# Atomic Orbitals

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```							Atomic Orbitals

Hydrogen Atom:

z

electron at coordinates
(x, y, z) or (r, θ, φ)

θ
r = (x2 +y2 + z2) 1/2

x

φ
y
proton at origin

Solution of the Schrödinger Equation results in:

Quantum numbers n, l, m
Energy equation: E n = -RH / n 2
Wavefunction: ψ n, l, m (r, θ, φ) = Rn, l (r) Yl,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 :
ml = 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 , p0, p+1 also called px , py, pz

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: p0 = 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 = (x2 +y2 + z2) 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) = Rn, l (r) Yl,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 Rn, l (r) found for many electron atoms
do not have a simple mathematical form. They are
simply plots of Rn, l (r) vs. r
For example: the H 1 s orbital
wavefunction:

(ψ n l m = ψ 1 0 0 = )   ψ1s= N e–r
(N is a constant)

1.0

0.8

1s =   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

-2exp(-r/2)
1.6
1.4
1.2
r exp(-r/2)
1.0
0.8
2s
0.6
0.4
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
0        2       4        6          8

r

The H 2pz atomic orbital.

ψ 2 p z = N z e – r /2 , ψ 2 2 p z = N2 z2 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.
2 p z
2
0.6
       2pz
0.5
0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6

-12   -10   -8   -6   -4   -2   0   2   4   6   8   10   12

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

1s   0.6

0.4

0.2

0.0
0        2         4         6    8

ar

Fit this by adding together some number of
Gaussian functions.
g p (α p, r ) = Np exp ( - α p r2 )

ψ 1S =         Σ p = 1 , L (d p g p(α p , r) )

a sum of L Gaussian functions. The dp
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.

1.2
2s
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4   electron close to nucleus       electron away from nucleus
-0.6                                   determines chemistry
determines energy
-0.8
-1.0
-1.2
-1.4
-1.6
-1.8
-2.0
-2.2
0            2            4        6           8

r - scaled by Z
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 ‘split-
valence’ sets.

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