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Methods in
Computational
Chemistry
Section 3
An introduction to quantum mechanics
David Chalmers
2013
INDEX
Chapter 1.
Quantum theory
1
1.1. Properties of particles
3
1.2. Properties of waves
3
1.2.1. Standing waves
5
1.2.2. Wave-particle duality
6
Chapter 2.
7
The Schrödinger equation
2.1.1. Partial differential equations
8
2.1.2. The Schrödinger equation in one dimension
8
2.1.3. Solution to a particle in an infinite potential well
10
2.1.4. Particle in a 2D well
13
2.1.5. Atomic orbitals
14
2.1.6. Atomic orbital wave functions
15
2.1.7. Molecular orbitals
16
Chapter 3.
18
Ab initio quantum chemical calculations
3.1. Approximations in ab initio methods
18
3.1.1. The Born-Oppenheimer approximation
18
3.1.2. The orbital approximation
19
3.1.3. The LCAO approximation
20
3.2. The Hartree-Fock method
21
3.3. Self consistent field (SCF)
22
3.4. Basis functions - General introduction
24
3.4.1. Minimal basis sets
27
3.4.2. Split basis functions
27
3.4.3. Polarisation functions
28
3.4.4. Examples of basis functions
29
3.5. Limitations of the Hartree-Fock method
29
3.6. Density functional theory (DFT) methods
30
3.7. Ab initio procedures
30
3.7.1. Single point calculations
30
3.7.2. Geometry optimisation calculations
31
3.7.3. Frequency calculations
31
3.8. Outline of a typical ab initio calculation
32
3.8.1. Read input & calculate a geometry
32
3.8.2. Assign basis set
32
3.8.3. Calculate nuclear repulsion energy
32
3.8.4. Calculate integrals
32
3.8.5. Assign electronic configuration
33
3.8.6. Generate initial guess
33
3.8.7. Self-consistent field iterations (electronic energy)
33
3.8.8. Calculate the total energy
33
3.8.9. Electron density analysis
33
3.9. Capabilities of ab initio quantum chemistry
33
3.10. Applications of ab initio quantum chemistry
34
Chapter 4.
36
Semiempirical MO calculations
4.1. Semiempirical MO Software
39
4.2. Strengths and limitations of semiempirical methods
42
Chapter 5.
44
Section summary
Chapter 1.
Quantum theory
References:
Atkins and Jones. Chemical principals: The quest for insight.
3rd Ed. Chapter 1, The Quantum World
Atkins and de Paula. Physical Chemistry. Chapter 8. Quantum
Theory: introduction and principles
Classical physics provides a good description of macroscopic
objects. However it became apparent in the 19th century, that
classical physics fails to provide an accurate description of very
small objects such as atoms and molecules.
The effects of quantum mechanics are evident in many physical
phenomena. In the late 1800’s and early 1900’s physicists
identified limitations in their theoretical understanding of a
number of phenomena.
These included:
 The failure to predict the wavelength distribution of light
emitted from a ‘black body’ (an object which does not
reflect light, but only emits light by absorbing it and reradiating it).
 The failure to predict heat capacities of solids (ie how
much energy is required to raise the temperature of a
substance by one degree).
 The failure to predict atomic and molecular spectra.
 The observation that small particles, like electrons and
atoms can act like waves. Eg. beams of atoms, when
interacting with one another can form interference
patterns.
These and many other problems lead to the development of new
theories which describe the behaviour of very small particles
such as atoms.
Methods in Computational Chemistry: Section 3
2
Figure 1. Electronic emission spectrum of hydrogen caused by excited
electrons changing from higher energy levels to lower ones and emitting
photons with specific wavelengths (source http://en.wikipedia.org/wiki/
Basic_concepts_of_quantum_mechanics)
Two fundamental features of these theories are:
 That the energies of small particles are quantised. That is
the energies of particles can have only discrete values and
that any energy changes can only occur in ‘packets’ of a
fixed size.
 That small particles have properties of both waves and
particles. This is called wave-particle duality. In some
situations, small particles (eg atoms) may behave like
particles (they can have a fixed location in space) and in
other situations they may behave like waves (a beam of
atoms can be diffracted).
Methods in Computational Chemistry: Section 3
3
1.1. Properties of particles




Particles are localised in space
Particles have a defined size
Particles can be reflected
Particles can be scattered
Figure 2. A particle (basketball) being reflected (off the ground).
Source: http://en.wikipedia.org/wiki/File: Bouncing_ball_strobe_edit.jpg
1.2. Properties of waves
 Waves are not localised in space
 Waves have the properties of wavelength, amplitude and
phase
 Waves can be reflected
 Waves can be diffracted
 Waves form interference patterns
Methods in Computational Chemistry: Section 3
Figure 3.
4
Reflection of water waves by a wall
Figure 4. Diffraction of water waves and wave interference patterns
produced by an island and headland.
Methods in Computational Chemistry: Section 3
5
1.2.1. Standing waves
Standing waves occur where waves travelling in different
directions interfere with each other - eg by reflecting off a hard
surface. Standing waves occur when a string of a guitar or
violin is plucked.
Figure 5. Standing
waves
on
http://en.wikipedia.org/wiki/Standing_wave
a
string.
Source:
Methods in Computational Chemistry: Section 3
6
1.2.2. Wave-particle duality
Wave particle duality proposes that quantum mechanical
particles such electrons or atoms exhibit properties of both
atoms and particles.
Wave-particle duality is demonstrated by Young’s double slit
experiment.
Figure 6. Young’s double slit experiment.
Particles (photons,
electrons, etc) are fired through a plate with a single slit (S1) and then
pass through a plate with two apetures (left). An interference pattern
results – even if the particles are fired one at a time (right). The particles
have wave-like behaviour and interfere with themselves. Source
http://en.wikipedia.org/wiki/Double-slit_experiment.
Methods in Computational Chemistry: Section 3
Chapter 2.
Figure 7. Erwin
Wikipedia)
7
The Schrödinger equation
Schrödinger and
Werner
Heisenberg
(source
In 1925 Erwin Schrödinger (1884-1961) and Werner Heisenberg
(1901-1976) independently formulated a general quantum
theory. Although the two formulations are mathematically
equivalent, Schrödinger presented his theory in terms of partial
differential equations and, within this framework, the energy of
an isolated molecule can be obtained by the solution of a wave
equation called the Schrödinger equation.
Schrödinger’s influence on science and extended well beyond
the development of quantum mechanics, he later published an
important, short book called ‘What is life’ which discussed
molecular physics, genetics, mutations and cellular
reproduction.
Methods in Computational Chemistry: Section 3
8
2.1.1. Partial differential equations
A partial differential equation (PDE) is one that has two or more
independent variables, an unknown function (which depends on
those variables) and partial derivatives of the unknown function.
The equation for a wave moving along a string is an example of
a PDE.
1 ¶ 2u
v ¶t
2
2
=
¶ 2u
¶x 2
Here t is time, u is the amplitude of the wave and v is the
velocity of the wave.
2.1.2. The Schrödinger equation in one dimension
Detailed study Schrödinger equation is beyond the scope of this
course – but it is useful to take a look at the equation itself to
understand the important features. We can simplify things a
little if we only consider one dimension (x). Similar results if we
consider two and three dimensions.
The Schrödinger equation for a one-dimensional system
containing a particle of mass m that is under the influence of an
energy potential (U(x,t)) that varies with the position (x) and
time (t) is shown below‡:
¶ 2 Y(x,t)
h ¶Y(x,t)
- 2
+U(x,t)Y(x,t)
=
i
Eqn 1.
2p ¶t
8p m ¶x 2
h
In this differential equation, h is Planck’s constant and ‘i’ is the
square root of -1 (i2 = -1).  represents the wave function. The
inclusion of ‘i’ shows that the wave function is a complex
quantity.
‡
A. Hinchcliffe, Molecular modelling for beginners. Chapter 11.
Methods in Computational Chemistry: Section 3
9
How do we interpret a wave function? The square of the wave
function |(x)|2 dx represents the probability that the particle
will be found between x and x + dx.
The Schrödinger equation can be rearranged as shown below.
æ
ö
h ¶2
h ¶Y(x,t)
+U(x,t)
Y(x,t)
=
i
ç- 2
÷
2
2p ¶t
è 8p m ¶x
ø
This form is often written as:
ˆ Y=EY
H
ˆ is an operator (a mathematical function which
In this form, H
modifies another function), known as the Hamiltonian operator.
This operator characterises how the energy changes with respect
to the positions of the atoms. E is the energy operator.
The time-independent Schrödinger equation
It can be shown that if the energy potential (U) does not change
with time – then the wave function also is not time-dependent.
Methods in Computational Chemistry: Section 3
10
2.1.3. Solution to a particle in an infinite potential well
It is possible to solve the time-independent Schrödinger
equation for a particle in simple, one-dimensional potential
wells. The derivation will not be given here (see Hinchliffe,
Molecular modelling for beginners, Chapter 11), but it is useful
to look at the solutions.
Imagine a one-dimensional energy well with infinite potential
energy outside the well and zero potential energy inside the well
(Figure 1). An infinite energy means that the particle has zero
probability of existing outside the region 0  1. The solutions
to the Schrödinger equation in this case are simple sine waves as
shown in Figure 2.
Energy (arbitrary units)
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
-0.1
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
1.1
x
Figure 8.
A potential well with infinite energy outside the
region 0  1 and zero potential energy inside this region
Methods in Computational Chemistry: Section 3
Wave function (arbitrary units)
1.5
11
n=1
n=2
n=3
1
0.5
0
-0.5
-1
-1.5
-0.1
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
1.1
x
Figure 9.
Solutions to the Schrödinger equation for the 1D
potential energy well shown in Figure 7.
Note that there are multiple, quantised solutions to the equation.
The solutions are differentiated by the integer variable n. This
value is the quantum number.
Square of wave function (arbitrary units)
Remember that the square of the wave-function describes the
probability that the particle will be found in a particular
position. The probability function for the 1D infinite well is
shown in Figure 3. Note that if n > 1 there are regions inside
the well with zero probability that the particle will be found in
this location (nodes) and that the number of nodes increases as
the quantum number increases.
1.2
n=1
n=2
n=3
1
0.8
0.6
0.4
0.2
0
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
x
Figure 10.
Squared wave functions for the 1D energy well.
Methods in Computational Chemistry: Section 3
12
For this example, the energy of each solution is given by the
equation below (L is the well length). An energy level diagram
is shown in Figure 4.
n2h2
E=
8mL2
18
16
Energy (h2/8mL2)
14
12
10
8
6
4
2
0
Figure 11.
Energy diagram for the 1D energy well.
Examining the solution to the equation reveals a number of
observations:
 The energy levels depend on the size of the well.
Narrower wells will have higher energies.
 The energy of the lowest state is not zero. This is
generally true for quantum mechanical systems. The
energy of the lowest energy state is known as the zeropoint energy.
Methods in Computational Chemistry: Section 3
13
2.1.4. Particle in a 2D well
Figure 12.
The wavefunction for a particle in a 2D well/.
Source http://en.wikipedia.org/wiki/Image:Particle2D.jpg
If we extend the case of a particle in a 1D well to two
dimensions, the solutions to the equation are of the type shown
in Figure 5. Note that the wavefunction has nodes running
parallel to the x and y axes. This system has two quantum
numbers that describe the number of nodes in each dimension.
Methods in Computational Chemistry: Section 3
14
2.1.5. Atomic orbitals
The solutions to the Schrödinger equation for the case of a
‘particle in a well’ can help us to understand the shapes of the
atomic orbitals.
Atomic orbitals are described using three quantum numbers: n, l
and ml. The principal quantum number, n, can take positive
integer values, and describes the shell. The quantum number l
takes values 0 l-1 and describes the subshell (s, p, d, f, etc).
The quantum number ml takes values –l  l. Together, l and ml
describe the angular momentum of the orbital.
As we increase the quantum numbers n and l we introduce more
nodes into the orbitals.
1s
2s
2px y z
3s
3pxy yz xz
Figure 13.
3dz2
3dx2-y2
Atomic orbitals. http://www.orbitals.com/orb
Methods in Computational Chemistry: Section 3
15
2.1.6. Atomic orbital wave functions
The wave functions for atomic orbitals arise from the interaction
between the negatively charged electron and the positively
charged atomic nucleus which is governed by Coulomb’s law
(where d is the distance from the nucleus).
V=
q1q2
4pe0 d
The resulting Schrödinger equation is shown below. This
equation is in three dimensions. We can denote this by using a
vector r = (ai + bj + ck) to denote position in three-dimensional
space. The highlighted portion of the equation is the externally
applied potential energy. In this case, this is due to the
electrostatic interaction between the electron and the nucleus.
-
¶Y(r,t)
h ¶Y(r,t)
+
V
(r,t)
=
i
8p 2 m ¶r 2
2p ¶t
h
The Schrödinger equation can be solved analytically (ie exactly)
only for atoms containing a single electron (hydrogen, He +, Li++,
etc). It can not be solved exactly for atoms or molecules
containing multiple electrons.
Methods in Computational Chemistry: Section 3
16
The mathematical solution for the hydrogen 1s orbital is:
 = Ner/
where  is a constant known as the Bohr radius and N is a factor
which normalises the probability of finding an electron in all
space, 2, to 1.
The shape of the 1s orbital is plotted below. Orbitals with this
shape are known as Slater-type orbitals.
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-3
-2
-1
0
1
2
3
Figure 14. A 1D slice through the hydrogen 1s wave function
2.1.7. Molecular orbitals
Molecular orbitals can be approximated by combining atomic
orbitals. The orbitals are combined by making linear
combinations of the atomic orbitals.
A linear combination is the sum of a number of elements (eg
vectors or mathematical functions) where each multiplied by a
constant. For example, if x, y and z are functions, then
ax + by + cz
is a linear combination of these functions.
Methods in Computational Chemistry: Section 3
17
0.7
0.6
0.5
0.4
H1
H2
0.3
Total
0.2
0.1
0
-3
-2
-1
0
1
2
3
0.5
0.4
0.3
0.2
0.1
H1
0
-3
-2
-1
-0.1
H2
0
1
2
3
Total
-0.2
-0.3
-0.4
-0.5
Figure 15. Linear combinations (top addition, bottom subtraction) of 1s
atomic orbitals from two hydrogen atoms to give a bonding (top) and
antibonding (bottom) molecular orbitals for H2.
Methods in Computational Chemistry: Section 3
18
Chapter 3. Ab initio quantum chemical
calculations
This is a simplified version of the teaching materials prepared
by Brian Salter-Duke for the ACCVIP project. A more detailed
version is available at http://hydra.vcp.monash.edu.au/modules/
Calculations which solve the Schrödinger equation are known as
ab initio calculations. The term ab initio is Latin meaning ‘from
first principles’.
3.1. Approximations in ab initio methods
The term ab initio does not mean that we are solving the
Schrödinger equation exactly. It is not possible to solve the
Schrödinger equation exactly for systems that have more than
one electron (ie virtually all molecules) and we must therefore
use a number of mathematical approximations.
This means that nearly all quantum
provide approximate results. When
calculations we must take care to use an
method and ensure that the effect of
minimised.
chemical calculations
performing ab initio
appropriate calculation
the approximations is
3.1.1. The Born-Oppenheimer approximation
Figure 16. Max Born and J. Robert Oppenheimer (around 1944).
Source Wickpedia. Oppenheimer was Born’s PhD student.
Methods in Computational Chemistry: Section 3
19
The mass of an electron is about 1/1836 of a proton. As a result,
the velocity of the nucleus is much smaller than that of the
surrounding electrons. It is therefore possible to simplify the
solution of the Schrödinger equation by separating it into two
parts, one describing the motions of the electrons in a field of
fixed nuclei and the other describing the vibration, rotation and
translations motions of the nuclei. This is known as the
adiabatic or Born-Oppenheimer approximation.
Etotal = Eelectronic + Evibrational, rotational, translational
Molecular orbital theory is concerned with finding approximate
solutions to the electronic Schrödinger equation. It does not
attempt to solve the vibrational, rotational and translational part.
3.1.2. The orbital approximation
The orbital approximation makes the assumption that each
electron occupies a separate one-electron wavefunction or spin
orbital. Each electron is treated independently.
This approximation neglects electron correlation – the fact that
electrons do not move independently from one another, but
move together in a correlated way.
Methods in Computational Chemistry: Section 3
20
3.1.3. The LCAO approximation
The molecular orbitals (i) are usually expressed as linear
combinations of one-electron functions. In the simplest case,
where these functions are the atomic orbitals of the constituent
atoms, this expansion is called a linear combination of atomic
orbitals (LCAO):
i = c1i1 + c2i2 + ... cNiN
Qualitatively, this is like saying that the two molecular orbitals
in H2 are linear combinations of the 1s atomic orbitals:

2 = 0.5 a - 0.5 b
1 = 0.5 a + 0.5 b
Figure 17. Linear combinations of atomic orbitals produce the bonding
and antibonding orbitals in H2
Figure 18.
The highest occupied molecular orbital (HOMO)
and lowest unoccupied molecular orbital (LUMO) for ethylene.
These can be understood as being combinations of the carbon atoms
atomic p-orbitals.
Methods in Computational Chemistry: Section 3
21
3.2. The Hartree-Fock method
The Hartree-Fock method is a widely used method for solving
the Schrödinger equation.
The essential idea of the Hartree-Fock or molecular orbital
method is that, for a closed shell system (i.e. a molecule each
orbital contains two electrons – not a radical), the electrons are
assigned two at a time to a set of molecular orbitals. This can be
represented by the simple picture:
Here a system of eight electrons occupies the four molecular
orbitals of lowest energy. Only one unoccupied molecular
orbital is shown. Note that the unoccupied molecular orbitals are
often called virtual orbitals. Open shell systems (radicals) will
not be covered here.
To give us freedom to vary the molecular orbitals to best suit the
molecule in question, we expand each molecular orbital in terms
of a set of basis functions which are normally centred on the
atoms in the molecule. This gives:
n
yi = å Cmifm
m =1
Here each molecular orbital i is now expanded as a linear
combination of basis functions,µ. Ci is the coefficient for
each basis function.
Methods in Computational Chemistry: Section 3
22
Our aim is to find the value of the coefficients Cµi that gives the
best molecular orbitals. The sum is over n basis functions. n is
the number of basis functions chosen for the system. We call
this the basis set size.
3.3. Self consistent field (SCF)
To overcome the many-body problem (that each electron
intereacts with, and is affected by the other electrons in the atom
or molecule) the Hartree-Fock approach usually determines the
wave function for each spin orbital using the average field of
the other electrons. This is done individually for each electron
in turn and repeated until the energy of the system converges to
a single value. This iterative process is called the Self
Consistent Field method.
The output below, from the program Jaguar, shows two
optimisation steps from a QM calculation. Note the energy
dropping in each SCF iteration (blue) and also the energy
difference between the two optimisation steps.
Figure 19.
(Next page) Output from a quantum mechanics
program showing optimisation of the electronic energy of the system
to obtain a selfconsistent field (SCF). The energies are highlighted
in blue
Methods in Computational Chemistry: Section 3
i
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d
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d
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g
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23
RMS
density
change
maximum
DIIS
error
etot
1 N N 5 M
-2210.62211396823
8.2E-03
etot
2 Y Y 6 M
-2215.27178729700 4.6E+00 9.0E-03
etot
3 Y Y 6 M
-2215.32908944686 5.7E-02 3.9E-03
etot
4 N Y 2 U
-2216.31385160808 9.8E-01 6.3E-03
etot
5 Y Y 6 M
-2216.41403872518 1.0E-01 3.6E-03
etot
6 N Y 2 U
-2216.42742120621 1.3E-02 7.6E-04
etot
7 Y Y 6 M
-2216.43548248944 8.1E-03 3.1E-04
etot
8 N Y 2 U
-2216.43603661334 5.5E-04 1.2E-04
etot
9 Y Y 6 M
-2216.43597127327 -6.5E-05 3.7E-05
etot 10 N Y 2 U
-2216.43609844096 1.3E-04 8.4E-06
etot 11 N Y 2 U
-2216.43609990927 1.5E-06 5.7E-06
etot 12 N Y 2 U
-2216.43610072367 8.1E-07 1.7E-06
etot 13 N N 2 U
-2216.43610063534 -8.8E-08 0.0E+00
scf (DFT) done.
der1a (1-e 1st deriv.) done.
rwr (1st deriv. Q) done.
der1b (2-e 1st deriv.) done.
geometry optimization step 1
energy:
-2216.43610063534 hartrees
gradient maximum:
2.7256E-02 . ( 4.5000E-04 )
gradient rms:
6.5478E-03 . ( 3.0000E-04 )
displacement maximum:
8.0701E-02 . ( 1.8000E-03 )
displacement rms:
1.7960E-02 . ( 1.2000E-03 )
2.0E-01
7.9E-02
1.5E-01
4.5E-02
1.7E-02
9.3E-03
1.3E-03
8.4E-04
4.0E-04
1.1E-04
2.4E-05
6.7E-06
0.0E+00
total energy
energy
change
predicted energy change: -1.7658E-02
step size:
0.30054
trust radius: 0.30000
---------------- geometry iteration
1 complete ---------------geopt (optimize geometry) done.
i
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p
d
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d
i
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r
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RMS
density
change
maximum
DIIS
error
etot
1 N N 2 U
-2216.44224698949
7.9E-04
etot
2 Y Y 6 M
-2216.44997260830 7.7E-03 1.7E-04
etot
3 N Y 2 U
-2216.44994474280 -2.8E-05 7.0E-05
etot
4 Y Y 6 M
-2216.45055643097 6.1E-04 2.0E-05
etot
5 N Y 2 U
-2216.45057070014 1.4E-05 8.1E-06
etot
6 N Y 2 U
-2216.45057429400 3.6E-06 4.3E-06
etot
7 N N 2 U
-2216.45057476447 4.7E-07 0.0E+00
scf (DFT) done.
der1a (1-e 1st deriv.) done.
rwr (1st deriv. Q) done.
der1b (2-e 1st deriv.) done.
geometry optimization step 2
energy:
-2216.45057476447 hartrees
energy change:
-1.4474E-02 . ( 5.0000E-05 )
gradient maximum:
6.4643E-03 . ( 4.5000E-04 )
gradient rms:
1.6979E-03 . ( 3.0000E-04 )
displacement maximum:
5.3949E-02 . ( 1.8000E-03 )
displacement rms:
7.8657E-03 . ( 1.2000E-03 )
7.1E-03
2.0E-03
2.5E-03
4.6E-04
1.7E-04
4.4E-05
0.0E+00
total energy
energy
change
Methods in Computational Chemistry: Section 3
24
3.4. The variational principle
The variational principle states that the energy for an
approximate wave function always lies above or is equal to the
exact solution of the Schrödinger equation for that system.
This means that if we have a wave function that contains
adjustable parameters (the coefficients Ci discussed above) and
we adjust them to minimise the expectation value of the energy,
then we are approaching the exact result.
3.5. Basis functions - General introduction
Basis functions are the mathematical functions that are used to
create each orbital.
There are two ways we can think about the basis functions:
 The first and simplest way is to think of basis functions as
the atomic orbitals.
 The second way is just to think of basis functions as a set
of mathematical functions which are designed to give the
maximum flexibility to the molecular orbitals. This leads
to what are often called extended basis sets. Where we can
add almost any other function we like.
Since the coefficients of the basis functions in the final
molecular orbitals are selected by the variational principle to
minimise the energy, if we make a bad guess for some basis
functions, they will simply appear with small or zero
coefficients. However we must include basis functions that
really do count for something and we must exclude poor basis
functions since they increase the computational cost for no real
gain.
Methods in Computational Chemistry: Section 3
25
The most widely used ab inito methods build basis functions is
by combining a number of Gaussian functions. A Gaussian is
also called a normal distribution.
A Gaussian curve is defined by the following equation:
-(x-b)2
f (x) = ae
2c 2
Where a is the height, b is the position of the peak maximum
and c is the width at half height.
Gaussian functions are used because they have mathematical
properties that simplify their use. Notably multiplying two
Gaussian functions together produces another Gaussian.
Methods in Computational Chemistry: Section 3
1
0.9
0.8
0.7
0.6
0.5
Slater orbital
0.4
Gaussian
0.3
0.2
0.1
0
-4
-2
0
2
4
1
0.9
0.8
0.7
0.6
Slater orbital
0.5
Gaussian 1
0.4
Gaussian 2
0.3
STO-2G
0.2
0.1
0
-4
-2
0
2
4
1
0.9
Slater orbital
0.8
Gaussian 1
0.7
Gaussian 2
0.6
0.5
Gaussian 3
0.4
Gaussian 4
0.3
Gaussian 5
0.2
Gaussian 6
0.1
STO-6G
0
-4
-2
0
2
4
Figure 20.
Figure showing how complex orbital shapes can
be build by addition of Gaussian functions. This figure shows how
Slater-type orbitals can be built from 2 or 6 Gaussian functions to
give the STO-2G and STO-6G basis functions
26
Methods in Computational Chemistry: Section 3
27
3.5.1. Minimal basis sets
A minimal basis set is one that has one basis function for every
occupied atomic orbital in each atom. We do however complete
all sub shells.
Therefore for hydrogen, the minimal basis set is just one 1s
orbital. For carbon, the minimal basis set consists of a 1s orbital,
a 2s orbital and the full set of three 2p orbitals.
The minimal basis set for the methane molecule consists of 4 1s
orbitals, one per hydrogen atom, and the set of 1s, 2s and 2p
described above for carbon. The total basis set comprises 9 basis
functions.
Several minimal basis sets are in common use, but by far the
most common are the STO-nG basis sets devised by John Pople
and his group. The most common of these is STO-3G, where a
linear combination of three Gaussain Type Orbitals (GTOs) are
used to make one orbital (known as a Slater-type orbital or
STO). The STO-nG basis sets are available for almost all
elements in the periodic table.
3.5.2. Split basis functions
In the very early calculations on the hydrogen molecule it was
discovered that Slater orbitals (which correspond to the
hydrogen 1s orbital found when solving the Schrödinger
equation) do not give the best result in the molecular
environment. Better results are obtained if the size of the
orbitals is scaled and the orbitals in the H2 molecule are slightly
contracted. This is a general penomonon and the generally used
solution is to replace each basis set orbital by two orbitals, one
large and one small. In each molecular orbital both orbitals of
the set appear and they will mix in the ratio that gives the lowest
energy. The combination of a large orbital and a small orbital is
essentially equivalent to an orbital of intermediate size.
Methods in Computational Chemistry: Section 3
28
Figure 21.
Combination of large and small s and p orbitals to
give in each case an orbital of intermediate size.
We can choose to scale only the valence orbitals of the minimal
basis set in this way, giving rise to the split valence basis set (eg
3-21 G or 6-31 G), or we can scale all the orbitals of the
minimal basis set in this way, giving rise to double-zeta basis
sets (these often have DZ in the basis set name).
3.5.3. Polarisation functions
Additional functions are necessary in order to allow distortion of
the orbitals to reflect polarisation and provide improved results.
For example on C, N, O it is common to add a single set of d
orbitals. On hydrogen, it is common to add a p orbital. These
are known as polarisation functions and are commonly denoted
with a ‘*’ or two. E.g. The 6-31G basis set with a set of dorbitals on carbon would become 6-31G* and if p-orbitals are
also added to hydrogen the basis set would be denoted 6-31G*.
Methods in Computational Chemistry: Section 3
29
3.5.4. Examples of basis functions
Very many sets of basis functions have been developed. Some
common ones are:
 STO-nG (n = 2-6 most commonly n = 3) a minimal basis
set which has orbitals built of n Gaussian functions per
orbital (s, p, d).
 6-31G, 6-311G ‘Split valence’ basis sets using six
gaussians for inner orbitals and more functions for the
valence orbitals.
 6-311G* A split valence basis set including polarisation
functions
3.6. Limitations of the Hartree-Fock method
A large number of methods have been used to improve the
Hartree-Fock method.
First, why is the Hartree-Fock method not capable of giving the
correct solution to the Schrödinger equation if a very large and
flexible basis set is selected? In passing, we note that the very
best Hartree-Fock wave function, obtained with just such a large
and flexible basis set, is called the "Hartree-Fock limit".
The problem is that electrons are not paired up in the way that
the Hartree-Fock method supposes. It suggests that the two
electrons have the same probability of being in the same region
of space as being in separate symmetry equivalent regions of
space. For example, in H2 it would give the same probability of
both electrons being near one atom as one being near one atom
and the other near the second atom. This is clearly wrong. The
Hartree-Fock method also only evaluates the repulsion energy as
an average over the whole molecular orbital.
The two electrons in a molecular orbital are in reality moving in
such a way that they keep more apart from each other than being
close. We call this effect correlation. The difference in energy
between the exact result and the Hartree-Fock limit energy is
called the correlation energy.
Methods in Computational Chemistry: Section 3
30
The electron correlation must be taken into account to improve
the accuracy of QM methods. A number of techniques are in
common use:
 variational methods (CISD, MCSCF, CASSCF)
 perturbation methods (MP2, MP4, CCSD(T) )
 density functional methods (B3LYP, BLYP, etc)
3.7. Density functional theory (DFT) methods
Density functional theory is an alternate approach to solving the
Schrödinger equation. Over recent years it has been developed
to give good quality results for many molecular systems. DFT
methods are relatively fast, and as a result have become widely
used. We will not cover DFT methods here, except to note that
the B3LYP functional is the most popular DFT method.
3.8. Ab initio procedures
There are several procedures that can be carried out with any ab
initio program. We will restrict ourselves here to three: Single point calculations
 Geometry optimisation calculations
 Frequency calculations
3.8.1. Single point calculations
This procedure simply calculates the energy, wave function and
other requested properties at a single fixed geometry. It is
usually done first at the beginning of a study on a new molecule
to check out the nature of the wavefunction. It is also frequently
carried out after a geometry optimisation, but with a larger basis
set or a more superior method than is possible with the basis set
and method used to optimise the geometry. Thus for a very large
system the geometry may be optimised at HF level with the 321G basis set, but energy differences between isomers are then
explored with the MP2 method and the 6-31G** basis set.
Methods in Computational Chemistry: Section 3
31
3.8.2. Geometry optimisation calculations
Geometry optimisation in ab initio calculations is essentially the
same as for molecular mechanics calculations, except that the
energy of the molecule at each step is calculated using QM
methods.
The procedure calculates the wave function and energy at a
starting geometry and then proceeds to move to a new geometry
which will give a lower energy. This is then repeated until we
have the lowest energy geometry close to the starting point.
Ideally this procedure calculates the forces on the atoms by
evaluating the gradient (first derivative) of the energy with
respect to atomic coordinates analytically. Sophisticated
algorithms (eg the BFGS algorithm) are used to select a new
geometry at each step, which gives rapid convergence to the
geometry with the lowest energy.
3.8.3. Frequency calculations
We carry out frequency runs for two reasons. First, we may
want to actually predict the frequencies and the I.R. and Raman
intensities. Here we note that the frequencies are harmonic
frequencies - they are those obtained by assuming the potential
energy surface is harmonic.
More importantly, frequency calculations can also be used to
determine if the structure is at a minimum or transition structure.
This process is equivalent to determining the gradient at the
minimum. At a minimum all vibrational frequencies will be real
and positive. If we have a transition structure or any stationary
point other than a minimum, some of the frequencies will be
complex. These are printed out as negative numbers and are
often called imaginary frequencies. A well-behaved transition
structure for a reaction will have one imaginary frequency. If we
have restrained the symmetry in the optimisation, we may get
more than one imaginary frequency.
Methods in Computational Chemistry: Section 3
32
3.9. Outline of a typical ab initio calculation
3.9.1. Read input & calculate a geometry
The geometry can usually be input in the form of a Z-matrix or
in Cartesian coordinates.
3.9.2. Assign basis set
When we are optimising the geometry, it is usually most
efficient to start with a fairly poor (but fast) basis set such as
STO-3G or 3-21G for organic compounds, and then use the
optimised geometry with this basis set as a starting geometry for
a better (but more computationally intensive) basis set.
3.9.3. Calculate nuclear repulsion energy
After getting the molecular geometry and the basis set, the
program will then most probably evaluate the nuclear repulsion
energy. This is just
E=å
Z AZ B
RAB
where the summation is over all pairs of atoms, A and B; ZA is
the atomic number of atom A; and RAB is the distance between
atoms A and B. 1
This energy will be added later to the electronic energy.
3.9.4. Calculate integrals
It is necessary to calculate integrals involving all the terms in
the Hamiltonian for the system and the basis set functions. The
integrals consist of two types.
One-electron integrals depend only on the coordinates of one
electron. These calculate the energy of a single electron due to
interaction with the nucleus, etc. These are simple and there are
n2 of them, where n is the number of basis functions.
1
Note that this is Coulomb’s law where all of the constants (40) are normalised to
1. In this case the energy unit is Hartrees.
Methods in Computational Chemistry: Section 3
33
Two electrons integrals calculate the energy due to electronelectron interactions. These integrals are more problematic.
Their number rises as n4. This is the first computationally
demanding part of any ab inito program.
3.9.5. Assign electronic configuration
We must specify the total charge of the molecule in the input
data, so the total number of electrons in the molecule can be
counted correctly and the electronic configuration. We select the
latter by first defining the multiplicity (eg singlet, doublet,
triplet). Most of the cases you will come across will be closed
shell singlets.
3.9.6. Generate initial guess
Here, a set of very rough molecular orbitals is selected as a first
guess to the molecular orbitals. These are then filled in
ascending order of energy until all the electrons are used.
3.9.7. Self-consistent field iterations (electronic energy)
The solution for the molecular orbitals and the total energy has
to be carried out iteratively starting from the initial guess.
3.9.8. Calculate the total energy
Total energy = nuclear repulsion + electronic
The total energy is just evaluated by adding the nuclear
repulsion energy to the electronic energy.
3.9.9. Electron density analysis
In the electron density analysis (often called the population
analysis) the electron density of the whole molecule is
partitioned in some way amongst all the orbitals and all the
atoms. In this way, atomic charges, dipole moments, multipole
moments and other properties can be calculated.
3.10. Capabilities of ab initio quantum chemistry
 Can handle any element
 Can calculate wavefunctions and detailed descriptions of
molecular orbitals
Methods in Computational Chemistry: Section 3
34
 Can optimise geometries
 Can be used for equilibrium structures, transition
structures, intermediates, and neutral and charged species
 Can handle any electron configuration (ground and excited
states)
 Can calculate atomic charges, dipole moments, multipole
moments, polarisabilities, etc.
 Can calculate vibrational frequencies, IR and Raman
intensities, NMR chemical shifts
3.11. Applications of ab initio quantum chemistry
Quantum chemistry has a very wide range of applications. It’s
strengths lie in the accurate prediction of molecular structure
(provided the molecules are small) and in the ability to provide
information about the electronic state of the molecule. It is also
the only way to investigate chemical reactions involving bond
formation.
Strengths of ab initio quantum chemistry
 No experimental bias
 Can improve a calculation in a logical manner (basis sets,
level of theory)
 Provides information on intermediate species, including
spectroscopic data
 Can calculate novel structures (no experimental data is
required)
 Can calculate any electronic state
 Can calculate all atomic elements
Limitations of ab initio quantum chemistry
 Calculations are more complex
 Requires more CPU time than empirical (Molecular
mechanics) or semi-empirical methods (MOPAC, etc)
 Can not be (easily) used for large system
Methods in Computational Chemistry: Section 3
35
Methods in Computational Chemistry: Section 3
Chapter 4.
36
Semiempirical MO calculations
Semiempirical MO methods are quantum mechanics
calculations that make several significant approximations
(ignoring the overlap of some orbitals) in order to speed up the
calculation.
Rather attempting to solve the Schrödinger
equation from first principles, they are parameterised to
reproduce experimental data.
Semiempirical calculations are much faster than ab initio
calculations but substantially slower than Molecular Mechanics
calculations.
Semiempirical calculations make a number of assumptions:
 The Born-Oppenheimer approximation is a fundamental
assumption (i.e. the nuclei remain fixed on the time scale
of electron movement). This assumption is also made for
ab initio calculations.
 Only valence electrons participate bonding and are
principally responsible for the physical properties of a
molecule. Ab initio programs spend a large amount of
time calculating the inner, non-valence electrons.
Therefore if the non-valence electrons are combined with
the nucleus to form a ‘core’ the calculation can be speeded
up greatly.
 Semiempirical methods make the assumption that the
some types of interactions between orbitals (particularly
between orbitals on different atoms) are small and can be
neglected. This significantly reduces the number of
calculations that must be performed. Various
simplifications have been made, giving rise to a number of
semiempirical methods such as CNDO (Complete Neglect
of Differential Overlap, developed by Pople et al in the
mid 1960s) and NDDO (Neglect of Diatomic Differential
Overlap, developed by Dewar and Thiel in the 1970s).
Methods in Computational Chemistry: Section 3
37
As stated above, semiempirical methods are parameterised to
reproduce experimental data. A set of parameters is required for
each atomic element.
Table 1. Parameters used for the RM1 semiempirical method
(Rocha et al. J. Comp. Chem. 1101, 27, 2006)
Uss
Upp
s
p
A
Gss
Gsp
Gpp
Gp2
Hsp
ai
bi
ci
s
p
s atomic orbital one-electron one-center integral
p atomic orbital one-electron one-center integral
s atomic orbital one-electron two-center resonance integral
term
p atomic orbital one-electron two-center resonance integral
term
atom A core-core repulsion term
s–s atomic orbitals one-center two-electron repulsion
integral
s–p atomic orbitals one-center two-electron repulsion
integral
p–p atomic orbitals one-center two-electron repulsion
integral
p–p� atomic orbitals one-center two-electron repulsion
integral
s–p atomic orbital one-center two-electron exchange
integral
Gaussian multiplier for the ith Gaussian of atom A
Gaussian exponent multiplier for the ith Gaussian of atom A
radial center of the ith Gaussian of atom A
s-type Slater atomic orbital exponent
p-type Slater atomic orbital exponent
Note that semiempirical methods often report the enthalpy
(heat) of formation as well as the total molecular energy. The
enthalpy of formation is defined as the energy produced (or
required) to form 1 mole of the compound its standard state
from its constituent elements in their standard states. The
standard state of an element is the most stable form at 101.325
kPa and 298 K)
Methods in Computational Chemistry: Section 3
38
The actual energy values calculated using semiempirical
methods often differ from ab initio due to the use of ‘frozen’
core orbitals in the semiempirical approach.
Methods in Computational Chemistry: Section 3
39
4.1. Semiempirical MO Software
As is the case for molecular mechanics software, semiempirical
methods are implemented in a number of different software
packages. The most widely used of these is MOPAC which is
written in FORTRAN and was developed by the Dewar group
and first released in 1981. MOPAC exists in a number of
versions, some of which are freely available (eg MOPAC 2007,
www.openmopac.net).
The current freely available MOPAC (MOPAC 2007) contains a
number of different semiempirical methods including:
MNDO Modified INDO
AM1
Austin Model 1
PM3
PM5
RM1
Parametric Model 3
Parametric Model 5
Reparameterization
of AM1
Developed by Michael Dewar and Walter Thiel
Developed by Michael Dewar and Andrew Holder
in 1986
Developed by Jimmy Stewart in 1988
Published J. Stewart et al in 2005
Methods in Computational Chemistry: Section 3
40
Example MOPAC output for carbamic acid
*********************************************************************
FRANK J. SEILER RES. LAB., U.S. AIR FORCE ACADEMY, COLO. SPGS., CO.
********************************************************************
AM1 CALCULATION RESULTS
*********************************************************************
*
MOPAC: VERSION 6.00
CALC'D. Mon Apr 28 10:35:06 2008
* MMOK
- APPLY MM CORRECTION TO CONH BARRIER
* T=
- A TIME OF 3600.0 SECONDS REQUESTED
* DUMP=N - RESTART FILE WRITTEN EVERY 3600.0 SECONDS
* AM1
- THE AM1 HAMILTONIAN TO BE USED
************************************************************050BY050
AM1 T=3600 MMOK
/san1/vcp1/people/david/mopacfile.dat
ATOM CHEMICAL BOND LENGTH BOND ANGLE
TWIST ANGLE
NUMBER SYMBOL (ANGSTROMS)
(DEGREES)
(DEGREES)
(I)
NA:I
NB:NA:I
NC:NB:NA:I
NA NB NC
1
C
2
O
1.37680 *
1
3
O
1.24488 *
117.32253 *
1 2
4
N
1.36457 *
114.51033 * 179.99077 * 1 2
5
H
.97136 *
107.28397 *
-.10502 * 2 1
6
H
.98763 *
118.69101 * 179.99923 * 4 1
7
H
.98705 *
120.34405 *
.00000 * 4 1
CARTESIAN COORDINATES
NO.
ATOM
X
Y
Z
1
C
.0000
.0000
.0000
2
O
1.3768
.0000
.0000
3
O
-.5714
1.1060
.0000
4
N
-.5661 -1.2416 -.0002
5
H
1.6654
.9275 -.0017
6
H
-1.5511 -1.3136 -.0002
7
H
.0021 -2.0487 -.0003
H: (AM1): M.J.S. DEWAR ET AL, J. AM. CHEM.
C: (AM1): M.J.S. DEWAR ET AL, J. AM. CHEM.
N: (AM1): M.J.S. DEWAR ET AL, J. AM. CHEM.
O: (AM1): M.J.S. DEWAR ET AL, J. AM. CHEM.
SOC.
SOC.
SOC.
SOC.
107
107
107
107
3
3
2
2
3902-3909
3902-3909
3902-3909
3902-3909
RHF CALCULATION, NO. OF DOUBLY OCCUPIED LEVELS = 12
MOLECULAR MECHANICS CORRECTION APPLIED TO PEPTIDELINKAGE
GRADIENTS WERE INITIALLY ACCEPTABLY SMALL
SCF FIELD WAS ACHIEVED
(1985)
(1985)
(1985)
(1985)
Methods in Computational Chemistry: Section 3
41
AM1 CALCULATION
VERSION 6.00
Mon Apr 28 10:35:06 2008
FINAL HEAT OF FORMATION =
-98.20079 KCAL
TOTAL ENERGY
=
-1018.12384 EV
ELECTRONIC ENERGY
=
-2616.73556 EV
CORE-CORE REPULSION
=
1598.61172 EV
IONIZATION POTENTIAL =
11.01791
NO. OF FILLED LEVELS =
12
MOLECULAR WEIGHT
=
61.040
SCF CALCULATIONS =
2
COMPUTATION TIME = .000 SECONDS
ATOM CHEMICAL BOND LENGTH BOND ANGLE
TWIST ANGLE
NUMBER SYMBOL (ANGSTROMS)
(DEGREES)
(DEGREES)
(I)
NA:I
NB:NA:I
NC:NB:NA:I
NA NB NC
1
C
2
O
1.37680 *
1
3
O
1.24488 *
117.32253 *
1 2
4
N
1.36457 *
114.51033 * 179.99077 * 1 2 3
5
H
.97136 *
107.28397 * -.10502 * 2 1 3
6
H
.98763 *
118.69101 * 179.99923 * 4 1 2
7
H
.98705 *
120.34405 *
.00000 * 4 1 2
H 7 .000000
NET ATOMIC CHARGES AND DIPOLE CONTRIBUTIONS
ATOM NO. TYPE
CHARGE
ATOM ELECTRON DENSITY
1
C
.3961
3.6039
2
O
-.3226
6.3226
3
O
-.4137
6.4137
4
N
-.4130
5.4130
5
H
.2585
.7415
6
H
.2506
.7494
7
H
.2441
.7559
DIPOLE
X
Y
Z
TOTAL
POINT-CHG.
.328 -2.567
-.002
2.588
HYBRID
.004
.010
-.001
.011
SUM
.332 -2.556
-.003
2.578
NO.
1
2
3
4
5
6
7
CARTESIAN COORDINATES
ATOM
X
Y
Z
C
.0000
.0000
.0000
O
1.3768
.0000
.0000
O
-.5714 1.1060
.0000
N
-.5661 -1.2416 -.0002
H
1.6654
.9275 -.0017
H
-1.5511 -1.3136 -.0002
H
.0021 -2.0487 -.0003
ATOMIC ORBITAL ELECTRON POPULATIONS
1.20196 .81168 .84439 .74591 1.86312 1.21335 1.35377
1.91988
1.72615
1.21863
1.54906
1.43270
1.11115
1.81264
.74149 .74937 .75588
== MOPAC DONE ==
1.89239
1.05648
Methods in Computational Chemistry: Section 3
4.2. Strengths
methods
and
limitations
of
42
semiempirical
Semiempirical MO methods are much faster than ab initio MO
methods. They can be used with quite large molecular systems,
including proteins (see J Mol Model (2009) 15:765–805).
In general, semiempirical methods are known to have a number
of limitations, particularly in the calculation of some barriers to
rotation and in the calculation of weak interactions (eg hydrogen
bonding).
For example, the AM1 and PM3 methods grossly underestimate
the barrier to rotation about an amide bond. This has
necessitated the addition of an extra molecular mechanics force
for compounds that contain amide bonds to produce improved
results.
Continued development of semiempirical methods has resulted
in a steady improvement in the quality of the results.
Methods in Computational Chemistry: Section 3
43
Methods in Computational Chemistry: Section 3
Chapter 5.
44
Section summary
Below are the key concepts from this section.
Quantum mechanics




Properties of particles
Properties of waves
Quantisation
Wave-particle duality
The Schrödinger equation




Key features of the Schrödinger equation
Solutions to the particle in an infinite potential well
Quantum numbers
Relationship between the simple 1D particle example and
atomic orbitals
Ab initio QM calculations
 The Hartree-Fock method
 Basis sets
 QM calculations make a number of assumptions:
o That the Born-Oppenheimer approximation holds
(ie. that the nuclei remain fixed on the scale of
electron movement).
o That the basis sets adequately represent molecular
orbitals.
o That electron correlation is adequately included.
 Strengths and weaknesses of ab initio QM calculations
Semiempirical MO calculations
 The basis of semiempirical MO calculations
 Strengths and weaknesses of semiempirical
calculations
MO