Introduction to Computational
Chemistry
10/8/2019 1
Prepared by
Cina Foroutan-Nejad
and
Martin Novak
What is what? Who is who?
Reference:
Essentials of Computational Chemistry,
Theories and Models
By
Christopher J. Cramer
10/8/2019 2
Theory
• A theory is one or more rules that are
postulated to govern the behavior of physical
systems
• These rules are quantitative in nature and
expressed in the form of a mathematical
equation
• The quantitative nature of scientific theories
allows them to be tested by experiment
• Are you familiar with BIG THEORIES?
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How far a Theory Goes?
• If a sufficiently large number of interesting
systems falls within the range of the theory,
practical reasons tend to motivate its
continued use.
• Which examples do you know for the above
mentioned statement?
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Rule
• Occasionally, a theory has proven so robust
over time, even if only within a limited range
of applicability, that it is called a law
• Which one is a law?
Coulomb Law
Huckel Rule
• Which laws you know?
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Model
• A model, on the other hand, typically involves the
deliberate introduction of simplifying
approximations into a more general theory so as
to extend its practical utility.
• Another feature sometimes characteristic of a
quantitative model is that it incorporates certain
constants that are empirically determined
• How many models in chemistry you know?
• What is the golden rule for successful application
of a model?
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Computation
• Computation is the use of digital technology
to solve the mathematical equations defining
a particular theory or model
• In general three groups of people can be listed
as “Computational Chemists”. Usually they
collaborate together and their disciplines
overlap to a great extent
• Do you know who they are?
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Who Are You?
• Theorists tend to have as their greatest goal the development of
new theories and/or models that have improved performance or
generality over existing ones
• Researchers involved in molecular modeling tend to focus on target
systems having particular chemical relevance (e.g., for economic
reasons) and to be willing to sacrifice a certain amount of
theoretical rigor in favor of getting the right answer in an efficient
manner.
• Computational chemists may devote themselves not to chemical
aspects of the problem, per se, but to computer-related aspects,
e.g., writing improved algorithms for solving particularly difficult
equations, or developing new ways to encode or visualize data,
either as input to or output from a model
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Our Sacred Temple;
Quantum Mechanics
• The ultimate goal of a computational chemist
is to extract all information about his/her
model system purely from high-level quantum
mechanic computations
• Among several version of quantum mechanics,
wave equation introduced by Erwin
Schrödinger is the preferred version among
chemists.
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Schrödinger Equation
• Time dependent non-relativistic Schrödinger
equation
• Time independent non-relativistic Schrödinger
equation
10/8/2019 10
What a Computational Chemist
Can Do?
• Schrödinger equation provides us information
about “Energy” of a system. All molecular
properties that are related to energy are
within the realm of computational chemistry.
• Which properties of matter are related to the
energy?
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Potential Energy (Hyper)-Surface
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H+H–
H+H–
H2
H2
Potential Energy (Hyper)-Surface
10/8/2019 13
PES is a hyper-surface defined by the
potential energy of a group of atoms in a
particular electronic state over all possible
atomic arrangements.
Every atom in space has 3 degrees of
freedom, corresponding to three dimensions.
An N-atomic molecule has 3N–6 (3N–5 for
linear systems) internal degrees of freedom
beside 3 degrees of rotational as well as 3
degrees of translational freedom. PES of a
molecule denotes variation of energy by
changing the degrees of freedom of that
molecule.
Stationary Points on a PES
• An Stationary Point is a point at which
𝜕𝐸
𝜕𝑥
= 0
• If for all degrees of freedom
𝜕2 𝐸
𝜕𝑥2 > 0 then the stationary point
is a Local Minimum
• If on a PES we find the most stable local minimum then we
can call it the Global Minimum
• If for n degrees of freedom in a stationary point
𝜕2 𝐸
𝜕𝑥2 < 0 then
the stationary point is an nth order saddle point
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PES for Diatomics
10/8/2019 15
• Morse Potential:
𝑉 𝑟 = 𝐷𝑒(1 − 𝑒−𝑎 𝑟−𝑟𝑒 )2
• 𝐷𝑒 is the depth of potential well
• a controls the width of potential
• r is the interatomic separation
• And re is the equilibrium distance
PES for Diatomics
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• Lenard-Jones Potential:
𝑉𝐿𝐽 = 4𝜀[
𝜎
𝑟
12
−
𝜎
𝑟
6
]
• 𝜀 is the depth of potential well
• σ is the finite distance at which the
inter-particle potential is zero
• r is the interatomic separation
PES for Polyatomics
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• On a 2-dimensional surface, only a 2D PES can be projected.
• For a reaction we conventionally
depict reaction
coordinates which is
the minimum energy
path of the reaction.
This may involve
rearrangement of several
degrees of freedom
though.
Reaction Coordinates
• A reaction coordinates can be roughly
presented as the following.
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How a Software Optimized a
Molecule?
• Software is STUPID! Watch out!
10/8/2019 19
Digging Schrödinger Equation
Cina Foroutan-Nejad
References:
Introduction to Computational Chemistry
Frank Jensen
Essentials of Computational Chemistry, Theories and Models
Christopher J. Cramer
10/8/2019 20
Schrödinger Equation
• The wave function or state function (Ψ) describes
state of a system in quantum mechanics.
• For a one particle, one dimensional system, Ψ is a
function of time and coordinate, Ψ = Ψ(x,t)
• Newton’s second law tells us the future of a classical
system but for a quantum system we need
Schrödinger Equation
−
ℏ
𝑖
𝜕Ψ 𝑥, 𝑡
𝜕𝑡
= −
ℏ2
2𝑚
𝜕2Ψ 𝑥, 𝑡
𝜕𝑥2
+ 𝑉(𝑥, 𝑡)Ψ 𝑥, 𝑡
10/8/2019 21
Time-Independent Schrödinger
Equation
• If Potential energy does not change by time then 𝑉 =
𝑉 𝑥 and then
−
ℏ
𝑖
𝜕Ψ 𝑥,𝑡
𝜕𝑡
= −
ℏ2
2𝑚
𝜕2Ψ 𝑥,𝑡
𝜕𝑥2 + 𝑉(𝑥)Ψ 𝑥, 𝑡 (1)
Furthermore Ψ 𝑥, 𝑡 can be written as product of
two functions: Ψ 𝑥, 𝑡 = 𝜓 𝑥 𝑓 𝑡
Now, considering partial derivations one can write
•
𝜕Ψ 𝑥,𝑡
𝜕𝑡
=
𝑑𝑓 𝑡
𝑑𝑡
𝜓 𝑥 ,
𝜕2Ψ 𝑥,𝑡
𝜕𝑥2 =
𝜕2 𝜓 𝑥
𝜕𝑥2 𝑓(𝑡) (2)
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Time-Independent Schrödinger
Equation
• By substituting 2 in 1 we have
−
ℏ
𝑖
𝑑𝑓 𝑡
𝑑𝑡
𝜓 𝑥 = −
ℏ2
2𝑚
𝑓(𝑡)
𝑑2 𝜓 𝑥
𝑑𝑥2 + 𝑉 𝑥 𝑓(𝑡)𝜓 𝑥 (3)
−
ℏ
𝑖
1
𝑓 𝑡
𝑑𝑓 𝑡
𝑑𝑡
= −
ℏ2
2𝑚
1
𝜓 𝑥
𝑑2 𝜓 𝑥
𝑑𝑥2 + 𝑉 𝑥 𝜓 𝑥 (4)
Now we have two equal functions while the left hand side of
equation is a function of time and the right hand side is a
function of x. This is only possible if both sides are equal to a
constant, called E.
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Time-Independent Schrödinger
Equation
• The left hand side is not of our interest but
the right hand side of equation 4 is the time
independent Schrödinger Equation for a one
particle, one dimensional system
−
ℏ2
2𝑚
𝑑2
𝜓 𝑥
𝑑𝑥2
+ 𝑉 𝑥 𝜓 𝑥 = 𝐸𝜓 𝑥
෡𝐻𝜓 𝑥 = 𝐸𝜓 𝑥
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Acceptable Wave Function
• A wave function 𝜓 is called well behaved (or
acceptable) if
1. Is Continues all over space
2. Its first derivative is Continues over space
3. Is Single Valued
4. Is Quadratically Integrable, i.e. ‫׬‬ 𝜓𝜓∗
𝑑𝜏 over
space exist
Since we do not know what 𝜓 is, any function that
satisfies these conditions is a Suitable First Guess
for 𝜓.
10/8/2019 25
Molecular Hamiltonian
• The Hamiltonian operator of a molecule,
neglecting relativistic effects and spin-orbit
can be written as the following
෡𝐻 = −
ℏ2
2
෍
𝛼
1
𝑚 𝛼
𝛻𝛼
2
−
ℏ2
2𝑚 𝑒
෍
𝑖
𝛻𝑖
2
+
෍
𝛼
෍
𝛽>𝛼
𝑍 𝛼 𝑍 𝛽 𝑒2
𝑟 𝛼𝛽
− ෍
𝛼
෍
𝑖
𝑍 𝛼 𝑒2
𝑟𝑖𝛼
+ ෍
𝑖
෍
𝑗>𝑖
𝑒2
𝑟𝑖𝑗
10/8/2019 26
Born-Oppenheimer Approximation
• Concerning the fact that nuclei are way heavier
than electrons one can imagine that nuclei
almost remain still while electrons circle around
the molecule. Therefore, nuclear kinetic energy
can be separated from the Electronic Hamiltonian
෡𝐻 = −
ℏ2
2𝑚 𝑒
෍
𝑖
𝛻𝑖
2
+
− ෍
𝛼
෍
𝑖
𝑍 𝛼 𝑒2
𝑟𝑖𝛼
+ ෍
𝑖
෍
𝑗>𝑖
𝑒2
𝑟𝑖𝑗
+ ෍
𝛼
෍
𝛽>𝛼
𝑍 𝛼 𝑍 𝛽 𝑒2
𝑟 𝛼𝛽
10/8/2019 27
Atomic Units
• Atomic Units are units of nature
• Schrödinger Equation written in au looks much
less complicated
Energy: 1 Hartree = Eh = e2/a0 = 27.2114 eV
=4.35974417(75)×10−18 J = 627.5095 kcal.mol-1
Length: a0 = 5.2917721092(17)×10−11 m = 0.52917721092 Å
Charge: e = 1.602176565(35)×10−19 C
Mass: me = 9.10938291(40)×10−31 kg ≠ 1 amu (Atomic
Mass Unit)
10/8/2019 28
Schrödinger Equation; A Differential
Equation
Schrödinger Equation is a differential equation. This means that
Schrödinger Equation has more than one acceptable
eigenfunctions, ψ, for one particular problem, i.e. molecule.
Each ψ is associated with a particular eigenvalue that is a
particular energy. All these eigenfunctions for a one-particle
system in three dimensions are orthonormal. Orthonormal
means:
න 𝜓𝑖 𝜓𝑗 𝑑𝜏 = 𝛿𝑖𝑗 ൝
𝛿𝑖𝑗 = 1 𝑖𝑓 𝑖 = 𝑗
𝛿𝑖𝑗 = 0 𝑖𝑓 𝑖 ≠ 𝑗
(5)
𝛿𝑖𝑗 is called Krönecker Delta
10/8/2019 29
Schrödinger Equation; A Differential
Equation
• Regarding (5) one can propose a way for evaluation of Energy
associated with a wave function. We knew ෡𝐻𝜓𝑖 = 𝐸𝜓𝑖;
multiplying this equation by ψj and integrating both sides
gives:
න 𝜓𝑖 𝐻𝜓𝑗 𝑑𝜏 = න 𝜓𝑖 𝐸𝜓𝑗 𝑑𝜏
Since E is a constant, we have:
න 𝜓𝑖 𝐻𝜓𝑗 𝑑𝜏 = 𝐸𝑖 𝛿𝑖𝑗
10/8/2019 30
Solving an Equation with Two Missing
Variables!
• We have no clue about the nature of 𝜓
• We do not know the energy of the system
• Don’t panic! There is a solution:
The Variational Principle
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The Variational Principle
• Let’s assume an arbitrary function, Φ, to be an
eigenfunction of Schrödinger Equation. Φ
should be a linear combination of some of
infinite number of real eigenfunctions, ψ, that
is:
Φ = ෍
𝑖
𝑐𝑖 𝜓𝑖 (6)
Unfortunately yet we don’t know anything
about ψi and of course Ci!
10/8/2019 32
The Variational Principle
• Fortunately, normality of Φ imposes a
constraint on the coefficients of (6).
න Φ2
𝑑𝜏 = 1 = න ෍
𝑖
𝑐𝑖 𝜓𝑖 ෍
𝑗
𝑐𝑗 𝜓𝑗 𝑑𝜏 =
෍
𝑖𝑗
𝑐𝑖 𝑐𝑗 න 𝜓𝑖 𝜓𝑗 𝑑𝜏 = ෍
𝑖𝑗
𝑐𝑖 𝑐𝑗 𝛿𝑖𝑗 = ෍
𝑖
𝑐𝑖
2
10/8/2019 33
The Variational Principle
• For measuring the Energy associated with
wave function Φ, one can write:
න Φ𝐻Φ𝑑𝜏 = න ෍
𝑖
𝑐𝑖 𝜓𝑖 𝐻 ෍
𝑗
𝑐𝑗 𝜓𝑗 𝑑𝜏 =
෍
𝑖𝑗
𝑐𝑖 𝑐𝑗 න 𝜓𝑖 𝐻𝜓𝑗 𝑑𝜏 = ෍
𝑖𝑗
𝑐𝑖 𝑐𝑗 𝐸𝛿𝑖𝑗 = ෍
𝑖
𝑐𝑖
2
𝐸
10/8/2019 34
The Variational Principle
The Variational Principle states that there is a lower limit for the
energy of a quantum system that is the energy obtained form its
exact wave function. Therefore, energy obtained form any
arbitrary wave function is always higher than that of exact wave
function.
‫׬‬ Φ𝐻Φ𝑑𝜏
‫׬‬ Φ2 𝑑𝜏
≥ 𝐸0
Here, E0 is the energy of exact wave function. The Variational
Principle enable us to start from any primary guess for wave
function and try to minimize energy as much as possible close
the energy of the exact wave function.
10/8/2019 35
Definition of Terms:
Ab Inotio and DFT Methods
• The word ab initio means from the beginning. In a nutshell,
ab initio methods are different approximations for solving the
Schrödinger Equation. Ab initio methods are NOT
parameterized based on experiments.
• Density Functional Theory or DFT is an old method for finding
the energy of a molecular system based on Hohenberg-Kohn
Theorem as the following:
𝑭 𝝆 𝒓 = 𝑬 𝒆𝒍𝒆𝒄
• Unfortunately, because the ideal functional is not known, DFT
methods are parameterized to be as consistent as possible
with experiments.
10/8/2019 36
Suitable Guess For Wave Function
• An approximation used widely in computational chemistry
is application of a Basis set to simulate the exact wave
function as much as possible.
• A basis set is a limited number of functions which are used
to construct a first guess for the molecular wave function.
• Two types of basis sets are Slater and Gaussian basis sets.
• Computational cost for the fastest ab initio or DFT method
increases by increasing the number of basis functions, M,
by M4 . So, an ideal basis set provides a reasonable solution
to a problem with least number of basis functions.
10/8/2019 37
10/8/2019 38
STO and GTO
STO and GTO; Mathematical Form
• An STO has the following form:
𝑠 𝑟 = 𝑁𝑠 𝑒−𝜁𝑟
• A GTO has the following form:
𝑔 𝑟 = 𝑁𝑔 𝑒−𝜁𝑟2
𝑔 𝑟 = 𝑥 𝑎
𝑦 𝑏
𝑧 𝑐
𝑁𝑔 𝑒−𝜁𝑟2
Ns /Ng is the normalization constant and ζ is orbital
exponent. 𝑥 𝑎
𝑦 𝑏
𝑧 𝑐
is used for defining orbital type
for example for a p orbital a = 1, b = c = 0.
10/8/2019 39
Gaussian Basis Sets
• Every GTO is consisted of one or more
Primitive GTOs
• The degree of contraction is the number of
PGTOs that enter a CGTO or Contracted GTO
10/8/2019 40
Basis Sets
• Pople Type Basis Sets
STO-nG→ n1s,n2s,n2p,… Minimal Basis set
for second row elements
STO-3G→ (6s,3p) → (2s,1p)
k-nlmG→ k core, 2 (nl) or 3 (nlm) zeta, n inner
valence, l middle valence, and m outer valence
for second row elements
6-311G→ (11s,5p) → (4s,3p)
10/8/2019 41
Basis Sets
• Correlation Consistent Basis Sets
s and p optimized at HF, polarization
optimized at CISD
10/8/2019 42
How a Basis Set Looks Like?
6-311+G(3df,3pd)
C 0
S 6 1.00
4563.2400000 0.00196665
682.0240000 0.0152306
154.9730000 0.0761269
44.4553000 0.2608010
13.0290000 0.6164620
1.8277300 0.2210060
SP 3 1.00
20.9642000 0.1146600 0.0402487
4.8033100 0.9199990 0.2375940
1.4593300 -0.00303068 0.8158540
SP 1 1.00
0.4834560 1.0000000 1.0000000
SP 1 1.00
0.1455850 1.0000000 1.0000000
SP 1 1.00
0.0438000 1.0000000 1.0000000
D 1 1.00
2.5040000 1.0000000
D 1 1.00
0.6260000 1.0000000
D 1 1.00
0.1565000 1.0000000
F 1 1.00
0.8000000 1.0000000
Normalization & Contraction constants
Aug-cc-pV5Z
C 0
S 10 1.00
96770.0000000 0.0000250
14500.0000000 0.0001900
3300.0000000 0.0010000
935.8000000 0.0041830
306.2000000 0.0148590
111.3000000 0.0453010
43.9000000 0.1165040
18.4000000 0.2402490
8.0540000 0.3587990
3.6370000 0.2939410
S 10 1.00
96770.0000000 -0.0000050
14500.0000000 -0.0000410
3300.0000000 -0.0002130
935.8000000 -0.0008970
306.2000000 -0.0031870
111.3000000 -0.0099610
43.9000000 -0.0263750
18.4000000 -0.0600010
8.0540000 -0.1068250
3.6370000 -0.1441660
S 1 1.00
1.6560000 1.0000000
S 1 1.00
0.6333000 1.0000000
.
.
.
Normalization & Contraction constants
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Introduction to Ab Initio
Methods
Cina Foroutan-Nejad
Reference:
Introduction to Computational Chemistry
Frank Jensen
10/8/2019 44
Hatree Product Wave Function
Neglecting electron-electron repulsion one can write a molecular
Hamiltonian in terms of one-electron Hamiltonians.
𝐻 = ෍
𝑖
𝑁
ℎ𝑖
ℎ𝑖 = −
1
2
𝛻𝑖
2
− ෍
𝑘=1
𝑀
𝑍 𝑘
𝑟𝑖𝑘
All eigenfunctions of one-electron Hamiltonian must satisfy the oneelectron
Schrödinger equation.
ℎ𝑖 𝜓𝑖 = 𝜀𝑖 𝜓𝑖
If Hamiltonian is separable then many-electron eigenfunction should
be a product of one-electron eigenfunctions:
Ψ 𝐻𝑃 = 𝜓1 𝜓2 … 𝜓 𝑁
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Hatree Product Wave Function
Then the energy of “Hartree Product” wave function is:
𝐻Ψ 𝐻𝑃 = 𝐻𝜓1 𝜓2 … 𝜓 𝑁 = ෍
𝑖=1
𝑁
ℎ𝑖 𝜓1 𝜓2 … 𝜓 𝑁 =
ℎ1 𝜓1 𝜓2 … 𝜓 𝑁 + 𝜓1(ℎ2 𝜓2) … 𝜓 𝑁 + 𝜓1 𝜓2 … (ℎ 𝑁 𝜓 𝑁 )
= 𝜀1 𝜓1 𝜓2 … 𝜓 𝑁 + 𝜓1(𝜀2 𝜓2) … 𝜓 𝑁 + 𝜓1 𝜓2 … (𝜀 𝑁 𝜓 𝑁 )
= ෍
𝑖
𝑁
𝜀𝑖 𝜓1 𝜓2 … 𝜓 𝑁 = (෍
𝑖
𝑁
𝜀𝑖)Ψ 𝐻𝑃
If one-electron wave functions are normal, then Hartree Product
wave function is normal.
|Ψ 𝐻𝑃|2
= |𝜓1|2
|𝜓2|2
… |𝜓 𝑁|2
10/8/2019 46
Hatree Product Hamiltonian
In the one-electron Hatree Hamiltonian electron electron
repulsion is considered as an average as the following:
ℎ𝑖 = −
1
2
𝛻𝑖
2
− ෍
𝑘=1
𝑀
𝑍 𝑘
𝑟𝑖𝑘
+ ෍
𝑖≠𝑗
න
𝜌𝑗
𝑟𝑖𝑗
𝑑𝑟
The last term measures interaction potential between all other
electrons (except i) that occupy the same orbital. 𝜌𝑗 represents
the electron density of electron j. Second and third terms on the
r.h.s. of the equation are essentially the same but electron is
treated as a wave so its probability density is integrated over
space but nucleus is treated as a point charge.
10/8/2019 47
Hatree Product Hamiltonian
To calculate energy form Hartree Hamiltonian one should be
aware that electron-electron repulsion for each pair of electrons
is two times considered in the Hartree Hamiltonian. So, energy
is:
𝐸 = ෍
𝑖
𝜀𝑖 −
1
2
෍
𝑖≠𝑗
ඵ
|𝜓𝑖|2
|𝜓𝑗|2
𝑟𝑖𝑗
𝑑𝑟𝑖 𝑑𝑟𝑗
Hartree approach has a major limitation!
It does not consider a major property of electrons; the electron
SPIN
10/8/2019 48
Electron Spin
• Spin is a fundamental property of matter that
is NOT conceivable for human being based on
his/her daily experiences.
• We are living in a 3D world so all we need is 3
numbers, corresponding to 3 dimensions of
space to describe any object around us.
• Spin is an additional index that is necessary for
describing electrons in an atom or molecule.
10/8/2019 49
Slater Determinants
A real wave function must include the effect of spin so must be
antisymmetric that is the wave function must change sign if a
pair of electrons are replaced between two one-electron
orbitals. A mathematical trick to have such wave function is using
a matrix. In a matrix changing any two rows (or columns ) change
the sign of the matrix.
10/8/2019 50
Slater Determinants
A Slater determinant can be abbreviated for spin-orbit spatial
products, 𝜒 𝑁(𝑁). Furthermore, in SD all electrons are permitted
to be in all orbitals. This is a very important property that
satisfies “indistinguishability” principle.
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Quantum Mechanical Exchange
Electron spin has a unique effect on the energy
of a multi-electron system. Since electrons with
same spin exclude each other, the energy of a
molecular system decreases proportional to the
degree of electron exclusion, i.e. electrons with
same spin do not meet each other and do not
repel each other!
This unique property can be calculated based on
a Slater determinant.
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Hartree-Fock Approach
• Hartree was the first man who used selfconsistent
field approach to reduce the energy
of his wave function
• Fock for the first time used Hartree’s SCF
method for minimizing the energy of a wave
function based on Slater determinant.
• Roothaan years later introduced matrix
algebra necessary for HF calculations using a
“basis set” representation for MOs.
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Electron Correlation
We saw that exchange decreases energy of a
molecule by keeping electrons with the same
spin away. Electron correlation is another
phenomenon that decreases the energy of a
molecule by keeping electrons with the different
spin as far as possible. In fact, electron motions
in a molecular system is always synchronized to
some degrees to keep the energy as low as
possible! Nevertheless, HF approach did not
provide us a way to evaluate this energy.
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Correlated Mthods
HF method is most accurate single-determinant approach. We
cannot lower the energy of a molecule by a single determinant
more that HF approach. But what if we use more than a single
determinant to construct our wave function?
In fact wave function can be made based on linear combination
of different wave functions as the following:
Ψ = 𝑐0Ψ 𝐻𝐹 + 𝑐1Ψ1 + 𝑐2Ψ2 + ⋯
Coefficients reflect the weight of every wave function.
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Dynamic vs. Static Correlation
Empirical studies suggest that weight of HF wave function is
usually the most in final wave function. In such cases the
electron correlation originates from so called Dynamic
Correlation. Dynamic correlation is result of instantaneous
electron-electron interactions.
However, in some cases several wave functions have exactly the
same or very close weights in the final wave function. In such
cases with “degenerate” or “near degenerate” energy levels we
are dealing with a so-called “multi-reference” system. The
correlation in such systems is of static correlation type. Although,
there is no clear distinction between the mechanism of these
two types of correlation, conventionally this distinction is made.
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Static Correlation
An example
of systems with
static correlation
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CASSCF and Full CI
• Complete Active Space-SCF is a method for considering static
correlation to a great extent by considering a limited number
of orbitals called “active space”.
• A CASSCF approach uses HF wave function and is denoted as
CAS(m,n) were m and n are number of electrons and orbitals
in the active space.
• Each single wave function that in CAS approach is used is
called a “Configuration State Function” and number of all CSFs
for a typical CAS(m,n) is as the following:
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CASSCF and Full CI
• A CASSCF for all orbitals of a system is called Full CI.
• Full CI approach is the most accurate approach for solving
non-Born-Oppenheimer non-relativistic time-independent
Schrödinger equation!
• Unfortunately, full CI is so EXPENSIVE that only is doable for
smallest molecules.
• This because the number of CSFs increase unbelievably by
increasing the number of electrons and available orbitals that
is the basis function.
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More Correlated Methods
• CASSCF/Full CI are not the only correlated methods. There are
several methods based on Perturbation theory. Here we do
not talk about those approaches but to be familiar with them
it is useful to know that their accuracy and unfortunately
computational cost increases as the following:
HF < MP2 ~ MP3 ~ CCD < CISD < MP4SDQ ~ QCISD ~
CCSD < MP4SDTQ (MP4) < QCISD(T) ~ CCSD(T)
• Number of basis functions with computational cost of HF has
M4 relationship; computational cost of CCSD(T) increases by
M7.
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Density Functional Theory
• DFT is an alternative approach based on the
Kohn-Sham theorem.
𝐹 𝜌 𝑟 = 𝐸
• Unfortunately, the exact functional is not known
to us! Nevertheless, many functional are
suggested. All these functionals are empirically
corrected by comparison with high-level ab initio
approaches or based on experimental data.
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Density Functional Theory
• Although the exact functional is not known,
efficiency of DFT attracted many researchers.
Therefore, DFT approaches are in fact the
most widely used approaches worldwide.
• There are four different types of functionals:
LDA, GGA, Hybrid-GGA and meta-Hybrid-GGA.
• Some of popular functionals of each type are
LSDA, BLYP, B3LYP and M062X.
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(Hopefully) The
End!
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