## Atomic Orbitals

Electrons exist in different energy levels. Electrons have wave properties. The kinetic energy of an electron corresponds to a wavelength and thus a frequency. The position of an electron is described in terms of the probability of finding it in a certain position. The probability corresponds to the amplitude of the associated wave. The Schrodinger equation relates the energy of a system to its wave properties. The equation can be solved exactly for the hydrogen atom with one proton and one electron. This yields a description of quantized energy levels and orbital configurations. This scheme is used for larger atoms with certain adjustments. Electrons are added to the various orbitals, filling the higher energy levels as the size of the atom increases.

The are three orbital quantum numbers that
describe an orbital. They are called *n, l* and *m _{l}*. The principal
quantum number

*n*determines the size of the atom. The second orbital quantum number

*l*determines the shape of the orbital in a general way. The third quantum number

*m*determines the orientation, for example, with respect to an applied magnetic field.

_{l}The value of *n* can be any positive integer
(1, 2, 3, 4, etc.) The value of *l* is from zero to *n*-1. Thus for *n *=
1, the value of *l* is 0. For *n* = 2, *l* can be 0 or 1. For *n* = 3,
*l* can be 0, 1 or 2. For *n* = 4, *l* can have values of 0, 1, 2 or 3. The
various shapes that correspond to the values of l are referred to as *s, p, d* and *f*
orbitals.

l |
0 | 1 | 2 | 3 |

orbital | s |
p |
d |
f |

A given orbital can be described by combining the
first and second quantum numbers *n* and *l*.

1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p, 4d, 4f, 5s, etc.

The third quantum number has the range -*l*
to *l*. Thus for *l* = 0 there is only one value for *m _{l}* (0).
For

*l*= 1 there can be three values for

*m*(-1, 0 or 1). For

_{l}*l*= 2,

*m*can have five values (-2, -1, 0, 1 or 2). For

_{l}*l*= 3,

*m*can have seven values (-3, -2, -1, 0, 1, 2 or 3). Thus for the

_{l}*s*orbital there is only one value for

*m*. For a

_{l}*p*orbital there are 3 orientations. For a

*d*orbital there are five orientations. For an

*f*orbital there are 7 orientations.

orbital | s |
p |
d |
f |

orientations | 1 | 3 | 5 | 7 |

There is a fourth quantum number *m _{s}*.
This is called the spin quantum number. It can have two values, 1/2 and -1/2. This is
because an electron can orient in two ways in an applied magnetic field. You might imagine
an electromagnet with the wire coiled clockwise or counter clockwise. Thus an orbital
specified by the first three quantum numbers can be occupied by only two electrons.

The shapes or orbitals are described by surfaces
through regions of equal probability. The shapes most commonly used are those that include
99% probability. An *s* orbital is spherical. An *s* orbital is shown below.

A *p* orbital has two parts separated by a
nodal plane where the probability is zero. There are three orientations available for a *p*
orbital. They are named *p _{z}*,

*p*and

_{y}*p*. These are shown below.

_{x}A *d *orbital has four lobes. The
probability is zero between the lobes. There are five possible orientations. These are
shown below.

The first quantum number determines the size,
i.e., a 1*s* orbital is smaller than a 2*s* orbital which is smaller than a 3*s*
orbital, etc. The energy of an orbital is determined mostly by the first two quantum
numbers. The order in increasing energy is given below.

1s, 2s, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f @ 5d, 6p, 7s, 5f @ 6d

When building the electronic configuration of
many electron atoms we start by placing electrons in the lowest energy orbitals and work
up from there. For hydrogen the single electron goes into the 1*s* orbital. This is
the ground state, the state of lowest energy for the hydrogen atom. The electron may be
elevated into higher energy orbitals by absorbing the energy of a photon of the right
frequency. As the electron falls back to the ground state energy is released. The energy
of the photon emitted depends on the energy difference between the orbits involved in the
transition.

For helium the two electrons, one of spin 1/2 and
the other of spin -1/2, are placed in the 1*s* orbital. The 1*s* orbital is
filled with two electrons. This is represented as 1*s*^{2}. The next electron
(lithium) goes into the orbital with the next highest energy, the 2*s* orbital. This
is written 1*s*^{2}2*s*^{1}. The 2*s* orbital is filled at
beryllium with two electrons. The electron configuration of the beryllium atom in the
ground state is represented in the form 1*s*^{2}2*s*^{2}. With
boron the next electron is placed in one of the 2*p* orbitals. They are all
equivalent in energy. This is shown as 1*s*^{2}2*s*^{2 }2*p*^{1}.

The carbon atom has six electrons. The lowest
energy is acieved by placing the next electron in one of the unoccupied 2*p*
orbitals. Furthermore the lowest energy is achieved if both the 2*p* electrons have
the same spin (parallel spin). Such electrons are called *unpaired*. The lowest
energy carbon atom electron configuration can be represented as 1*s*^{2}2*s*^{2
}2*p*^{2 }or 1*s*^{2}2*s*^{2 }2*p _{x}*

^{1 }2

*p*

_{y}^{1}

_{.}

For a list of electronic configurations for all the elements go to Electronic Configurations.