Computational Analysis

IntroductionIntroduction toto ComputationalComputational ChemistryChemistry

David Tur, PhD

Scientific Applications expert

[email protected]

1. Introduction

2. Molecular mechanics

3. Quantum chemistry methods

I. Wave function based methods

Hartree-Fock

Post-HF Methods

II. Semi-empirical methods

III. DFT

4. Basis sets

5. Computational chemistry methods in solid state6. Conclusions


IntroductionIntroduction

“Computational chemistry simulates chemical structures

and reactions numerically, based in full or in part on the fundamental laws of physics.”

– Foresman and Frisch

“The underlying physical laws necessary for the mathematical theory of a large part of physics and

the whole of chemistry are thus completely known,

and the difficulty is only that the exact application of

these laws leads to equations much too complicated

to be solvable.”– Paul Dirac

Theoretical Chemistry is defined as the mathematical description of chemistry. When these mathematical methods are sufficiently well developed to be automated and implemented on a computer, we can talk about Computational Chemistry.


Very few aspects of chemistry can be computed exactly, but almost every aspect of chemistry has been described in a qualitative or approximate quantitative computational scheme.

The theoretical chemist must keep in mind that numbers obtained from theoretical calculations are not exact (they use many approximations), but they can offer an useful insight into real chemistry.

� Different models use different approximations (or levels of theory) to produce results of varying levels of accuracy.

� There is a trade off between accuracy and computational time.


�There are two main types of models depending on the starting point of the theory:

Classical methods, are those methods that use Newton mechanics to model molecular systems.

Quantum Chemistry methods, which makes use of quantum mechanics to model the system. These methods use different type of approximation to solve the Schrödinger equation.

What can we predict with modern Theoretical Chemistry:

� Geometry of a molecule� Dipole moment� Energy of reaction� Reaction barrier height� Vibrational frequencies� IR spectra� NMR spectra� Reaction rate� Partition function� Free energy� Any physical observable of a small molecule


Classical methods

Quantum methods

DFT

Wavefunction

based methods

Post-HF

Molecular MechanicsSemi-empirical

methods

LDA LSDA

GGA

hybrid-GGA meta-GGA

Hybrid-meta-GGA MP2

CoupledCluster

FCI

HF&MCSCF

MRCI

CASPT2

SR MR


1. Introduction




Hartree-Fock

Post-HF Methods


III. DFT

4. Basis sets



� Molecular Mechanics (MM) use the laws of classical physics to

predict structures and properties of molecules

� The motions of the nuclei are studied and the electrons are not

explicitly treated (Born-Oppenheimer approximation)

� Molecules are seen as a mechanical assemblies made up of simple

elements like balls (atoms), sticks (bonds) and flexible joints (bond

angles and torsion angles)

� MM treats molecules as a collection of particles held together by a

simple harmonic forces.

� These harmonic forces are described in terms of individual potential

functions.

� The overall molecular potential energy or steric energy of the molecule

is the sum of the potential functions of its constituents.

Molecular Molecular MechanicsMechanics

With this assumptions, the Mechanics equation can be simply written as:

E=EB+E

A+E

D+E

NB

where EB

is the energy involved in the deformation of a bond, either by

stretching or compression, EA

is the energy involved in angle bending, ED

is

the torsional angle energy, and ENB

is the energy involved in interactions

between atoms that are not directly bonded.

*Picture from the NIH site of Molecular modelinghttp://cmm.cit.nih.gov/modeling/

Molecular MechanicsMolecular Mechanics

� The exact functional form of the potential function of Force Field

depends on the program being used.

� Bond and angle terms are generally modeled as harmonic potentials

centered around equilibrium bond-length values (derived from exp. or

ab initio calculations). Morse potential is an alternative that results in

more accurate results of vibrational spectra but at higher computational

cost. The dihedral terms shows multiple minima and thus can not be

modeled as harmonic oscillators.

� The Non Bonded interactions are much more computationally costly to

calculate, and different approaches are used to model them. This term

is divided between short range interactions (VdW) usually modeled

using Lennard-Jones potential and long range or electrostatic

interactions, whose basic functional is the Coulomb potential.


General form of the Molecular Mechanics equations:


*Picture from the Wikipedia: Molecular Modelling


Molecular Mechanics computations are quite inexpensive (compared to ab initio

methods), and they allow to be used to compute properties for very large

systems containing many thousands of atoms such as:

� Energy optimization (combined with simulated annealing, Metropolis, or

other MC methods)

� Calculation of binding constants

� Simulating of protein folding kinetics

� Examination of active site coordinates

� Design of binding sites


However it carries two main limitations:

� Each force field achieves good results only for limited class of molecules

related to those for which it was parameterized.

� The neglecting of electrons means that MM methods can not treat chemical

problems where electronic effects are dominant (bond formations, bond

breaking…)

1. Introduction




Hartree-Fock

Post-HF Methods


III. DFT

4. Basis sets


Introduction to Computational ChemistryIntroduction to Computational Chemistry

Classical methods

Quantum methods

DFT

Wavefunction

based methods

Post-HF

Molecular MechanicsSemi-empirical

methods

LDA LSDA

GGA

hybrid-GGA meta-GGA

Hybrid-meta-GGA MP2

CoupledCluster

FCI

HF&MCSCF

MRCI

CASPT2

SR MR

Quantum Chemistry MethodsQuantum Chemistry Methods

Where did ab initio methods finished in this scheme?

Ab initio is Latin for ‘from the beginning’, and indicates that the

calculation is from first principles and that no empirical data is used.

Are DFT ab initio methods?

Rigorously speaking DFT should be considered an ab initio method,

but as the most common functionals use parameters derived from

empirical data, or from more complex calculations, it has historically

been grouped apart from ab initio methods.


Here we will difference between wave function based methods and

Density functional Theory (within ab initio methods).

Quantum Chemistry is a branch of theoretical chemistry that applies Quantum

Mechanics in order to mathematically describe the fundamental properties

of atoms and molecules.

The complete knowledge of the chemical properties of the system implies

computing the wave function that describes the electronic structure of these

atoms and molecules.

In 1925 Erwin Schrödinger analyzed what an electron would look like as a

wavelike particle around the nucleus of the atom. From this model he

formulated his equation for particle waves, which is the starting point of the

quantum mechanical study of the properties of an atom or molecule:

where H is the Hamiltonian and Ψ is the wavefunction associated with the state

of the system.


i Ht

∧∂Ψ= Ψ

∂h

�The main problem now is the solution of the electronic Schrödinger

equation.

�These methods are based on theories which range from highly

accurate, but suitable only for small systems, to very approximate,

but suitable for very large systems.

�Quantum chemistry addresses the solution of this problem in different

ways, depending on the mathematical approaches used.

�The exact solution to this equation is not known (apart from

monoelectronic systems), numerical methods must be used to solve it.


The Schrödinger model is based on the six postulates of quantum mechanics:

1. Associated with any particle moving in a conservative force field is a wavefunction , which contains all information that can be known about the system.

2. For every observable in classical mechanics, a linear Hermitian operator is defined3. When measuring the observable associated with the operator A in , the only values

that will ever be observed are the eigenvalues a which satisfy 4. The average value of the observable corresponding to operator is given by:

5. The wavefunction evolves in time according to the time-dependent Schrödinger equation:

6. The total wavefunction must be antisymmetric with respect to the interchange of all coordinates of one fermion with those of another. The electronic spin must be included in this set of coordinates. The Pauli Exclusion Principle is a direct result of this antisymmetryprinciple.


( ),x tΨ

*

*

Â drÂ

dr

Ψ Ψ=

Ψ Ψ

∫

∫

( ),H r t it

∧ ∂ΨΨ =

∂h

Â aΨ = ΨÂ aΨ = Ψ

The most commonly used quantum chemistry methods are:

1.- Ab initio methods, where the solution of the Schrödinger equation

is obtained from first principles of quantum chemistry using rigorous

mathematical approximations, and without using empirical data.In the frame of

ab initio methods there are two strategies to solve equation:

� Wavefunction based methods, which are based on obtaining the

wavefunction of the system,

� Density functional based methods, that consist in the study of the

properties of the system through its electronic density, but avoiding

the explicit determination of the electronic wavefunction.

2.- Semi-empirical methods, which are less accurate methods that

use experimental results to avoid the solution of some terms that appear in

the ab initio methods.


�The first and most relevant ab initio method is the Hartree-Fock theory, which was first

introduced in 1927 by D.R. Hartree.

�The procedure, which he called self-consistent field (SCF), is used to calculate approximate wavefunctions and energies for atoms and ions.

�The HF method assumes that the exact, N-body wavefunction of the system can

be approximated by a single Slater determinant (fermions) or by a single permanent (bosons) of N spin-orbitals.

�The starting point for the HF method is a set of approximate one-electron wavefunction, (orbitals). For a molecular or crystalline calculation the initial approximate one-electron wavefunctions are typically a LCAO (linear combination of atomic orbitals)

�Using variational principle (HF upper bound to true ground state energy), we can

derive a set of N-coupled equations for the N spin orbitals. The minimization of the HF energy expression with respect to changes in the orbtials, by applying Langrange method of undetermined multiplieres, yields the HF equations defining the orbitals.

Wave function based methodsWave function based methods

( )1 2det , ... Nφ φ φΨ =

1-Wavefunction written as a single determinant

Brief mathematical exposition of HF theory:

2-The electronic Hamiltonian can be written as:

( ) ( ),eli i j

H h i v i j∧

<

= +∑ ∑

where h(i) and ν(i,j) are the one and two electrons operator defined as:

21( )

2

Ai

A iA

Zh i

r= − ∇ −∑ ( )

1,

ij

v i jr

=

elelE H∧

= Ψ Ψ

3-The electronic energy of the system is given by :

4-The resulting HF equations from minimization of the energy:

( ) ( ) ( )1 1 1i i if x x xχ ε χ∧

=

where εi is the energy value associated with orbital χi and f is defined as

( ) ( ) ( ) ( )1 1 1 1jj

j

f x h x J x K x∧ ∧ ∧ ∧

= + −∑



where Ji found in the second term of the equation is the so-called Coulomb term that gives the average local potential at point x due to the charge distribution from the electron in orbital χi and is defined as:

( ) ( ) ( ) ( )1 1 1 1jj

j

f x h x J x K x∧ ∧ ∧ ∧

= + −∑

( ) ( ) ( )( )

( ) ( ) ( )2

2

1 1 2 2 2 2 1

12 12

1j i i j j i

xJ x x x dx x x x

r r

χχ χ χ χ χ

∧

= =∫

( ) ( ) ( )( ) ( )

( ) ( ) ( )*

2 2

1 1 1 2 2 2 1

12 12

1j ij i j j i i

x xK x x x dx x x x

r r

χ χχ χ χ χ χ

∧

= =∫

and Ki third operator term of the equation, is the exchange operator that is defined as:

The Hartree-Fock equations can be solved numerically or in the space spanned by a set of basis functions (Hartree-Fock-Roothan equations) Some guess of the initial orbitals is required, and then theses orbitals are refined iteratively (self-consistent-field approach, SCF), finally obtaining the form and energy.


Greatly simplified algorithmic flowchart illustrating the Hartree-Fock Method.

*Flowchart from Wikipedia: Hartree-Fock

The error in the determination of the total energy due to the use of the Hartree-Fock method is the so-called correlation energy:

corr o HFE E E= −

where E0 is the is the exact nonrelativistic energy of the system and EHF is the energy in the Hartree-Fock limit (this limit is obtained by carrying out HF

calculations using an infinite basis set).

There have been a large number of methods developed to improve the Hartree-Fock results, all accounting for the correlation energy in one way or another, the so-called Post-HF methods.


Hartree-Fock method makes five major simplifications in order to deal with this

task:

� The Born-Oppenheimer approximation is inherently assumed. The full

molecular wavefunction is actually a function of the coordinates of each of the

nuclei, in addition to those of the electrons.

� Typically, relativistic effects are completely neglected. The momentum

operator is assumed to be completely non-relativistic.

� The variational solution is assumed to be a linear combination of a finite

number of basis functions, which are usually (but not always) chosen to be

orthogonal. The finite basis set is assumed to be approximately complete.

� Each energy eigenfunction is assumed to be describable by a single Slater

determinant, an antisymmetrized product of one-electron wavefunctions (i.e.,

orbitals).

� The mean field approximation is implied. Effects arising from deviations

from this assumption, known as electron correlation, are completely neglected.

Relaxation of the last two approximations give rise to many post-Hartree-Fock

methods.


Quantum methods

Wavefunction

based methods

Post-HFMP2

CoupledCluster

FCI

HF&MCSCF

MRCI

CASPT2

SR MR

Single reference (SR) methods: use a single Slater determinant as a zero order wavefunction or

starting point to generate the excitation states used to describe the system.

Multireference methods (MR): where the systems

need to be described by more than one electronic configuration (e.g. for molecular ground states that are quasi-degenerate with low-lying excited states, or in bond-breaking situations)

While the HF wavefunction is uniquely defined by specifying the number of occupied orbitals in each symmetry, in the MCSCF (multi configurationalSCF) the electronic state of the system is approximated by a multi-configuration

wavefunction


Single reference Post-HF methods:

Configuration Interaction (CI): a variational method that accounts for the correlation energy using a variational wavefunction, which is a linear combination of determinants or

configuration state functions built form spin orbitals (SO).

If the expansion includes all possible configurations of the appropriate symmetry, then this is a full configuration interaction (FCI) procedure which exactly solves the electronic Schrödinger equation within the space spanned by the one-particle basis set.In practice not all the unoccupied Hartree-Fock orbitals can be computed, The expansion in must be truncated, not considering any excitations above a given order. When the

expansion is truncated at the zeroth order, the Hartree-Fock method is recovered. At first order truncation the ‘Configuration Interaction with only Single excitations’ (CIS) is obtained, at second order ‘CI with Single and Double excitations’ (CISD), and so on: CISDT (third order), CISDTQ (fourth order), etc.



Møller-Plesset (MP): This is a perturbational method. Møller-Plesset perturbation theory adds electron correlation to the Hartree-Fock method by means of Rayleigh-Schrödinger perturbation theory (RSPT). In RS-PT one considers an unperturbed Hamiltonian operator H0 to which is added a small (often external) perturbation V:

where λ is an arbitrary real parameter. In MP theory the zeroth-order wave function is an exact eigenfunction of the Fock operator, which thus serves as the unperturbed operator. The

perturbation is the correlation potential.In RS-PT the perturbed wave function and perturbed energy are expressed as a power series in λ:

Substitution of these series into the time-independent Schrödinger equation gives a new equation:

Equating the factors of λk in this equation gives an kth-order perturbation equation.



Coupled Cluster (CC) theory: is another numerical technique used for describing many-body systems starting from HF molecular orbital and adding correcton terms to take into account electron correlation. The coupled cluster methodology employs an excitation operator T in a analogous form as C in CI theory that has the form:

operator used to construct the new molecular wavefunction starting from HF MO

used to find an approximate solution to the Schrödigner equation

After some farragous algebre, the correlatino energy is obtainded from the CC equations:

Depending on the highest number of excitations allowed in the definition of T we obtain different CC methods: CCSD, CC3, CCSD(T), CCSDTQ…


FCIFCI/

STO-3G

FCI/

3-21G

FCI/

6-31G*

FCI/

6-311G(2df)EXACT

···

CCSD(T)CCSD(T)/

STO-3G

CCSD(T)/

3-21G

CCSD(T)/

6-31G*

CCSD(T)/

6-311G(2df)

CCSD(T)CBS

CCSDCCSD/

STO-3G

CCSD/

3-21G

CCSD/

6-31G*

CCSD/

6-311G(2df)

CCSDCBS

MP2MP2/

STO-3G

MP2/

3-21G

MP2/

6-31G*

MP2/

6-311G(2df)

MP2CBS

HFHF/

STO-3G

HF/

3-21G

HF/

6-31G*

HF/

6-311G(2df)

HF CBS

STO-3G 3-21G 6-31G* 6-311G(2df) ··· CBS

*Head-Gordon, J. Phys. Chem. Vol. 100, No, 31, 1996

Single reference Post-HF methods: Chart arranged in order of increasing

accuracy, when increasing the level of correlation, and the size of the basis sets.


Multi-reference Post-HF methods:

The most commonly used MCSCF approach, which simplifies the selection of

configurations needed to construct a proper wavefunction, is the so-called

Complete Active Space Self Consistent Field method (CASSCF). In a CASSCF

wavefunction the occupied orbital space is divided into a set of inactive orbitals

and a set of active orbitals. All inactive orbitals are doubly occupied (closed

shell) in each Slater determinant. In contrast, the active orbitals have varying

occupations, and all possible Slater determinants are taken into account

distributing electrons in all possible ways among the active orbitals

corresponding to a full CI in the active space.

Starting from MCSCF, are obtained MRCI methods (analogous to CI using HF),

or CASPT2 (analogous to MP2 for SR methods).


� Semi-empirical quantum methods, represents a middle road between the

mostly qualitative results available from molecular mechanics and the high

computationally demanding quantitative results from ab initio methods

� Semi-emipirical methods attempt to address two limitations of the Hartree-

Fock calculations, such as slow speed and low accuracy, by omitting or

parameterizing certain integrals

� Integral approximation: there are three principal levels of integral

approximations:

• Complete Neglect of Differential overlap (CNDO)

• Intermediate Neglect of Differential Overlap (INDO)

• Neglect of Diatomic Differential Overlap (NDDO) (Used by PM3, AM1…)

� the integrals at a given level of approximation are either determined directly

from experimental data or calculated from corresponding analytical formula with ab

initio methods or from suitable parametric expressions.

SemiSemi--empirical Methodsempirical Methods

Semi-empirical methods are very fast, give accurate results when applied to

molecules that are similar to those used for parametization, and are applicable

to very large molecular systems

� Density functional theory (DFT): is an alternative approach to study the electronic

structure of many-body systems that over the past 10-20 years has strongly influenced the evolution of Quantum Chemistry. � DFT expresses the ground state energy of the system in terms of total one-electron density rather than making use of the wavefunction.

� The DFT formalism has the same starting point as the wavefunction based methods, that is the Born-Oppenheimer approximation where the nuclei of the treated molecules are seen as fixed, coupled with an extenal potential , Vext, where the electrons are moving. � The Hamiltonian can now be rewritten as

( ) ( )2

0 0 0 0 1 2 3

2

, , , ,N

i Ni

r dr r r r rρ ρ∧

=

= Ψ Ψ = Ψ∏∫ K

�In the DFT approach, the key variable is the particle density, and for this Hamiltonian, the ground state gives rise to a ground state electronic density ρ0(r) defined as:

extH F V∧ ∧ ∧

= +

where F is the sum of the kinetic energy of the electrons, and the electron-electron Coulomb interaction

Thus the ground state wavefuntion and density ρ0(r), are both functionals of the external potential and the number of electrons and N.

Density Functional TheoryDensity Functional Theory

� Starting from here Hohenberg-Kohn (HK) made two remarkable statements that are the basis of the modern employed DFT methods:

Theorem 1: The external potential Vext, and hence the total energy, is a unique functional of the electron density ρ(r)

Theorem 2: The groundstate energy can be obtained variationally: the density that minimises the total energy is the exact groundstate density.

� Appling these theorems, and after some more farragoes mathematics, the form of the

Schrödinger equation is:

( ')( ) ' ( ) ( )

'KS XC ext

rV r dr V r V r

r r

ρ= + +

−∫

21( ) ( ) ( )

2KS i i iV r r rψ ε ψ

− ∇ + =

where the Khon-Sham potential, VKS, has the

Since the Kohn-Sham potential VKS(r) depends upon the density, it is necessary to solve these equations self-consistently as in the Hartree-Fock scheme


There are many different ways to approximate this functional VXC,

and to do this EXC is generally divided into two separate terms:

[ ] [ ] [ ]XC X CE E Eρ ρ ρ= +

where EX is the exchange (int. e- same spin) term and EC the correlation

(int. e- opposite spin) term.

It is important to note that although Kohn-Sham is an exact method

in principle, because of the unknown exchange-correlation

functional VXC it turns out to be approximate.

FINDING THE APPROPIATE FUNCTIONAL: From LDA (local density

approximation), to modern meta-hybrid-GGA (Generalized Gradient

approximation)


The development of GGA functionals has followed two main lines, the

non-empirical and the semi-empirical approach. The typical non-empirical

approach, favored in physics, is to construct a functional subject to several

exact constraints. This strategy can be viewed as a ladder with five rungs,

from the Hartree theory (the earth) to the exact exchange and correlation

functional (heaven), as follows:

EARTH (Hartree theory)

local density onlyLDArung 1

explicit dependence on density gradientsGGAsrung 2

explicit dependence on kinetic energy densitymeta-GGAsrung 3

explicit dependence on occupied orbitalshybrid functionalsrung 4

explicit dependence on unoccupied orbitalsfully nonlocalrung 5

HEAVEN (chemical accuracy)

Although “Jacob’s ladder” is historically presented as the starting point for

the formulation of the non-empirical approaches, both semi-empirical and

non-empirical functionals can be assigned to various rungs of the “ladder”.


1. Introduction




Hartree-Fock

Post-HF Methods


III. DFT

4. Basis sets



� In Mathematics: a basis set is a collection of vectors that spans a vectorial space where a numerical problem is solved.

� In quantum chemistry: the wavefunctions are the mathematical description of where an electron or group of electrons are, and basis sets represent the wavefunction that allow the Schrödinger equation via any of the methods previously explained

Types of basis functions:� Slater type basis set (STO): a set of functions which decayed exponentially with distance

from the nuclei.

� Gaussian type basis sets (GTO): STO are approximated as linear combination of gaussian type orbitals.

GTO have the advantage over STO in that it is much easier to calculate integrals analytically, and thus are huge computational savings, but the exponential dependence on r2

means that in contrast to STO, the maximum at the nucleus is not in fact well described

( ) ( )1

, , , ,, , ,STO n rn l m l mr Nr e Yξ

ξφ θ ϕ θ ϕ− −=

( ) ( )2(2 1)

, , , ,, , ,GTO n n rn l m l mr Nr e Yξ

ξφ θ ϕ θ ϕ− − −=

Basis setsBasis sets

There are two main families of basis set used in modern computational chemistry calculations:

� Pople family of basis set: “X-Y’Y’’…(+)G(p)”

where Ys indicates the number of Gaussians that compose the valence orbitals, and the Y itself represents the number of linear combinations of primitive Gaussians that compose each Gaussians. The number of polarized functions added is given by the p in brackets after the G, and these basis set can also be improved by adding diffuse

functions for non-hydrogen atoms (+) or also for hydrogen (++). e.g.: 3-21G, 6-31G(d), 6-311+G(d,p)…

� Correlation consistent basis set: “cc-pVXZ”

where X indicated the number of basis functions composing the valence orbitals(X=D,T,Q… are double, triple or quadruple zeta respectively). Difuse functions introduced with aug- prefixe.g.: cc-pVDZ, aug-cc-pVQZ…

Basis setsBasis sets

1. Introduction




Hartree-Fock

Post-HF Methods


III. DFT

4. Basis sets



Computational chemistry methods in solid stateComputational chemistry methods in solid state

Computational chemistry methods in solid state follow the same approach as they do for molecule but can introduce two different approaches:

� Translation symmetry:

The electronic structure of a crystal is in general described by a band structure, which defines the energies of electron orbitals for each point in the Brillouin zone. Ab initio and semi-empirical calculations yield orbital energies, therefore they can be applied to band structure calculations.

� Plane waves basis set:

Completely delocalized basis functions as an alternative to the molecular atom-centered basis functions. Common choice for prediction properties in crystals

1. Introduction




Hartree-Fock

Post-HF Methods


III. DFT

4. Basis sets



� small systems

� systems involving

electronic transitions

�molecules or systems

without available

experimental data ("new" chemistry)

� systems requiring

rigorous accuracy

�computationally

expensive

� useful for a broad

range of systems

� does not depend on

experimental data

� capable of calculating

transition states and excited states

�uses quantum

physics

� mathematically

rigorous, no empirical

parameters

�uses approximation extensively

Ab Initio

�medium-sized systems (hundreds of atoms)

�systems involving

electronic transitions

� requires

experimental data (or data from ab

initio) for parameters

�less rigorous than

ab initio) methods

� less demanding

computationally than abinitio methods

� capable of calculating

transition states and

excited states

� uses quantum

physics and experimentally derived

empirical parameters

� uses approximation

extensively

Semi-Empirical

�large systems

(thousands of atoms)

�systems or processes

with no breaking or forming

of bonds

�particular force

field applicable only

for a limited class of

molecules

�does not calculate electronic properties

�requires

experimental data

(or data from ab

initio) for parameters

� Computationally least

intensive - fast and

useful with limited

computer resources

� can be used for molecules as large as

enzymes

� uses classical

physics

�relies on force-field

with embedded

empirical parameters

Molecular Mechanics

Best for Disadvantages Advantages Method Type

ConclusionsConclusions

7.7.-- ConclusionsConclusions

50-200 atoms (50-300)M2-M3HF, KS-DFT

25-50 atoms (50-150)M5MP2

10-15 atoms (100)M6CCSD

8-12 atoms (20-50)M7CCSD(T)

2 atoms (15)FactorialFCI

Current estimate of maximum feasible

molecular size

Current

computational

dependence on

molecular size, M

Method

*Head-Gordon, J. Phys. Chem. Vol. 100, No, 31, 1996

‘Current scalings of electronic structure methods with molecular size, M and estimates of the

maximum molecular sizes (in terms of numbers of first-row non-hydrogen atoms) or which energy and gradient evaluations can be tackled by each method at

present’*

In blue (in brackets) actualization of this estimations to 2009 computational available resources.

These results must be taken into account due to the dependence of the calculation with the size of the basis set used.

ConclusionsConclusions

� Overall, the main conclusion that emerges from this talk is that it is possible to simulate different

properties of interest for chemists (biologists, pharmaceutics, physics…) applying computational chemistry, with reasonable cost and saving much experimental time in the laboratory.

� The key point is to select which is the best method to be used depending on the computational resources available and the desired accuracy.

Thank you for your attention!!!

QUESTIONS????

David Tur, PhD

Scientific Applications expert

[email protected]

Introduction to Computational ChemistryIntroduction to Computational Chemistry

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