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CFM
centro de física de materiales
Daniel Sánchez Portal
Ricardo Díez Muiño
Centro de Física de Materiales
Centro Mixto CSIC-UPV/EHU
Electronic structure calculations:
Methodology and applications to nanostructures
Electron correlation methods in
Quantum Chemistry
Electronic structure calculations:
Methodology and applications to nanostructures
Lectures on Quantum Chemistry:
Tuesday March 17th: 9.45 --> 12.30
Theoretical background
Wednesday March 18th : 9.45 --> 12.30
Practical exercise
Electronic structure calculations:
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Outline
• Brief introduction
• Hartree- Fock
• Basis sets
• Configuration Interaction
• Many-body perturbation theory
• Coupled-cluster methods
Electronic structure calculations:
Methodology and applications to nanostructures
Outline
Post
Hartree
Fock
• Brief introduction
• Hartree- Fock
• Basis sets
• Configuration Interaction
• Many-body perturbation theory
• Coupled-cluster methods
Electronic structure calculations:
Methodology and applications to nanostructures
Every attempt to employ mathematical methods in the
study of chemical questions must be considered
profoundly irrational and contrary to the spirit of
chemistry. If mathematical analysis should ever hold a
prominent place in chemistry—an aberration which is
happily almost impossible–it would occasion a rapid
and widespread degeneration of that science.
Auguste Comte, 1830.
Electronic structure calculations:
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In conclusion, I would like to emphasize my belief that
the era of computing chemists, when hundreds if not
thousands of chemists will go to the computing
machine instead of the laboratory, for increasingly
many facets of chemical information, is already at
hand. There is only one obstacle, namely, that
someone must pay for the computing time.
Robert Mulliken.
Nobel Prize address, 1966.
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Quantum chemistry is quantum mechanics applied to
the electrons in atoms and molecules.
Used to determine:
1.- Structure of the molecule (bond lengths, angles)
2.- Electronic energy
(bond energies, enthalpies of formation, etc)
3.- Spectra (electronic, vibrational, rotational, etc)
4.- Electrical properties (dipole moment, polarizability)
5.- Molecular orbitals and derived properties
such as effective charges, bond orders.
6.- Barriers to reaction and other rate properties.
Electronic structure calculations:
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Goal of quantum chemistry methods:
Multi-electron atoms and molecules
(r1,r2,...,rN )
H n (r1 , r2 ,..., rN )  En (r1 , r2 ,..., rN )
H 
ZI
1
1
2



 i 

2 i
r
iI
i  j rij
iI
Wave function of many electrons in an external potential (Borh-Oppenheimer)
Finite system (no periodic boundary conditions)

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Key quantity: The wave function
• Fundamental object in quantum mechanics: wave function
(r1,r2,...,rN )
• We want to find special wave functions such that
H n (r1 , r2 ,..., rN )  En (r1 , r2 ,..., rN )
where
H 
ZI
1
1
2



 i 

2 i
r
iI
i  j rij
iI
• This is a fundamentally many-body equation!
• A large variety of methods have been proposed and are
being used to solve this problem.
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Variational principle
This will be one of our main tools today.
It states that the energy calculated from an approximation to the true
wavefunction will always be greater than the true energy:
Thus, the better the wavefunction, the lower the energy.
At a minimum, the first derivative of the energy will be zero.
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Quantum Chemistry: Learning a new language
CIST, MP2, CC, CSF, TZV, ...
Electronic structure calculations:
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First (simple) approach: Hartree approximation
As a first guess, one may try to write the many-electron wavefunction as
a product of one-electron spin-orbitals ji(ri,si):
An important feature of the Hartree description is that the probability of
finding one electron at a particular point in space is independent of the
probability of finding any other electron at that point in space.
Thus, due to the independent particle model, the motion of the electrons in
the Hartree approximation is uncorrelated.
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Spin-orbitals
One-electron spin-orbitals ji(ri,si) are constructed as the product of a
spatial orbital and a spin function. In general, they are molecular orbitals.
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Hartree energy
Applying the variational principle to the above wave function, one can find
the single-particle Hartree equations:
self-consistent equations
mean field
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Antisymmetric wave function: Slater determinant
The main problem with Hartree’s wave function is that it violates Pauli´s
principle. The wave function of fermions must be antisymmetric and
therefore two fermions cannot be in the same quantum state.
Slater determinant
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Antisymmetric wave function: Slater determinant
-Exchanging any two rows of a determinant (exchanging two electrons)
leads to a change in sign  antisymmetry.
-Two electrons in the same quantum state  two identical rows 
the determinant is zero.
Slater determinant
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Variational principle on a Slater determinant:
Hartree-Fock
The minimization of the energy <|H|> assuming that the wave function  is a
Slater determinant leads to the Hartree-Fock approximation. The corresponding
single-particle Hartree-Fock equations are the following:
Electronic structure calculations:
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Variational principle on a Slater determinant:
Hartree-Fock
Coulomb term J
Identical to Hartree
Exchange term K
New exchange term
Again, this is a mean field self-consistent model.
Electronic structure calculations:
Methodology and applications to nanostructures
Variational principle on a Slater determinant:
Hartree-Fock
Notice that adding the term i=j in the sums modifies nothing:
It cancels out.
Electronic structure calculations:
Methodology and applications to nanostructures
Hartree-Fock equations: Fock operator
2
2




x
 *  x  j  x 


2
2
2

 j  x   e   dx '
1  Vext  r   e   dx

r  r 
r  r
 2me


 x    j j  x 
We can define the following operators by their action on an orbital:
J  x j j x    dx 
j x 
r r
2
K x j j x    dx 
j j x 
and from them we define the Fock operator:
1
F    2  Vext r    J  x   K  x 
2

so that the HF equations can be written as:
Fj j x    jj j x 
j* x j j x 
r r
j x 
Electronic structure calculations:
Methodology and applications to nanostructures
Restricted Hartree-Fock (RHF) and
Unrestricted Hartree-Fock (UHF)
RHF: the spatial part of the one-electron spinorbitals ji(ri,si) is identical for spin-up and spindown (closed-shell)
UHF: the spatial part of the one-electron spinorbitals ji(ri,si) depend on the spin-orientation.
Here, the wavefunction may be not a proper spin
eigenfunction (spin contamination).
The energy of a UHF wave function is always lower than (or equal to)
the corresponding RHF wave function (there is more flexibility in the
former).
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Accuracy of HF: Spin issues
Slater determinants are always eigenfunctions of Sz.
However, they are not necessarily eigenfunctions of S2.
For the general case there are always linear
combinations of determinants that are eigenfunctions of
Sz and S2 at the same time.
Such spin-adapted linear combination of determinants
(configurations) are needed to describe open-shell
systems.
Electronic structure calculations:
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Accuracy of Hartree-Fock
Hartree-Fock calculations often account
for ~99% of the total energy of the
system.
The problem is that the remaining ~1%
can determine the physical and
chemical properties of the system.
Electronic structure calculations:
Methodology and applications to nanostructures
Accuracy of Hartree-Fock
Hence, we have to improve over HF:
How to do that?
Electronic structure calculations:
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Basis sets
Orbitals are usually expanded in basis sets.
Electronic structure calculations:
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Basis sets
Slater-type orbitals (STOs)
n,l,m (r,,) = Nn,l,m, Yl,m (,) rn-1 e-r
are characterized by quantum numbers n, l, and m and
exponents (which characterize the radial 'size' ) .
Slater-type orbitals are similar to Hydrogenic orbitals
in the regions close to the nuclei.
Specifically, they have a non-zero slope
near the nucleus on which they are located
d/dr(exp(-r))r=0 = -
so they can have proper electron-nucleus cusps.
Electronic structure calculations:
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Basis sets
Cartesian Gaussian-type orbitals (GTOs)
a,b,c (r,,) = N'a,b,c, xa yb zc exp(-r2),
are characterized by quantum numbers a, b, and c,
which detail the angular shape and direction of the
orbital, and exponents  which govern the radial 'size’.
GTOs have zero slope near r=0
because
d/dr(exp(-r2))r=0 = 0.
The Coulomb cusp at the origin is not properly described.
But, computationally, multi-center integrals are much more
efficiently obtained.
Electronic structure calculations:
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Basis sets
To overcome the cusp weakness of
GTO functions, it is common to
combine two, three, or more GTOs,
with combination coefficients that are
fixed (and not treated as parameters),
into new functions called contracted
GTOs or CGTOs. However, it is not
possible to correctly produce a cusp by
combining any number of Gaussian
functions because every Gaussian has
a zero slope at r = 0 as shown here.
tight Gaussian
orbital with cusp at r = 0
loose Gaussian
medium Gaussian
r
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Basis sets: what to do with these building bricks
Minimum basis set: the number of basis functions is equal to the number of
core and valence electrons in the atom.
Double zeta (DZ): there are twice as many basis functions as there are core
and valence electrons.
Triple zeta (TZ): there are three times as many basis functions as the
number of core and valence electrons.
Quadruple zeta (QZ), Pentuple Zeta (PZ or 5Z), etc.
In any of them: split valence basis means that only the number of basis
functions representing the valence electrons is increased.
H
C
N
HCN molecule:
DZ basis allows for
different bonding in
different directions
Electronic structure calculations:
Methodology and applications to nanostructures
Basis sets: what to do with these building bricks
Polarization functions: a basis function with a higher component of angular
momentum is added, p-functions to s-based orbitals, d-functions to p-based
orbitals, etc.
Double Polarization functions: basis functions with two higher components of
angular momentum are added.
For instance, double zeta with polarization (DZP), triple zeta plus double
polarization (TZDP), etc.
Polarization functions give angular flexibility in forming molecular orbitals
between valence atomic orbitals.
Polarization functions also allow for angular correlations in describing the
correlated motions of electrons.
H
C
N
Electronic structure calculations:
Methodology and applications to nanostructures
Electron correlation
Hartree-Fock is an approximation:
It replaces the instantaneous
electron-electron repulsion by an
average repulsion term.
Strictly speaking, electron correlation energy is defined
as the difference between the HF energy and the lowest
possible energy that one can obtain within a given basis
set.
Physically, it corresponds to the fact that, on average, the
electrons are further apart than the situation described by
the (R)HF wave function.
A clear example in RHF: electrons are paired in
molecular orbitals and the spatial overlap between the
orbitals of such pair-electrons is exactly one!
Electronic structure calculations:
Methodology and applications to nanostructures
Electron correlation methods: Post Hartree-Fock
To improve over Hartree-Fock and include electron-correlation,
the easiest way is to start from the Hartree-Fock approximation
and ADD new things.
Different methodologies will be defined by the different ways to
‘add’ things to Hartree-Fock. Typically, they fall into two classes:
• Wavefunction expansion: The most common
approaches are Configuration Interaction (CI) and
Coupled-Cluster Methods (CC, CCSD).
• Perturbation theory: The most common approach is
Møller-Plesset (MP2 or MP4).
Electronic structure calculations:
Methodology and applications to nanostructures
Configuration Interaction
CI has many variants but is always based on the idea of
expanding the wavefunction as a sum of Slater determinants.
Electronic structure calculations:
Methodology and applications to nanostructures
Configuration Interaction
CI has many variants but is always based on the idea of
expanding the wavefunction as a sum of Slater determinants.
where we are adding new Slater determinants that are singly (s), doubly
(d), triply (t), quadruply (q), etc. Excited relative to the original HF
determinant.
These determinants are often referred to as Singles (S), Doubles (D), Triples
(T), Quadruples (Q), etc.
Electronic structure calculations:
Methodology and applications to nanostructures
Configuration Interaction
These determinants are often referred to as Singles (S), Doubles (D), Triples
(T), Quadruples (Q), etc.
Electronic structure calculations:
Methodology and applications to nanostructures
Configuration Interaction (CI)
CI has many variants but is always based on the idea of
expanding the wavefunction as a sum of Slater determinants.
S
CI=a0HF+aSS+aDD+…= aii
Again we use the variational principle and look for the ai
coefficients that make minimal the wave function energy.
Löwdin (1955): Complete CI gives exact wavefunction for the
given atomic basis. For an infinite basis, it provides the exact
solution.
Orbitals are NOT reoptimized in CI!
Electronic structure calculations:
Methodology and applications to nanostructures
Configuration Interaction (CI)
S
CI=a0HF+aSS+aDD+…= aii
Structure of the CI matrix
Brillouin’s theorem: Matrix elements between the HF
reference determinant and singly excited states are
zero.
Electronic structure calculations:
Methodology and applications to nanostructures
Configuration Interaction (CI)
S
CI=a0HF+aSS+aDD+…= aii
In order to develop a computationally tractable model, the
number of excited determinants in the CI expansion must be
reduced.
Truncating the expansion at one (s) does not improve the HF
result because of Brillouin’s theorem.
The lowest CI level that improves over HF is CI with Doubles
(CID).
The number of singles is much lower than the number of
Doubles. Therefore, including singles is not a big deal: CI with
Singles and Doubles (CISD).
Also with Triples: CISDT. Also with Quadruples (CISDTQ).
Electronic structure calculations:
Methodology and applications to nanostructures
Configuration Interaction (CI)
S
CI=a0HF+aSS+aDD+…= aii
The lowest CI level that improves over HF is CI with Doubles (CID).
Weights of excited configuration in the Ne atom. Doubles have the highest weight!
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Methodology and applications to nanostructures
Example: correlation in the H2 dissociation problem.
Let’s try to illustrate how CI accounts for electron
correlation taking as an example the dissociation of
the hydrogen molecule H2
Take two 1s orbitals, one in each center of the molecule, A and B
A
B
HF
Electronic structure calculations:
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Example: correlation in the H2 dissociation problem.
The basis determinants for a full CI calculation are the following:
Double
Single
2+3 triplet SZ=0
Single
2-3 singlet
Single : triplet SZ=1
Single : triplet SZ=-1
Electronic structure calculations:
Methodology and applications to nanostructures
Example: correlation in the H2 dissociation problem.
The ground state 0 and the doubly excited 1 can be expanded in terms of
the atomic orbitals:
ionic
covalent
Now, if we increase the bond length towards infinity, the HF wave
function is still a mixture of ionic and covalent components and, in the
dissociation limit will be 50% H+H- and 50% H0H0.
This is totally wrong!!
Electron correlation is missing: electrons try to avoid each other!
Electronic structure calculations:
Methodology and applications to nanostructures
Example: correlation in the H2 dissociation problem.
We can solve that by using full CI. The full CI matrix can be shown to be:
For 1Sg symmetry
only these terms
matter
The variational parameters allow us to
choose the best combination for each
bonding distance. For instance, the ionic
component disappears for a1=-a0
Electronic structure calculations:
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Example: correlation in the H2 dissociation problem.
The problem can also be treated with a UHF wave function.
Although the UHF wave function does not solve everything: spin contamination.
We introduce a variational parameter c in the
definition of the molecular orbitals. Now they
are different for spin-up and spin-down.
ionic
covalent
but now we have
an additional
triplet component
Electronic structure calculations:
Methodology and applications to nanostructures
Example: correlation in the H2 dissociation problem.
All this is conspicuous in the energy diagram:
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Multi-Reference calculations
For almost degenerate levels it is crucial to optimize the orbitals as well:
Multi-Reference Self-Consistent Field (MRSCF): a kind of CI in which the orbitals,
as well as the coefficients, are optimized. Configurations included in MCSCF are
defined by the active space.
Multi-Reference Configuration Interaction (MRCI): A MRSCF function is chosen as
reference. Singles, doubles, etc., are generated out of all the determinants that
enter the MRSCF.
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Methodology and applications to nanostructures
Active space in CI
Reduced CI methods
Idea:
Not all determinants are equally important.
Ansatz:
Only allow excitations from a subset of orbitals
into a subset of virtual orbitals (active space).
Allow only a maximal number of excitations.
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Methodology and applications to nanostructures
Many-body perturbation theory
In perturbation theory, the Hamiltonian splits into:
Perturbation, i.e., its effect should be small!
Unperturbed Hamiltonian
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Møller-Plesset Perturbation Theory
First, let us remember the Hartree-Fock equations:
2
2




x
 *  x  j  x 


2
2
2

 j  x   e   dx '
1  Vext  r   e   dx


r r 
r  r
 2me


 x    j j  x 
These are the self-consistent equations that the single-particle wave
functions should fulfill to obtain the minimum energy for a singledeterminant many-body wave function.
We have the Fock operator, defined as:
1
F    2  Vext r    J  x   K  x 
2

so that the HF equations can be written as:
Fj j x    jj j x 
Electronic structure calculations:
Methodology and applications to nanostructures
Møller-Plesset Perturbation Theory
The exact Hamiltonian He is:
1 M 2 M
1
1
H e     k Vext rk   
2 k 1
2 k   rk  r
k 1
We define an unperturbed Hamiltonian H0 as the sum of Fock operators:
M
H 0   Fk
k 1
1 M 2 M
H 0     k Vext rk    J  x k   K  x k 
2 k 1
k 1
k ,
Electronic structure calculations:
Methodology and applications to nanostructures
Møller-Plesset Perturbation Theory
The exact Hamiltonian He is:
1 M 2 M
1
1
H e     k Vext rk   
2 k 1
2 k   rk  r
k 1
We define an unperturbed Hamiltonian H0 as the sum of Fock operators:
M
H 0   Fk
k 1
1 M 2 M
H 0     k Vext rk    J  x k   K  x k 
2 k 1
k 1
k ,
In H0 we are summing up
twice the electron-electron
Coulomb interaction
Electronic structure calculations:
Methodology and applications to nanostructures
Møller-Plesset Perturbation Theory
The exact Hamiltonian He is:
1 M 2 M
1
1
H e     k Vext rk   
2 k 1
2 k   rk  r
k 1
We define an unperturbed Hamiltonian H0 as the sum of Fock operators:
M
H 0   Fk
k 1
1 M 2 M
H 0     k Vext rk    J  x k   K  x k 
2 k 1
k 1
k ,
In H0 the electron-electron
interaction is considered in
an average way.
Electronic structure calculations:
Methodology and applications to nanostructures
Møller-Plesset Perturbation Theory
The exact Hamiltonian He is:
1 M 2 M
1
1
H e     k Vext rk   
2 k 1
2 k   rk  r
k 1
We define an unperturbed Hamiltonian H0 as the sum of Fock operators:
M
H 0   Fk
k 1
1 M 2 M
H 0     k Vext rk    J  x k   K  x k 
2 k 1
k 1
k ,
The perturbation is therefore H’=He-H0:
H 
1
1
  J  x k   K x k 

2 k   rk  r
k

Not that small!
Electronic structure calculations:
Methodology and applications to nanostructures
Møller-Plesset Perturbation Theory
M
H 0   Fk
k 1
1 M 2 M
H 0     k Vext rk    J  x k   K  x k 
2 k 1
k 1
k ,
The unperturbed Hamiltonian H0 is a sum of one-electron operators without
coupling. Therefore the solution will be the sum of all energies of the
independent systems and the wavefunction will be the (antisymmetrized)
product of the one electron wavefunctions (orbitals).
In other words, the ground state wave function will be the Hartree-Fock
wavefunction 0HF.
The unperturbed energy is E0=< 0HF | H0 | 0HF > ≠ EHF
H 
1
1
  J  x k   K x k 

2 k   rk  r
k

The energy adding the first-order correction is E1=< 0HF | H0+ H’ | 0HF > = EHF!!
(it is the matrix element that we have varied to obtain the HF equations)
Electronic structure calculations:
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Møller-Plesset Perturbation Theory (MP)
Electron correlation corrections start at second order with this choice of the
unperturbed Hamiltonian.
MP2 means second-order in Moller-Plesset expansion.
The perturbative correction is:
2 
E ( MP2) 

i excited
i 
 0 H  0
2
E  0   E i 
where the excited states (i)0. are eigenstates of the unperturbed Hamiltonian H0,
i.e., they are determinants in which excitations have been created.
Actually, doubly-excited (not single-excited) determinants are the first contribution!
Following contributions to the perturbative expansion are
MP3, MP4, etc.
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Convergence of MP perturbative series
No smooth convergence, or no convergence at all is possible!!
In practice, only low orders of the expansion can be included.
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Coupled-cluster methods
The basic ansatz of coupled cluster theory is that the exact many-electron
wavefunction  may be generated by the operation of an exponential operator on
a single determinant.
0 is a single determinant wave function
(usually, the Hartree-Fock wave function 0 = HF is used)
T is an excitation operator. The excitation operator can be written as a linear
combination of single, double, triple, etc excitations, up to N fold excitations for an
N electron system:
Electronic structure calculations:
Methodology and applications to nanostructures
Coupled-cluster methods
and the action of each Ti operator is to create the full set of i-excitations
there is only one way to have a single
excitation T1, but two ways to generate double
excitations: a double excitation (T2) and two
consecutive single excitations (T1 T1).
etc.
Electronic structure calculations:
Methodology and applications to nanostructures
Coupled-cluster methods
and the action of each Ti operator is to create the full set of i-excitations
there is only one way to have a single
excitation T1, but two ways to generate double
excitations: a double excitation (T2) and two
consecutive single excitations (T1 T1).
etc.
To construct the coupled cluster wavefunction one must then determine
the various amplitudes t through a system of coupled equations.
Electronic structure calculations:
Methodology and applications to nanostructures
Coupled-cluster (CC) methods… in practice
In coupled cluster methods, for a given type of corrections (say, single
excitations, for instance), all required terms are included.
As a consequence, the method scales like M2N+2 for M basis functions
and N electrons  a very expensive scaling!!
In practice, only corrections up to a given term are included:
If higher-order terms are calculated in perturbation theory, they are indicated with
Parentheses. For instance CCSD(T) means that the triples are obtained perturbatively.
Electronic structure calculations:
Methodology and applications to nanostructures
Coupled-cluster (CC) methods… in practice
In coupled cluster methods, for a given type of corrections (say, single
excitations, for instance), all required terms are included.
As a consequence, the method scales like M2N+2 for M basis functions
and N electrons  a very expensive scaling!!
In practice, only corrections up to a given term are included:
Almost the only one used in practice.
Electronic structure calculations:
Methodology and applications to nanostructures
Coupled-cluster methods
Electronic structure calculations:
Methodology and applications to nanostructures
Coupled-cluster methods
Electronic structure calculations:
Methodology and applications to nanostructures
In summary:
• Quantum chemistry methods are very accurate but
computationally expensive.
• Key points: choice of basis set and methodology
used to improve over Hartree-Fock.
Electronic structure calculations:
Methodology and applications to nanostructures
Electronic structure calculations:
Methodology and applications to nanostructures
Electronic structure calculations:
Methodology and applications to nanostructures
Electronic structure calculations:
Methodology and applications to nanostructures
Electronic structure calculations:
Methodology and applications to nanostructures
Electronic structure calculations:
Methodology and applications to nanostructures
Electronic structure calculations:
Methodology and applications to nanostructures
Electronic structure calculations:
Methodology and applications to nanostructures
Electronic structure calculations:
Methodology and applications to nanostructures
Electronic structure calculations:
Methodology and applications to nanostructures
Electronic structure calculations:
Methodology and applications to nanostructures