically symmetric.
8.4 Representations of orbitals
8.3.2 Radial distribution functions
The radial distribution function
P gives the probability that the
electron will be found anywhere
in a shell of radius r. For a 1s
(l=0) electron in hydrogen, P is
a maximum when r is equal to
the Bohr radius a0. The value
of P is equivalent to the reading
that a detector shaped like a
spherical shell would give as its
radius is varied.
8.4.1 The boundary surfaces of orbitals
Representation of the 1s and 2s hydrogenic atomic orbitals in terms
of their electron densities (represented by the density of shading)
The boundary surface of
an s orbital within which
there is a 90%
probability of finding the
electron.
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8.4.2 The boundary surfaces of p orbitals.
8.4.3 The boundary surfaces of d (l=2) orbitals.
The boundary surfaces of p (l=1) orbitals. A nodal plane passes
through the nucleus and separates the two lobes of each orbital.
8.5 The energy eigenvalues
The first term in Veff is the Coulomb potential energy of the
electron in the field of the nucleus. The second term stems from
the centrifugal force that arises from the angular momentum of
the electron around the nucleus.
Veff  
e2
l (l  1) 2

4 0 r
2 r 2
When l=0, the electron has no angular momentum, and the
effective potential energy is purely Coulombic and attractive at all
radii.
The allowed energy values depend only on the quantum
number n:
En  
e 4
32 2 02 2 n 2
8.6 Hydrogenic atoms
A hydrogenic atom is a one-electron atom or ion of general atomic
number Z. Examples : H, He+, Li2+ and U91+ .
The Coulomb interaction of the electron with the nucleus is then
given by
V (r )  
Ze 2
4 0 r
and the eigenvalues are
En  
Z 2 e 4
32 2 02  2 n 2
with n=1, 2, 3,…
with n=1,2,…
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8.7 Atomic orbitals and quantum numbers
An atomic orbital is a one electron wavefunction for an
electron in an atom. Each hydrogenic atomic orbital is
defined by three quantum numbers, designated n, l, and ml.
When an electron is described by one of these
wavefunctions, we say that it ‘occupies’ that orbital.
The quantum number n is called the principal quantum
number. It determines the energy of the electron (see
above).
An electron in an orbital with quantum number l has an
angular momentum of magnitude {l(l+1)}1/2 
with
l=0, 1, 2, …, n-1.
An electron in an orbital with quantum number ml has a zcomponent of angular momentum ml 
with
ml =0, ±1, ±2, …, ±l.
The two other quantum numbers, l and ml, come from the
angular solutions, and specify the angular momentum of the
electron around the nucleus.
8.8 Shells and subshells
All the orbitals of a given value of n form a single shell of the
atom. In a hydrogenic atom all electrons belonging to the
same shell have the same energy.
Shells are referred to by letters:
n=1
2
3
4…
K
L
M
N…
Orbitals with the same value of n but different values of l form
a subshell.
Subshells are also referred to by letters:
l=0
1
2
3
4…
s
p
d
f
g…
The subshell with l=0 contains only one orbital (ml=0). The
one with l=1 contains three (ml=-1,0,1), and with l=2 five etc.
In consequence, the K-shell (n=1) contains 1 orbital, the Lshell (n=2) contains one s-orbital and three p-orbitals, i.e.,
four orbitals, the M-shell then contains four orbitals plus five
d-orbitals, i.e., nine orbitals altogether.
Generally speaking, the shell with the principal quantum
number n contains n2 orbitals. In a hydrogenic atom each
energy level is n2–fold degenerate.
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8.9 Organization of orbitals into subshells and
shells
The organization of
orbitals (white squares)
into subshells
(characterized by l)
and shells
(characterized by n)
8.10 Energy levels of the hydrogen atom
The energy levels of the
hydrogen atom showing
the subshells and (in square
brackets) the numbers of
orbitals in each subshell. In
hydrogenic atoms, all
orbitals of a given shell have
the same energy.
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