Chemistry Reference
In-Depth Information
where V is the potential coulombian energy of the electron, E is the total energy of the
system,
ћ
is a convenient modification of Planck's constant, and μ is its reduced mass:
2
Ze
V
=−
(2.2)
4πε
r
0
h
2π
(2.3)
=
11 1
µ
=+
(2.4)
mm
e
N
In the potential energy equation, r is the distance between the electron and the
nucleus, and ε
0
is the permittivity in vacuum (Atkins 1994; Atkins and Beran 1990;
Massey 1990).
A simpler representation of Schrödinger's equation is:
Hψ = Eψ
(2.5)
In this equation, H is the operator of the kinetic and potential energy of the electron-
nucleus system and it describes the position of the electron in the three-dimensional
Cartesian space (Equation [2.6]).
2
2
2
2
+
h
8m x
∂
∂
∂
∂
∂
∂
H
=−
π
+
+
V(x, y,z)
(2.6)
2
2
y
2
z
2
2
2
2
∂
∂
∂
∂
∂
∂
where
are partial differentials, and V(x, y, z) is the potential
,
and
x
2
y
2
z
2
energy in the Cartesian system.
Schrödinger's equation cannot be solved exactly, especially for multi-electron
atoms, mainly because of the effects determined by the attraction and rejection of
electrons phenomena. However, by using different methods of approximation, one
can obtain satisfactory results.
The wave function that indicates the charge density or electronic cloud provides
a space model of the movement of the electrons. Specific electron clouds are posi-
tioned like concentric layers, at different energy levels around the nucleus.
Each atomic orbital is described by a unique set of three quantum numbers: n,
l, and ml.
l
. Also, each electron is described by a fourth unique quantum number ml.
s
,
which gives information regarding the spin quantum number.
The principal quantum number n indicates on which layer the electron is posi-
tioned and is represented by a positive integer number: 1, 2, 3, 4, 5, … ∞. It can also
be represented with the capital letters K, L, M, N, O, P, and Q according to Bohr's
system. The energy value of these levels increases with distance from the nucleus;
therefore, the lowest energy level is the level with the principal quantum number 1.
The next quantum number (l) is called the secondary quantum number, orbital, or
azimuthal quantum number. For each principal quantum number there are values of
