Introduction to Computational Quantum Chemistry
Introduction to Computational Chemistry
(Prepared by Radek Marek Research Group)
Lesson 01 - Introduction to Computational Chemistry 1
Reference
Essentials of Computational Chemistry, Theories and Models
by Christopher J. Cramer
(Prepared by Radek Marek Research Group)
Lesson 01 - Introduction to Computational Chemistry 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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Lesson 01 - Introduction to Computational Chemistry 3
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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Lesson 01 - Introduction to Computational Chemistry 4
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’s Rule
which laws you know?
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Lesson 01 - Introduction to Computational Chemistry 5
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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Lesson 01 - Introduction to Computational Chemistry 6
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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Lesson 01 - Introduction to Computational Chemistry 7
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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Lesson 01 - Introduction to Computational Chemistry 8
the core: 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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Lesson 01 - Introduction to Computational Chemistry 9
The Schrödinger Equation
Time dependent non-relativistic Schrödinger equation
i
dt
ψ(r, t) =
− 2
2µ
2
+ V (r, t) ψ(r, t)
Time independent non-relativistic Schrödinger equation
Eψ(r) =
− 2
2µ
2
+ V (r) ψ(r)
(Prepared by Radek Marek Research Group)
Lesson 01 - Introduction to Computational Chemistry 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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Lesson 01 - Introduction to Computational Chemistry 11
Potential Energy (Hyper)-Surface
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Lesson 01 - Introduction to Computational Chemistry 12
Potential Energy (Hyper)-Surface
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Lesson 01 - Introduction to Computational Chemistry 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 transnational 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
∂E2
∂2x
> 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
∂E2
∂2x
, 0
then the stationary point is an nth
order saddle point
(Prepared by Radek Marek Research Group)
Lesson 01 - Introduction to Computational Chemistry 14
PES for Diatomics
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 inter-atomic separation
re, is the equilibrium distance
(Prepared by Radek Marek Research Group)
Lesson 01 - Introduction to Computational Chemistry 15
PES for Diatomics
Lenard-Jones Potential:
VLJ (r) = 4ε (
σ
r
)12
− (
σ
r
)6
ε, is the depth of potential well
σ, controls the width of potential
r, is the inter-atomic separation
(Prepared by Radek Marek Research Group)
Lesson 01 - Introduction to Computational Chemistry 16
PES for Polyatomics
on a 2-dimensional surface, only a 2D PES can be
projected, but for a reaction we conventionally depict react
coordinates which is the minimum energy path of the
reaction. This may involve rearrangement of several
degrees of freedom.
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Lesson 01 - Introduction to Computational Chemistry 17
Reaction Coordinates
a reaction coordinates can be roughly presented as the
following.
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Lesson 01 - Introduction to Computational Chemistry 18
How a Software Optimized a Molecule
Software is NOT that SMART! Watch Out!
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Lesson 01 - Introduction to Computational Chemistry 19
Digging Schrödinger Equation
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Lesson 01 - Introduction to Computational Chemistry 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
−
i
∂Ψ(x, t)
∂t
= −
2
2m
∂2Ψ(x, t)
∂x2
+ V (x, t)Ψ(x, t)
(Prepared by Radek Marek Research Group)
Lesson 01 - Introduction to Computational Chemistry 21
Time-Independent Schrödinger Equation
if potential energy does not change by time then V=V(x)
and then
−
i
∂Ψ(x, t)
∂t
= −
2
2m
∂2Ψ(x, t)
∂x2
+ V (x, t)Ψ(x, t)
furthermore Ψ(x,t) can be written as product of two
functions: Ψ(x,t) = ψ(x) f(t)
now considering partial derivatives one can write
∂Ψ(x, t)
∂t
=
df(t)
dt
ψ(x),
∂2Ψ(x, t)
∂x2
=
∂2ψ(x)
∂x2
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Lesson 01 - Introduction to Computational Chemistry 22
Time-Independent Schrödinger Equation
by substituting 2 in 1 we have
−
i
df(t)
dt
Ψ(x) = −
2
2m
f(t)
∂2Ψ(x, t)
∂x2
+ V (x)f(t)Ψ(x)
−
i
1
f(t)
df(t)
dt
= −
2
2m
1
ψ(x)
d2Ψ(x)
dx2
+ V (x)Ψ(x)
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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Lesson 01 - Introduction to Computational Chemistry 23
Time-Independent Schrödinger Equation
the left hand side is not of our interest but the right hand
side of equation 4 is the the time independent Schrödinger
Equation for a sone particle, one dimensional system
−
i
d2ψ(x)
dx2
+ V (x)ψ(x) = Eψ(x) ˆHψ(x) = Eψ(x)
(Prepared by Radek Marek Research Group)
Lesson 01 - Introduction to Computational Chemistry 24
Acceptable Wave Function
a wavefunction ψ is called well behaved (or acceptable) if
1. is Continues all over space
2. it’s first derivative is Continues over space
3. is Single Valued
4. is Quadratically Integrable, i.e. ψ ψ d over space exist
since we do not know what ψ is, any function that satisfies
these conditions is a Suitable First Guess for ψ.
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Lesson 01 - Introduction to Computational Chemistry 25
Molecular Hamiltonian
the Hamiltonian operator of a molecule, neglecting
relativistic effects and spin-orbit can be written as the
following
ˆH = −
2 α
1
mα
2
α −
2me
ß
2
i
+
α β>α
ZαZβe2
rαβ
−
α i
Zαe2
riα
+
i j>i
e2
rij
(Prepared by Radek Marek Research Group)
Lesson 01 - Introduction to Computational Chemistry 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
ˆH = −
2me i
2
i
−
α i
Zαe2
riα
+
i j>i
e2
rij
+
α β>α
ZαZβe2
rαβ
(Prepared by Radek Marek Research Group)
Lesson 01 - Introduction to Computational Chemistry 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)1018 J = 627.5095 kcal.mol−1
Length: a0 = 5.2917721092(17)10−11m = 0.52917721092
Å
Charge: e = 1.602176565(35)10−19 C
Mass: me = 9.10938291(40)10−31 kg = 1 amu (Atomic
Mass Unit)
(Prepared by Radek Marek Research Group)
Lesson 01 - Introduction to Computational Chemistry 28
Schrödinger Equation; A Differential Equation
Schrödinger Equation is a differential equation. This
means that Schrödinger Equation has more than one
acceptable eigenfunctions, , for one particular problem, i.e.
molecule. Each 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:
ψiψjdτ = δij =
1, if i = j,
0, if i = j.
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Lesson 01 - Introduction to Computational Chemistry 29
Schrödinger Equation; A Differential Equation
one can propose a way for evaluation of Energy associated
with a wavefunction. We knew ˆHψi = Eψi; multiplying this
equation by ψj and integrating both sides gives:
ψiHψjdτ = ψiEψjdτ
since E is a constant, we have:
ψiHψjdτ = Eiδij
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Lesson 01 - Introduction to Computational Chemistry 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
the solution:
The Variational Principle
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Lesson 01 - Introduction to Computational Chemistry 31
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, Φ, that is:
Φ =
i
ciψi
unfortunately yet we don’t know anything about ψi and
ofcourse Ci!
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Lesson 01 - Introduction to Computational Chemistry 32
The Variational Principle
fortunately, normality of Φ imposes a constraint on the
coefficients.
Φ2
dτ = 1 =
i
ciψidτ
j
cjψjdτ =
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Lesson 01 - Introduction to Computational Chemistry 33
The Variational Principle
for measuring the Energy associated with wave function Φ,
one can write:
ΦHΦdτ =
i
ciψi H
j
cjψj dτ =
ij
cicj ψiHψjdτ =
ij
cicjEδij =
i
c2
i E
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Lesson 01 - Introduction to Computational Chemistry 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 wavefunction. Therefore, energy
obtained form any arbitrary wave function is always higher
than that of exact wave function.
ΨHψdτ
Φ2dτ
≥ 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
wavefunction.
(Prepared by Radek Marek Research Group)
Lesson 01 - Introduction to Computational Chemistry 35
Ab Initio 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)] = Eelec
unfortunately, because the ideal functional is not known,
DFT methods are parameterized to be as consistent as
possible with experiments.
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Lesson 01 - Introduction to Computational Chemistry 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.
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Lesson 01 - Introduction to Computational Chemistry 37
STO and GTO
atomic orbitals are basis functions that have this property and
come in two forms: Slater type orbitals (STO) and Gaussian type
orbitals (GTO)
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Lesson 01 - Introduction to Computational Chemistry 38
STO and GTO; Mathematical Form
an STO has the following form:
s(r) = Ne−ζr
s
a GTO has the following form:
g(r) = Ne−ζr2
g
g(r) = x2
y2
zc
Ne−ζr2
g
Ns/Ng is the normalization constant and ζ is orbital
exponent. xa yb zc is used for defining orbital type for
example for a p orbital a = 1, b = c = 0.
(Prepared by Radek Marek Research Group)
Lesson 01 - Introduction to Computational Chemistry 39
Gaussian Basis Sets
every GTO is consisted of one or more Primitive GTOs
χ(CGTO) =
k
i
aiχi(PGTO)
the degree of contraction is the number of PGTOs that
enter a CGTO or Contracted GTO
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Lesson 01 - Introduction to Computational Chemistry 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)
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Lesson 01 - Introduction to Computational Chemistry 41
Basis Sets
Correlation Consistent Basis Sets s and p optimized at HF,
polarization optimized at CISD
Composition in terms of contracted and primitive basis functions for the correlation consistent basis sets
Basis Hydrogen First row elements Second row elements
Contracted Primitive Contracted Primitive Contracted Primitive
cc-pVDZ 2s1p 4s 3s2p1d 9s4p 4s3p2d 12s8p
cc-pVTZ 3s1p1d 5s 4s3p2d1f 10s5p 4s3p2d 12s8p
cc-pVQZ 4s3p2d1f 6s 5s4p3d2f1g 12s6p 4s3p2d 12s8p
cc-pV5Z 5s4p3d2f1g 8s 6s5p4d3f2g1h 14s8p 4s3p2d 12s8p
cc-pV6Z 6s5p4d3f2g1h 10s 7s6p5d4f3g2h1i 16s10p 4s3p2d 12s8p
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Lesson 01 - Introduction to Computational Chemistry 42
How a Basis Set Looks Like?
Anatomy of Popple basis set (left) and Duning Basis Set (right)
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Lesson 01 - Introduction to Computational Chemistry 43
Introduction to Ab Initio Methods
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Lesson 01 - Introduction to Computational Chemistry 44
Hartree Product WaveFunction
neglecting electron-electron repulsion one can write a molecular
Hamiltonian in terms of one-electron Hamiltonians.
H =
N
i
hi
hi = −
1
2
2
i −
M
k=1
Zk
rik
all eigenfunctions of one-electron Hamiltonian must satisfy the oneelectron
Schrödinger equation.
hiψi = εiψi
if Hamiltonian is separable then many-electron eigenfunction should be
a product of one-electron eigenfunctions:
ΨHP = ψ1ψ2...ψN
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Lesson 01 - Introduction to Computational Chemistry 45
Hartree Product WaveFunction
then the energy of "Hartree Product" wavefunction is:
HΨHP = Hψ1ψ2...ψN =
N
i
hiψ1ψ2...ψN =
(hiψ1)ψ2...ψN + ψ1(h2ψ2)...ψN + (hN ψN ) =
(εiψ1)ψ2...ψN + ψ1(ε2ψ2)...ψN + (εN ψN )
=
N
i
εiψ1ψ2...ψN =
N
i
εi ΨH P
if one-electron wave functions are normal, then Hartree Product wave
function is normal.
|ΨHP |2
= |ψ2|2
|ψ2|2
...|ψN |2
(Prepared by Radek Marek Research Group)
Lesson 01 - Introduction to Computational Chemistry 46
Hartree Product Hamiltonian
in the one-electron Hatree Hamiltonian electron electron repulsion is
considered as an average as the following:
hi = −
1
2
2
i −
M
k=1
Zk
rik
+
i=j
ρj
rij
dr
the last term measures interaction potential between all other electrons
(except i) that occupy the same orbital. j 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.
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Lesson 01 - Introduction to Computational Chemistry 47
Hartree Product Hamiltonian
to calculate energy form Hartree Hamiltonian on 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
|ψi|2
|]ψj|2
rij
dridrj
Hartree approach has a major limitation! Its does not consider a major
property of electrons; the electron
SPIN
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Lesson 01 - Introduction to Computational Chemistry 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.
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Lesson 01 - Introduction to Computational Chemistry 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.
3
ΨSD =
1
√
2
ψa(1)α(1) ψb(1)α(1)
ψa(2)α(2) ψb(2)α(2)
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Lesson 01 - Introduction to Computational Chemistry 50
Slater Determinants
a Slater determinant can be abbreviated for spin-orbit spatial
products, χN (N). Furthermore, in SD all electrons are permitted
to be in all orbitals. This is a very important property that
satisfies “indistinguishability” principle.
ΨSD =
1
√
N!
χ1(1) χ2(1) .. χN (1)
χ1(2) χ2(2) .. χN (2)
: : : :
χ1(N) χ2(N) .. χN (N)
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Lesson 01 - Introduction to Computational Chemistry 51
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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Lesson 01 - Introduction to Computational Chemistry 52
Hartree-Fock Approach
Hartree was the first man who used self- consistent 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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Lesson 01 - Introduction to Computational Chemistry 53
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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Lesson 01 - Introduction to Computational Chemistry 54
Correlated Methods
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
coefficients reflect the weight of every wave function.
Ψ = c0ΨHF + c1Ψ1 + c2Ψ2 + ...
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Lesson 01 - Introduction to Computational Chemistry 55
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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Lesson 01 - Introduction to Computational Chemistry 56
Static Correlation
an example of systems with static correlation
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Lesson 01 - Introduction to Computational Chemistry 57
CASSCF and Full CI
Complete Active Space-SCF (CASSCF) 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:
N = n!(n+1)! m
2 ! m
2 +1 ! n− m
2 ! n− m
2 +1 !
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Lesson 01 - Introduction to Computational Chemistry 58
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.
(Prepared by Radek Marek Research Group)
Lesson 01 - Introduction to Computational Chemistry 59
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
.
(Prepared by Radek Marek Research Group)
Lesson 01 - Introduction to Computational Chemistry 60
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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Lesson 01 - Introduction to Computational Chemistry 61
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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Lesson 01 - Introduction to Computational Chemistry 62
END
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Lesson 01 - Introduction to Computational Chemistry 63