nnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnn

1. 3/9/2024
1

2. THEORETICAL BACKGROUND OF COMPUTATIONAL
CHEMISTRY
Overview of computational chemistry
 Computational chemistry is a branch of chemistry that uses computer
simulation to assist in solving problems in chemistry.
 The term computational chemistry is generally used when a
mathematical method is sufficiently well developed that it can be
automated for implementation on a computer.
 It uses methods of theoretical chemistry, incorporated with efficient
computer programs, to calculate the structures and properties of
molecules.
 The methods used to cover both static and dynamic situations.
 Computational chemistry methods range from very approximate to
highly accurate; the latter is usually feasible for small systems only.
3/9/2024
2

3.  Ab-initio methods are based entirely on quantum mechanics and
basic physical constants.
 Other methods are called semi-empirical because they use
additional empirical parameters.
 Both ab-initio and semi-empirical approaches involve
approximations. These range from simplified forms of the first-
principles equations that are easier or faster to solve, to
approximations limiting the size of the system (for example, periodic
boundary conditions), to fundamental approximations to the
underlying equations that are required to achieve any solution to
them at all.
3/9/2024
3

4.  For example, most ab-initio calculations make the Born–
Oppenheimer approximation, which greatly simplifies the underlying
Schrödinger equation by assuming that the nuclei remain in place
during the calculation.
 In principle, ab-initio methods eventually join to the exact solution of
the underlying equations as the number of approximations is
reduced.
 In practice, however, it is impossible to eliminate all approximations,
and residual error inevitably remains.
 The goal of computational chemistry is to minimize this residual error
while keeping the calculations tractable.
3/9/2024
4

5. Whereas experimental chemistry focuses on the synthesis of
molecules and materials or on the careful measurement of observable
properties, theoretical and computational chemistry aims to explain
chemical phenomena and to make predictions about experiments and
observations.
 The development of theoretical and computational chemistry are
inseparably linked with major experimental discoveries.
 In fact, the strong interplay between theoretical and experimental
chemistry led to the emergence of chemistry as a rigorous branch of
the physical sciences.
 In the broadest sense, theoretical and computational chemistry may
be thought of as being any aspect of chemistry that does not involve
direct experimentation or observation of chemical phenomena.
3/9/2024
5

6.  The term theoretical chemistry may be defined as the mathematical
description of chemistry.
 Theoretical chemistry provides the theoretical framework for making
predictions about experimental observations and for explaining
phenomena involving atoms, molecules, and materials.
 Computational chemistry uses computers to apply the methods of
theoretical chemistry to a broad range of topics in chemistry.
 Computational chemistry involves molecular modeling based on
theory.
 Starting from quantum mechanics, all chemical phenomena can be
calculated from theory but solving the exact equations directly is
infeasible except for very small systems.
3/9/2024
6

7.  To overcome this obstacle, many methods have been developed for
approximating the difficult equations of quantum mechanics, so that
they can be solved for molecular systems.
 The most promising of these methods rely on sophisticated
combinations
of two general strategies:
(i) use of compact models that capture the key many-particle effects
by construction and
(ii) (ii) efficient stochastic sampling of many dimensions. There are
many variations on these methods, which collectively make up the
toolbox of computational chemistry ( see below Figure ).
3/9/2024
7

8. Figure 1. Inter disciplinary field of computational chemistry [74].
 The primary objective of theoretical chemistry is to provide a
coherent account for the structure and properties of atomic and
molecular systems.
 Techniques adapted from mathematics and theoretical physics are
applied in attempts to explain and correlate the structures and
dynamics of chemical systems.
 In view of the immense complexity of chemical systems, theoretical
chemistry, in contrast to theoretical physics, generally uses more
3/9/2024
8

9.  Computational chemistry has become a useful way to investigate
materials that are too difficult to find or too expensive to purchase.
 It also helps chemists make predictions before running the actual
experiments so that they can be better prepared for making
observations.
 The Schrödinger equation is the basis for most of the computational
chemistry scientist’s use.
 This is because the Schrödinger equation models the atoms and
molecules with mathematics.
 For instance, you can calculate: electronic structure determinations,
geometry optimizations, frequency calculations, transition structures,
protein calculations, i.e. docking, electron and charge distributions,
potential energy surfaces (PES), rate constants for chemical reactions
(kinetics), thermodynamic calculations the heat of reactions, the energy
3/9/2024
9

10. Other major components include molecular dynamics, statistical
thermodynamics, and theories of electrolyte solutions, reaction
networks, polymerization, catalysis, molecular magnetism, and
spectroscopy ; unimolecular rate theory and met stable states;
condensed-phase and macromolecular aspects of dynamics.
They are used to identifying correlations between chemical structures
and properties. Computational approaches to help in the efficient
synthesis of compounds and to design molecules that interact in
specific ways with other molecules in drug design and catalysis.
3/9/2024
10

11.  Computational chemistry is a sophisticated combination of
chemistry, mathematics, physics, and computer science.
 Chemistry defines the question that rise in any chemical reaction,
physics defines the laws that are obeyed by the chemical system;
mathematics formulates a numerical representation of the problems
and computer science solves a mathematical model that provides
numbers that condense physical significance.
 Computational chemistry attempts to solve the Schrodinger equation
and its calculations verify the solution of the Schrodinger equation
provides quantitatively reproduced and experimentally observed
features of simple systems and approximate solutions for larger
systems.
3/9/2024
11

12. 3/9/2024
12
 There are two main branches of computational chemistry. These are
classical mechanics and quantum mechanics.
 Molecules are sufficiently small objects that, strictly speaking, the
laws of quantum mechanics must be used to describe them.
 However, under the right conditions, it is still sometimes useful and
much faster computationally to approximate the molecule using
classical mechanics. This approach is sometimes the ``molecular
mechanics'' (MM) or ``force-field'' method .
 All molecular mechanics methods are practical in the sense that the
parameters in the model are obtained by fitting to known
experimental data.
 Quantum mechanical methods usually classified either as ab
initio (hf) or semi-empirical.

13. 3.1.1. Schrödinger equation
3/9/2024
13
 In physics, specifically quantum mechanics, the Schrödinger equation,
formulated in 1926 by Austrian physicist Erwin Schrödinger, is an
equation that describes how the quantum state of a physical system
changes in time.
 It is a central to quantum mechanics. In the standard interpretation of
quantum mechanics, the quantum state is a wave function and the most
complete description that can be given to a physical system.
 Solutions to Schrödinger's equation describe molecular, atomic,
subatomic and the whole universe .
 The most general form is the time-dependent Schrödinger equation,
which gives a description of a system evolving with time.
 For systems in a stationary state, the time-independent Schrödinger
equation is sufficient.

14. 3/9/2024
14
 Approximate solutions to the time-independent Schrödinger equation are
commonly used to calculate the energy levels and other properties of
atoms and molecules.
 The Schrödinger equation describes time in a way that is inconvenient for
relativistic theories, a problem which is not as simple in matrix mechanics
and completely absent in the path integral formulation.
 The Schrödinger equation takes several different forms, depending on the
physical situation. This section presents the equation for the general
quantum system.
iħ
𝛛
𝛛𝐭
𝚿=ĤΨ, where 𝚿 is the wave function for different arrangements of the
system at dissimilar times ,iħ
𝛛
𝛛𝐭
is the energy operator (i is the imaginary unit
and ħ is the reduced Planck constant) and Ĥ is the Hamiltonian operator.
For a single particle with potential energy V in position space, the Schrödinger
equation receipts the form .

15. 3/9/2024
15
 iħ
𝛛
𝛛𝐭
𝚿 𝐫, 𝐭 = ĤΨ = (-
ħ𝟐
𝟐𝒎
∇2 + V(r) ) Ψ(r,t) = -
ħ𝟐
𝟐𝒎
∇2Ψ(r,t) + V(r)𝚿(r,t)
 Where -
ħ𝟐
𝟐𝒎
∇2 is the kinetic energy operator, m is the mass of the
particle, ∇2 is the Laplace operator given as
𝜵𝟐 =
𝝏𝟐
𝝏𝒙𝟐 +
𝝏𝟐
𝛛𝒚𝟐 +
𝝏𝟐
𝝏𝒛𝟐, where x, y and z are the Cartesian coordinates of
space,
 V(r) is the time independent potential energy at the position r, Ψ(r, t)
is the probability amplitude for the particle to be found at position r at
time t and
Ĥ = -
ħ𝟐
𝟐𝒎
∇2 + V(r) is the Hamiltonian operator for a single particle in a
potential.

16. 3/9/2024
16
 Time independent or stationary equation is also for a single particle
takes the form : E𝚿 𝐫 = (-
ħ𝟐
𝟐𝒎
∇2 + V(r))𝚿 𝐫 this equation describes
the standing solutions of the time dependent equation, which are the
states with definite energy.
3.1.2. Historical background and development
 Following Max Planck's quantization of light, Albert Einstein
interpreted Planck's quantum to be photons, particles of light, and
proposed that the energy of a photon is proportional to its frequency,
one of the first signs of wave–particle duality.
 Since energy and momentum are related in the same way as
frequency and wave number in special relativity, it followed that the
momentum p of a photon is proportional to its wavenumber 𝐯.
𝒉

17. 3/9/2024
17
 Louis de Broglie imagined that this is true for all particles, even particles
such as electrons.
 Assuming that the waves travel roughly along classical paths, he
showed that they form standing waves for certain discrete frequencies.
These parallel to discrete energy levels, which reproduced the old
quantum condition.
 Following up on these ideas, Schrödinger decided to find a proper wave
equation for the electron. He was guided by William R.
 Hamilton's similarity between mechanics and optics, determined in the
observation that the zero-wavelength limit of optics bring to mind a
mechanical system, the lines of light rays become sharp pathways
which obey Fermat's principle, an equivalent of the principle of least
action.
 A modern version of his reasoning is reproduced in the next section.

18. 3/9/2024
18
 iħ
𝛛
𝛛𝐭
𝚿 𝐗, 𝐭 = -
ħ𝟐
𝟐𝒎
∇2𝚿 𝐗, 𝐭 + V(X)𝚿(𝐱, 𝐭)
 Using this equation, Schrödinger calculated the hydrogen spectral
series by giving a hydrogen atoms electron as a wave Ψ(x, t), moving
in a potential well V, created by the proton.
 This calculation accurately reproduced the energy levels of the Bohr
model. However, by that time, Arnold Sommerfeld had developed the
Bohr model with relativistic corrections.
 Schrödinger used the relativistic energy momentum relation to find
what is now known as the Klein–Gordon equation in a coulomb
potential.
(E +
𝐞𝟐
𝒓
) 2𝚿 𝐗 = - ∇2𝚿 𝐗 + m2𝚿 𝐗

19. 3/9/2024
19
 He found the standing waves of this relativistic equation, but the
relativistic corrections not agreed with Sommerfeld's formula.
 Discouraged, he put away his calculations and out-of-the-way
himself in an isolated mountain small house with a lover .
 While at the small house, Schrödinger decided that his earlier non-
relativistic calculations were not enough to publish, and decided to
stop the problem of relativistic corrections for the upcoming.
 He set together his wave equation and the spectral analysis of
hydrogen in a paper in.
 The paper was strongly certified by Einstein, who saw the matter-
waves as a spontaneous description of nature, as opposed to
Heisenberg's matrix mechanics, which he considered exceedingly

20. 3/9/2024
20
 The Schrödinger equation details the behavior of 𝚿 but says nothing of
its nature. Schrödinger tried to interpret it as a charge density in his
fourth paper, but he was unsuccessful.
 In 1926, just a few days after Schrödinger's fourth and final paper was
published, Max Born successfully interpreted 𝚿 as probability
amplitude.
 Schrödinger, however, always opposed a statistical or probabilistic
method, with its related discontinuities, much like Einstein, who
believed that quantum mechanics was a statistical approximation to a
fundamental deterministic theory and never prepared to accept with the
Copenhagen interpretation.

21. 3.1.3. Experimental derivation and expressing the
wave function as a complex plane wave
3/9/2024
21
 Schrödinger's equation can be derived by the Assumptions in the
following short experimental way. It should be well-known that
Schrödinger's wave equation was a result of the imaginative
mathematical awareness of Erwin Schrödinger, and cannot be
derived independently.
 The total energy E of a particle is Et = K.E + V =
𝑷𝟐
𝟐𝑴
+ 𝑽
 This is the classical expression for a particle with mass m where the
total energy Et is the sum of the kinetic energy K.E, and the potential
energy V. P and M are respectively the momentum and the mass of
the particle.

22. 3/9/2024
22
 The de Broglie hypothesis of 1924, which states that any
particle can be related with a wave and that the momentum p
of the particle is related to the wavelength λ (or wave
number,𝑣) of such a wave by:
 P =
𝒉
𝛌
= ħ𝐯
 The three assumptions above agree one to derive the
equation for plane waves only.
 To conclude that it is true in general requires the superposition
principle, and thus, one must separately postulate that the
Schrödinger equation is linear.
 Schrodinger’s idea was to express the phase of a plane wave
as a complex phase factor.

23. 3/9/2024
23
 𝚿 𝐗, 𝐭 = Aei(k.x-ωt) and to realize that since
𝛛
𝛛𝐭
𝚿 = -iωΨ, then E𝚿 =
ħω 𝚿 = 𝐢ħ
𝛛
𝛛𝐭
𝚿 and similarly since
𝛛
𝛛𝐫
𝚿 = ikxΨ and
𝝏𝟐
𝝏𝒙𝟐 𝚿 =
−𝒌𝟐
𝐱𝚿 , we find:
 P2
X𝚿 = ( ħ𝐊 X)2𝚿 = − ħ2 𝝏𝟐
𝝏𝒙𝟐Ψ , so that again for a plane wave, he
obtained:
 P2Ψ = (p2
x + p2
y + p2
z)𝚿 = - ħ2(
𝝏𝟐
𝝏𝒙𝟐 +
𝝏𝟐
𝛛𝒚𝟐 +
𝝏𝟐
𝝏𝒛𝟐) 𝚿 = - ħ2∇2Ψ, and by
inserting these expressions for the energy and momentum into the
classical formula we started with, we get Schrodinger’s famous
equation, for a single particle in the three dimensional case in the
presence of a potential V.

24. 3.1.4. Time independent equation
3/9/2024
24
 This is the equation for the standing waves, the Eigen value equation
for Ĥ.
 The time independent Schrödinger equation may be gained from the
time dependent description by assuming unimportant time
dependence of the wave function of the form Ψ(x,t) = Ψ(x)𝒆−𝒊𝑬𝒕/ħ .
 This is possible only if the Hamiltonian is not a clear function of time,
as otherwise the equation is not separable into its spatial and
temporal parts.
 The operator iħ
𝛛
𝛛𝐭
can then be replaced by E. In mental form, for a
general quantum system, it is written by:
 ĤΨ = E Ψ, for a particle in one dimension,

25. 3/9/2024
25
 E Ψ = -
ħ𝟐
𝟐𝒎
𝝏𝟐𝚿
𝝏𝑿𝟐 = V(X) Ψ, But there is a further restriction the solution
must not grow at infinity, so that it has either a finite L2 -norm (if it is a
bound state) or a slowly diverging norm (if it is part of a continuum):
 𝚿 2 = ∫ 𝚿(𝐗) 2dX, if there is no potential, the equation reads:
 - E Ψ =
ħ𝟐
𝟐𝒎
𝝏𝟐𝚿
𝝏𝑿𝟐, which has oscillatory solutions for E > 0, the Cn are
arbitrary constants:
 Ψ E(X) = C1ei 𝟐𝒎𝐄/ħ𝟐
X + C2e-i 𝟐𝒎𝐄/ħ𝟐 X, and exponential solutions for
E < 0
 Ψ - 𝐄 (X) = C1e 𝟐𝒎𝐄/ħ𝟐 X + C2e- 𝟐𝒎𝐄/ħ𝟐 X

26. 3/9/2024
26
 The exponentially growing solutions have an infinite norm, and are
not physical.
 They are not allowed in a finite volume with periodic or fixed
boundary conditions. For a constant potential V the solution is
oscillatory for E > V and exponential for E < V, corresponding to
energies which are allowed or rejected in classical mechanics.
 Oscillatory solutions have a classically allowed energy and
correspond to actual classical motions, while the exponential
solutions have a disallowed energy and describe a small amount of
quantum bleeding into the classically disallowed region, to quantum
tunneling.
 If the potential V grows at infinity, the motion is classically confined to
a finite region, which means that in quantum mechanics every
solution becomes an exponential far enough away.

27. 3.1.5. Nonlinear equation and local conservation of
probability
3/9/2024
27
 The nonlinear Schrödinger equation is the partial differential equation
(in dimensionless form)[63]
 i∂tΨ=-
𝟏
𝟐
𝛛𝟐
XΨ+K 𝚿 2Ψ
 For the complex field Ψ(x,t), this equation arises from the
Hamiltonian[63]
 H = ∫ dx[
𝟏
𝟐 𝛛𝐱𝚿 2 +
𝑲
𝟐
𝚿 4], with the poisson brackets
 {Ψ (X), Ψ (Y)} = {Ψ* (X), Ψ* (Y)} =0
 {Ψ* (X), Ψ (Y)} = i∂ (X - Y). It must be noted that this is a classical
field equation. Unlike its linear counterpart, it never describes the time
evolution of a quantum state.

28. 3/9/2024
28
 The probability density of a particle is Ψ* (X, t) Ψ (X, t). The
probability flux is defined as [in units of (probability)/(area × time)]:
 j = -
ħ
𝟐𝒎
(Ψ*𝜵 Ψ - Ψ𝜵Ψ*) =
ħ
𝒎
Im(Ψ*𝜵 Ψ)
 The probability flux satisfies the continuity equation:
𝛛
𝛛𝐭
P (x.t) +𝜵. 𝒋 = 𝟎
 Where P(x.t) is the probability density measured in units of
probability/volume. This equation is the mathematical equivalent of
the probability conservation law. For a plane wave:
 Ψ (X, t) = Aei(kx-ωt)and j =(x,t) = A2ħ𝐊
𝒎

29. 3/9/2024
29
 So that not only is the probability of finding the particle the same
everywhere, but the probability flux is as expected from an object
moving at the classical velocity p/m. The reason that the Schrödinger
equation admits a probability flux is because all the hopping is local
and forward in time.
3.1.6 The Hartree–Fock method
 The origin of the Hartree–Fock method dates back to the end of the
1920s, soon after the discovery of the Schrödinger equation in 1926.
The earliest and widely used approximation was that of Hartree ,
which expresses the wave function of system as a product of one-
electron wave functions, so that the problem reduces to a one-
electron Schrodinger equation.

30. 3/9/2024
30
 Then, considerable improvement of the energy computation was
made by integrating the exchange effects with the so-called Hartree-
Fock approximation, which replaces the product of one-electron wave
functions by a linear combination of orbital’s.
 The Columbic electron-electron repulsion is implicitly taken into
account, and the average effect is included in the calculation.
 This is a variation calculation, which implies that the approximate
energies calculated are all equal to or greater than the exact energy
and the actual form of the single electronic molecular wave function
not known.
 Molecular orbital’s are the product of Hartree-Fock theory, and
Hartree-Fock is not an exact theory: it is an approximation to the
electronic Schrödinger equation.

31. 3.1.7. Density functional theory
3/9/2024
31
 The demanding improvements of the density-functional theory were
modeled by Hohenberg, Kohn and Sham that legitimized the model
spontaneously established by Thomas, Fermi and Dirac. In the
modern version of the DFT, self-consistent equations are solved for a
set of orbitals whose electron density is exactly that of the real
system.
 In practical calculations, the exchange and correlation contributions
are approximated. In the early 1990s, hybrids functional were
introduced by Becke.
 Among the hybrid functional, the most popular approximation in use
in chemistry today, with about 80% of the occurrences in the
literature, is the B3LYP (exchange of Becke and correlation of Lee-
Yang-Parr .
 For practical reasons, they replace the term Uee with Vex to indicate

32. 3/9/2024
32
 [T + Uee + Vex] Ψ el = Eel Ψ el, and the electronic energy is a
functional of the electron density profile n(r): Eel[n] = 〈Ψel⎟H⎟Ψel〉
=F[n] + ∫Vex(r)n(r)dr
 Where T = kinetic energy operator, Uee = electron- electron interaction
operator, Vex = external potential, Ψel= N – electron wave function and
H is Hamiltonian.
 F [n] = 〈Ψel⎟T⎟Ψ el〉 + 〈Ψ el⎟Uee⎟Ψ el〉 = T[n]+ Uee[n] is the so-called
universal energy functional, in the sense that it does not contain the
external potential Vex(r) and can be determined independently of R
Vex(r)n(r)dr.

33. 3.1.8. Semi empirical
3/9/2024
33
 are much faster than their ab-initio counter parts, mostly due to the
use of the zero differential overlap approximation.
 Within this calculation certain pieces of information are
approximated or completely omitted.
 Mostly the core electrons are not included in the calculation and
only a minimal basis set is used.
 The disadvantage is unpredictable results and fewer properties can
be predicted reliably.
 The commonly used semi empirical calculations are MINDO,
MNDO, MINDO/3, AM1, PM3 and SAM1

34. 3.1.9. Basis Sets (basis functions)
3/9/2024
34
 is a set of functions which are combined in a quantum chemical
calculation to create molecular orbitals and used to denote the
electronic wave functions in the HF or DFT in order to turn the partial
differential equations of the model into algebraic equations for
processing on a computer.
 In modern computational chemistry, quantum chemical calculations
are typically performed using a finite set of basis functions.
 The molecular spin- orbitals that are used in the Slater determinant
usually are expressed as a linear combination of basis set.
 It is important to choose a basis set large enough to give a good
description of the molecular wave function.
 Typically, the basis functions are centered on the atoms, and
sometimes they are called ``atomic orbitals.''

35. 3/9/2024
35
 The larger the basis set, however improves the accuracy of the
calculations by providing more variable parameters to produce a better
approximate wave function, but at the expense of increased
computational time.
 Likewise, even an exact treatment of electron correlation can give terrible
answers when paired with a very small basis set.
 Better and better results can be obtained when one increases the basis
set and improves the treatment of correlation.
 In the limit of an infinite basis set and an exact treatment of electron
correlation, the electronic Schrödinger equation would be solved exactly.
 From different types of basis set usually we use Pople basis set
developed by the late Nobel Laureate, John Pople, and popularized by
the Gaussian set of programs. Some of Pople basis sets are STO-3G
(minimal made by 3 Gaussian),6-31G (made of 6 Gaussian),6-31G* or 6-
31G(d) (for nonhydrogen atom) ,6-31G** or 6-31G(d,p) (for hydrogen

36. 3.2.0. Electronic correlation
3/9/2024
36
 Electronic correlation is the interaction between electrons in the electronic
structure 0f quantum system.
 The correlation energy is a measure of how much the movement of one
electron is influenced by the presence of all other electrons.
 Within the Hartree-fock method of quantum chemistry, the anti symmetric
wave function is approximated by a single Slater determinant.
 As a general the exact wave functions cannot expressed as single
determinants.
 The single-determinant approximation does not take into account Coulomb
correlation, leading to a total electronic energy different from the exact
solution of the non-relativistic Schrödinger equation with in the Born-
Oppenheimer approximation.
 Therefore, the Hartree-fock limit is always above this exact energy. The
difference is called the correlation energy, a term invented by Lowdin.
 The concept of the correlation energy was studied earlier by Wigner .

37. 3/9/2024
37
 A certain amount of electron correlation is already considered within
the HF approximation, found in the electron exchange term
describing the correlation between electrons with parallel spin.
 This basic correlation prevents two parallel-spin electrons from being
found at the same point in space and is often called Fermi
correlation.
 Coulomb correlation, on the other hand, describes the correlation
between the spatial position of electrons due to their Coulomb
repulsion, and is responsible for chemically important effects such as
London dispersion.
 Electron correlation is sometimes divided into dynamical and non-
dynamical (static) correlation.

38. 3/9/2024
38
 Dynamical correlation is the correlation of the movement of electrons
and is described under electron correlation dynamics and also with
the configuration interaction (CI) method.
 Static correlation is important for molecules where the ground state is
well described only with more than one (nearly-)degenerate
determinant.
 In this case the Hartree–Fock wave function (only one determinant)
is qualitatively wrong.
 The multi configurationally self-consistent field (MCSCF) method
takes account of this static correlation, but not dynamical correlation.

39. 3/9/2024
39
 To account for electron correlation there are many Post-Hartree-
fockmethods, including: CI is the most important methods for
correcting for the missing correlation.
 Starting with the Hartree–Fock wave function as the ground
determinant, one takes a linear combination of the ground and
excited determinants as the correlated wave function and optimizes
the weighting factors according to the variational principle.
 When taking all possible excited determinants one speaks of Full-CI.

40. 3/9/2024
40
 In a Full-CI wave function all electrons are fully correlated. For non-small molecules
Full-CI is much too computationally expensive.
 One truncates the CI expansion and gets well-correlated wave functions and well-
correlated energies according to the level of truncation.
 Moller-plesset perturbation theory, (MP2, MP3, MP4, etc.), Perturbation theory gives
correlated energies, but no new wave functions.
 PT is not variational. This means the calculated energy is not an upper bound for
the exact energy. It is possible to partition Møller–Plesset perturbation theory
energies via Interacting Quantum Atoms (IQA) energy partitioning (although most
commonly the correlation energy is not partitioned).
 This is an extension to the theory of atoms in molecules. IQA energy partitioning
enables one to look in detail at the correlation energy contributions from individual
atoms and atomic interactions.
 IQA correlation energy partitioning has also been shown to be possible with coupled
cluster methods.

41. 3.2.1. Electron-Nucleus Correlation
3/9/2024
41
 In probability theory, two events “A” and “B” are said independent,
that is to say, “uncorrelated”, if and only if the probability of observing
A and B is the product of the probability of observing A by the
probability of observing B: p(A ∧ B) = p(A) · p(B).
 According to the Born interpretation of quantum mechanics, the
square of the normalized wave function of a quantum system, |Ψ(r)|2=
p(r) is the probability of observing the system at configuration point r.
Putting these two elements together, we see that a Hartree product
wave function for a bipartite quantum system, Ψ H(r1, r2) = Ψ1(r1)
Ψ2(r2) is uncorrelated in the probabilistic sense, since |Ψ H(r1, r2)|2 =
|Ψ1(r1)|2 | Ψ2(r2)|2.
 The situation is somewhat less simple for a Fermionic system such as
the electrons of a molecular system. A Slater determinant type of

42. 3/9/2024
42
 ΨS(r1, r2) =
𝚿𝟏
(𝐫𝟏
)| 𝚿𝟐
𝐫𝟐
−|𝚿𝟏
(𝐫𝟐)𝚿𝟐
(𝐫𝟏)
𝟐
, which is the fermionic equivalent
of a Hartree product of distinguishable particles, has already some
built-in correlation, as in general, one has |ΨS (r1, r2)|2≠| Ψ1 (r1)|2 |
Ψ2(r2)|2.
 However, this correlation in the probabilistic sense is just Pauli spin
statistic correlation, as can be inferred from the wedge product
notation Ψ1∧Ψ2 of the same Slater determinant wave function, where
ant symmetry is built-in.
 So, in the Fermionic case, the “uncorrelated” reference is still chosen
to be the Slater determinant ansatz, and in quantum chemistry the
electronic correlation energy is defined as the difference between the
electronic full configuration interaction (E-FCI) energy and the lowest

43. 3/9/2024
43
 𝑬𝑐𝑜𝑟𝑟𝑒𝑙
𝑒𝑙
= 〈ΨE−FCI |H|ΨE−FCI 〉 − 〈Ψ HF |H|Ψ HF 〉
 Considering now a molecular system composed of a set of electrons,
with position and spin variables collectively denoted by 𝑅e and a set
of nuclei, with position and spin variables collectively denoted by 𝑅n,
the uncorrelated reference ansatz will be the Hartree product,
 ΨH(𝑅n, 𝑅e) = Ψ n(𝑅n) Ψ e(𝑅e), since both sets, electrons and nuclei
are clearly distinguishable. Then, it is natural to define the electron-
nucleus correlation energy by:
 Ecorre
EN
l = 〈ΨEN−F CI | H |ΨEN−F CI 〉−Ψ HMin〈Ψ H | H |Ψ H〉, Let us
emphasize that both the electronic, Ψe, and nuclear, Ψn, wave
functions are completely general functions of the electronic and
nuclear Hilbert spaces, respectively.

44. 3/9/2024
44
 That is to say, they are what would be called FCI-type wave functions
in a finite basis set context: Ψe fully accounts for the correlation of the
electrons in the mean field of Ψn. Conversely, Ψn fully accounts for
nuclear motion correlation in the mean field of Ψe. The fact that our
reference “uncorrelated” EN-wave function is the direct product of two
FCI-type wave functions, is not in contradiction with our claim of
generalizing the electronic correlation definition. Indeed, HF-
optimized orbital’s can also be regarded as “FCI” solutions of a one-
particle mean-field problem at every step of the self-consistent field
(SCF) process. The fundamental difference with respect to the Born-
Oppenheimer (BO) ansatz[79,80] , is that the electronic wave function
Ψe in ΨH does not depend parametrically upon the nuclear

45. 3/9/2024
45
 Algebraically, the “electronic” factor of a Born-Oppenheimer
type wave function Ψ BO (𝑅n, 𝑅e) = Ψn(𝑅n) Ψ can always be
decomposed as Ψe(𝑅n, 𝑅e) ) = 𝑖 𝚿1
𝑖
(𝑅n) ⊗𝚿2
𝑖
(𝑅e) and the
“nuclear” factor acts multiplicatively as an operator on the first
component:
 Ψn (𝑅n)[Ψe (𝑅n, 𝑅e)] = 𝑖(𝚿n(𝑅n)𝚿1
𝑖
(𝑅n)) ⊗𝚿2
𝑖
𝑅e),As a matter of fact,
we know, that in general, |Ψe (𝑅n, 𝑅e)|2≠= |Ψ1(𝑅n)|2 | Ψ2(𝑅e )|2 .
Moreover, minimizing the energy of such a BO-type wave function
would give nothing else but the EN-FCI result, EEN−F CI = 𝚿BO𝑀𝑖𝑛
〈Ψ
BO|H|Ψ BO〉, since the EN-FCI wave function can always be
expressed in this form.
 However, the electron-nucleus correlation energies calculated within