Atomic Orbitals
Hydrogen Atom: z electron at coordinates
(x, y, z) or (r, θ, φ)
θ r = (x 2 +y 2 + z 2 ) 1/2 x
φ y proton at origin
Solution of the Schrödinger Equation results in:
Quantum numbers n, l , m
Energy equation: E n
= -R
H
/ n 2
Wavefunction: ψ n, l , m
(r, θ, φ) = R n, l
(r) Y
l,m
(θ, φ)
Y
l, m
(θ, φ) – describes the rotation of a particle on the surface of a sphere (or radius r) and determines angular momentum properties.

Magnitude of angular momentum:
l = (l(l + 1)) 1/2 (h/2

)
Magnitude of z component of l : m l
= m (h/2

)
R n, l
(r) – describes the radial distribution of the electron and determines the energy equation.
R n, l
(r) = (polynomial in r) exp (- r /n)
Note: n = 2, l = 1 (p), m = -1, 0, 1 p
-1
, p
0,
p
+1 also called p x
, p y,
p z p
-1
= N r sin θ exp(-iφ) exp(-r/2) p
+1
= N r sin θ exp(iφ) exp(-r/2) p
+1
+ i p
-1
= N r sin θ (exp(iφ) + i exp(-iφ)) exp(-r/2)
= N r sin θ cos φ exp(-r/2) = N x exp(-r/2) similarly: p
+1
- i p
-1
= N r sin θ sin φ exp(-r/2)
= N y exp(-r/2) and: p
0
= N r sin θ exp(-r/2) = N z exp(-r/2) hence these are equivalent orbitals.
(The same approach gives the d orbitals etc.)

Many electron atom: Electron attracted to nucleus and repelled by (average) field of the other electrons. z electron at coordinates
(x, y, z) or (r, θ, φ) average spherical distribution of other electrons
θ r = (x 2 +y 2 + z 2 ) 1/2 x
φ y proton at origin
Solving the Schrödinger equation results in:
Energy equation: E n
= -R / n * 2 (see below)
Wavefunction: ψ n, l , m
(r, θ, φ) = R n, l
(r) Y
l,m
(θ, φ)
Angular part is identical. r part is similar but the scale changes as the electron is attracted to the more positive nucleus.
There is no exact polynomial solution, R n, l
(r) , the problem has to be solved by approximation methods.

e.g. The 1s orbital is not screened from the nuleus, and 1s electrons experience the full +Z charge of the nucleus and the radius decreases rapidly with increasing Z.
In fact r(1s) ~ 1/Z . r (1s) of H is 77 pm, 7.7x10
-11 m therefore r(1s) of U (Z=92) ~ 8x10 -13 m whilst the valence orbital, 5f, gives the U atom a radius of
~2x10 -10 m.
Other orbitals have smaller radii than the corresponding hydrogen orbital but are screened from the full effect of +Z by inner shells.
The energy depends on l as well as n. (Periodicity)
Quantum Chemical Calculations use atomic orbitals as the Basis Set for all methods. The functions R n, l
(r) found for many electron atoms do not have a simple mathematical form. They are simply plots of R n, l
(r) vs. r

For example: the H 1 s orbital wavefunction:
(ψ
n l m
= ψ
1 0 0
= ) ψ
1 s
= N e – r
(N is a constant)
1.0
0.8

1 s
=
0.6
exp( -r)
0.4
0.2
0.0
0 2 4 6 8 r
The solution of the Schrödinger equation for the H 2s orbital (n =2, l = 0, m = 0) is:
ψ
2 s
= N (r -2) e – r /2 = N( r e – r /2 - 2e – r /2 )
(N is a constant)

Consider graphs of the two parts of the function and their sum.
ψ
2s vs r
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
-1.2
-1.4
-1.6
-1.8
1.6
1.4
1.2
1.0
0.8
0.6
-2.0
-2.2
-2exp(-r/2)
r exp(-r/2)

2s
0 2 6 8 r
4
The H 2p z
atomic orbital.
ψ
2 p z
= N z e – r /2 , ψ 2
2 p z
= N 2 z 2 e – r has a Cartesian direction in the expression.
ψ
2 p z
is small for small z and large r and is negative for z negative. Probability distribution is always positive.

0.3
0.2
0.1
0.0
-0.1
0.6
0.5
0.4
-0.2
-0.3
-0.4
-0.5
-0.6
-12 -10 -8 -6 -4 -2 0 z
2 4 6 8 10 12

2 p z
 2
2 p z
\
Other atoms have similar a.o.s with r (or z) scaled appropriately.
Need a mathematical form for the a.o.s to use in calculations – to mathematically represent the Basis Set .
Only 2 representations are in common use: a) Slater Type Orbitals (STO) – only used extremely accurate calculations on very small molecules.
(b) Gaussian Type Orbitals (GTO) – used in every molecular orbitals package.

Gaussian Type Orbitals
Consider the 1s orbital of an atom – looks like the simple exponential function of H but over smaller r (scaled by a )
1.0
0.8

1 s
0.6
0.4
0.2
0.0
0 2 6 8 4 a r
Fit this by adding together some number of
Gaussian functions. g p
(α p
, r ) = N p exp ( - α p
r 2 )
ψ
1 S
= Σ p = 1 , L
(d p
g p
(α p
, r) )
a sum of L Gaussian functions. The d p coefficients and the α p
exponents are chosen for the best fit.

This is a ‘Contracted Gaussian Function’ as L, d and α values are fixed.
The details are considerably more complicated than this.
For general purposes the following summarises the most common basis sets:
Minimal Basis Sets
1) STO-NG sets are contracted sets of N
Gaussians. N=3-6
2) 4-31G sets use a contracted set of 4
Gaussians for inner orbitals and a contracted set of 3 Gaussians and a separate (long range)
1 Gaussian to represent valence orbitals. The proportions of the two parts of the valence orbitals is a variable of the calculation. Other similar sets 3-21G and 6-31G.(Split Valence)
Extended Basis Sets
3) * and ** as in 6-31G* or 4-31G** the set has a d -type orbital added to heavy atoms (*) also a p -type added to hydrogen atoms (**).
4) + and ++ similar but referred to as polarisation functions – not strictly atomic orbitals

Why treat the inner shells and valence orbitals differently?
Inner shells are hardly effected by chemical bonding but account for most of the electronic energy of the atom. More accurate the representation – the more accurate the total energy.
Why allow flexibility in the valence shell?
Consider the 2 s orbital.

2s
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
-1.2
-1.4
-1.6
-1.8
-2.0
-2.2
1.2
1.0
0.8
0.6
0.4
electron close to nucleus determines energy
0 electron away from nucleus determines chemistry
2 4 r - scaled by Z
6 8

Chemistry requires that the outer region be well described. Inner region rather less effected by bonding and can represented by a fixed contracted Gaussian function. The outer region needs to be different from molecule to molecule – hence the flexibility of ‘splitvalence’ sets.