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Methods and applications in Quantum Chemistry,
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Methods and applications in Quantum Chemistry,
Life Science and Drug Design software
I.
II.
III.
IV.
V.
Introduction
Theoretical Chemistry
Computational Chemistry Methods
From theory to practice: Software at CESCA
Drug Design software
Introduction
“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
I.
II.
III.
IV.
V.
Introduction
Theoretical Chemistry
Computational Chemistry Methods
From theory to practice: Software at CESCA
Drug Design software
Theoretical Chemistry
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
Theoretical Chemistry
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
Theoretical Chemistry
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
Theoretical Chemistry
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
Theoretical Chemistry
Theoretical Chemistry
Computational Chemistry methods
Theoretical Chemistry
Computational Chemistry methods
Theoretical Chemistry
Computational Chemistry methods
Theoretical Chemistry
Computational Chemistry methods
I.
II.
III.
IV.
V.
Introduction
Theoretical Chemistry
Computational Chemistry Methods
From theory to practice: Software at CESCA
Drug Design software
Computational Chemistry methods
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.
Computational Chemistry methods
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.
Computational Chemistry methods
Quantum methods
Classical methods
Molecular Mechanics
Semi-empirical
methods
GGA
DFT
Wavefunction
based methods
HF&MCSCF
LDA
LSDA
SR
hybrid-GGA
MR
meta-GGA
Hybrid-meta-GGA
Post-HF
MP2
Coupled
Cluster
FCI
CASPT2
MRCI
Computational Chemistry methods
Introduction
Molecular mechanics
Quantum chemistry methods
I. Wave function based methods
Hartree-Fock
Post-HF Methods
II. Semi-empirical methods
III. DFT
Basis sets
Computational chemistry methods in solid state
Conclusions
Computational Chemistry methods
Introduction
Molecular mechanics
Quantum chemistry methods
I. Wave function based methods
Hartree-Fock
Post-HF Methods
II. Semi-empirical methods
III. DFT
Basis sets
Computational chemistry methods in solid state
Conclusions
Molecular Mechanics
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.
Computational Chemistry methods
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 state
6. Conclusions
Molecular Mechanics
With this assumptions, the Mechanics equation can be simply written as:
E=EB+EA+ED+ENB
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 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.
Molecular Mechanics
General form of the Molecular Mechanics equations:
Molecular Mechanics
*Picture from the Wikipedia: Molecular Modelling
Molecular Mechanics
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…)
Introduction to Computational Chemistry
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 state
6. Conclusions
Quantum Chemistry Methods
Quantum methods
Classical methods
Molecular Mechanics
Semi-empirical
methods
GGA
DFT
Wavefunction
based methods
HF&MCSCF
LDA
LSDA
SR
hybrid-GGA
MR
meta-GGA
Hybrid-meta-GGA
Post-HF
MP2
Coupled
Cluster
FCI
CASPT2
MRCI
Quantum 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 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:
∂Ψ ∧
ih
=HΨ
∂t
where H is the Hamiltonian and Ψ is the wavefunction associated with the state
of the system.
Quantum Chemistry Methods
The main problem now is the solution of the electronic Schrödinger
equation.
The exact solution to this equation is not known (apart from
monoelectronic systems), numerical methods must be used to solve it.
Quantum chemistry addresses the solution of this problem in different
ways, depending on the mathematical approaches used.
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 Methods
The Schrödinger model is based on the six postulates of quantum mechanics:
1.
2.
3.
4.
Associated with any particle moving in a conservative force field is a wavefunction Ψ ( x, t ) ,
which contains all information that can be known about the system.
For every observable in classical mechanics, a linear Hermitian operator is defined
When measuring the observable associated with the operator A in Â Ψ = aΨ , the only values
that will ever be observed are the eigenvalues a which satisfy Â Ψ = aΨ
The average value of the observable corresponding to operator is given by:
∫ Ψ ÂΨdr
=
∫ Ψ Ψdr
*
Â
*
5.
The wavefunction evolves in time according to the time-dependent Schrödinger equation:
∧
H Ψ ( r , t ) = ih
6.
∂Ψ
∂t
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 antisymmetry
principle.
Quantum Chemistry Methods
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.
Wave function based 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 methods
Brief mathematical exposition of HF theory:
1-Wavefunction written as a single determinant
Ψ = det (φ1 , φ2 ...φN )
2-The electronic Hamiltonian can be written as:
∧
H el = ∑ h ( i ) + ∑ v ( i, j )
i
i< j
where h(i) and ν(i,j) are the one and two electrons operator defined as:
Z
1
h(i) = − ∇i2 − ∑ A
2
A riA
v ( i, j ) =
1
rij
3-The electronic energy of the system is given by :
∧
Eel = Ψ H el Ψ
4-The resulting HF equations from minimization of the energy:
∧
f ( x1 ) χ i ( x1 ) = ε i χ i ( x1 )
where εi is the energy value associated with orbital χi and f is defined as
∧
∧
∧
∧
f ( x1 ) = h ( x1 ) + ∑ J j ( x1 ) − K j ( x1 )
j
Wave function based methods
∧
∧
∧
∧
f ( x1 ) = h ( x1 ) + ∑ J j ( x1 ) − K j ( x1 )
j
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:
∧
J j ( x1 ) χ i ( x1 ) = χ i ( x2 ) ∫
χ ( x2 )
r12
2
dx2 = χ j ( x2 )
1
χ j ( x2 ) χi ( x1 )
r12
and Ki third operator term of the equation,
defined as:
∧
K j ( x1 ) χ i ( x1 ) = χ j ( x1 ) ∫
χ *j ( x2 ) χi ( x2 )
r12
is the exchange operator that is
dx2 = χ j ( x2 )
1
χ i ( x2 ) χ i ( x1 )
r12
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.
Wave function based methods
Greatly simplified algorithmic flowchart illustrating the Hartree-Fock Method.
*Flowchart from Wikipedia: Hartree-Fock
Wave function based methods
The error in the determination of the total energy due to the use of the HartreeFock method is the so-called correlation energy:
Ecorr = Eo − EHF
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.
Wave function based 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.
Wave function based methods
Quantum methods
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.
Wavefunction
based methods
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 configurational
SCF) the electronic state of the system is
approximated by a multi-configuration
wavefunction
HF&MCSCF
SR
MR
Post-HF
MP2
Coupled
Cluster
FCI
CASPT2
MRCI
Wave function based methods
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.
Wave function based methods
Single reference Post-HF methods:
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.
Wave function based methods
Single reference Post-HF methods:
Coupled Cluster (CC) theory: is another numerical technique used for describing manybody 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…
Wave function based methods
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.
FCI/
STO-3G
FCI/
3-21G
FCI/
6-31G*
FCI/
6311G(2df)
EXACT
CCSD(T)
CCSD(T)/
STO-3G
CCSD(T)/
3-21G
CCSD(T)/
6-31G*
CCSD(T)/
6311G(2df)
CCSD(T)
CBS
CCSD
CCSD/
STO-3G
CCSD/
3-21G
CCSD/
6-31G*
CCSD/
6311G(2df)
CCSD
CBS
MP2
MP2/
STO-3G
MP2/
3-21G
MP2/
6-31G*
MP2/
6311G(2df)
MP2
CBS
HF
HF/
STO-3G
HF/
3-21G
HF/
6-31G*
HF/
6311G(2df)
HF
CBS
STO-3G
3-21G
6-31G*
6311G(2df)
FCI
···
*Head-Gordon, J. Phys. Chem. Vol. 100, No, 31, 1996
···
CBS
Wave function based methods
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 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 HartreeFock 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.
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
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
∧
∧
∧
H = F + V ext
where F is the sum of the kinetic energy of the electrons, and the electron-electron
Coulomb interaction
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:
∧
N
ρ0 ( r ) = Ψ 0 ρ Ψ 0 = ∫ ∏ dri Ψ 0 ( r1 , r2 , r3 ,K , rN )
2
i =2
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 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:
 1 2

−
∇
+
V
(
r
)
KS
 2
ψ i ( r ) = ε iψ i ( r )
where the Khon-Sham potential, VKS, has the
VKS (r ) = ∫
ρ (r ')
r−r '
dr ' + VXC (r ) + Vext (r )
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
Density Functional Theory
There are many different ways to approximate this functional VXC,
and to do this EXC is generally divided into two separate terms:
E XC [ ρ ] = E X [ ρ ] + EC [ 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)
Density Functional Theory
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:
HEAVEN (chemical accuracy)
rung 5
fully nonlocal
explicit dependence on unoccupied orbitals
rung 4
hybrid functionals
explicit dependence on occupied orbitals
rung 3
meta-GGAs
explicit dependence on kinetic energy density
rung 2
GGAs
explicit dependence on density gradients
rung 1
LDA
local density only
EARTH (Hartree theory)
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”.
Introduction to Computational Chemistry
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 state
6. Conclusions
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.
n −1 −ξ r
φξSTO
e Yl ,m (θ , ϕ )
, n ,l , m ( r , θ , ϕ ) = Nr
Gaussian type basis sets (GTO): STO are approximated as linear combination of
gaussian type orbitals.
2
(2 n − n −1) −ξ r
e Yl , m (θ , ϕ )
φξGTO
, n ,l , m ( r , θ , ϕ ) = Nr
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
Basis 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- prefix
e.g.: cc-pVDZ, aug-cc-pVQZ…
Introduction to Computational Chemistry
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 state
6. Conclusions
Computational 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
Introduction to Computational Chemistry
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 state
6. Conclusions
Conclusions in Computational Chemistry Methods
Method Type
Advantages
Disadvantages
Best for
Molecular
Mechanics
uses classical
physics
relies on force-field
with embedded
empirical parameters
Computationally least
intensive - fast and
useful with limited
computer resources
can be used for
molecules as large as
enzymes
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
large systems
(thousands of atoms)
systems or processes
with no breaking or forming
of bonds
Semi-Empirical
uses quantum
physics and
experimentally derived
empirical parameters
uses approximation
extensively
less demanding
computationally than ab
initio methods
capable of calculating
transition states and
excited states
requires
experimental data
(or data from ab
initio) for parameters
less rigorous than
ab initio) methods
medium-sized systems
(hundreds of atoms)
systems involving
electronic transitions
Ab Initio
uses quantum
physics
mathematically
rigorous, no empirical
parameters
uses approximation
extensively
useful for a broad
range of systems
does not depend on
experimental data
capable of calculating
transition states and
excited states
computationally
expensive
small systems
systems involving
electronic transitions
molecules or systems
without available
experimental data ("new"
chemistry)
systems requiring
rigorous accuracy
Conclusions in Computational Chemistry Methods
Method
Current
computational
dependence on
molecular size, M
Current estimate of
maximum feasible
molecular size
FCI
Factorial
2 atoms (15)
CCSD(T)
M7
8-12 atoms (20-50)
CCSD
M6
10-15 atoms (100)
MP2
M5
25-50 atoms (50-150)
HF, KS-DFT
M2-M3
50-200 atoms (50-300)
‘Current scalings of electronic
structure methods with molecular
size, M and estimates of the
maximum molecular sizes (in terms
of numbers of first-row nonhydrogen 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.
*Head-Gordon, J. Phys. Chem. Vol. 100, No, 31, 1996
Conclusions in Computational Chemistry Methods
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.
I.
II.
III.
IV.
V.
Introduction
Theoretical Chemistry
Computational Chemistry Methods
From theory to practice: Software at CESCA
Drug Design software
From theory to practice: available software
Package
Altix
CP4000
NovaScale
Altix UV
License
Allowed users
ADF
x
x
x
x
Commercial
Academic
Amber
x
x
x
x*
Commercial
Academic
AutoDock
x
x
x
x
GNU GPL
All
CPMD
x
x
x
Free (non-profit)
Academic
DL_POLY
x
x
x
x*
Free academic lic.
Academic
Gamess (US)
x
x
x
x
Free academic lic.
Academic
Gaussian
x
x
x
x
Commercial
All
GROMACS
x
x
x
x*
Free
All
Jaguar
x
x
x*
Commercial
All(1)
MOPAC
x
x
x*
Free non-profit
Academic
NAMD
x
x
x*
Free
All
NWChem
x
x
x
Free academic lic.
Academic
POLYRATE
x
x
x
Free
All
x
x*
Commercial
Acad.(3 Univ.)
x
x
Free academic lic.
Academic
TURBOMOLE
SIESTA
* Coming Soon!!!!
x
x
x
Molecular mechanics
Uses classical physics
Relies on force-field with embedded empirical parameters
Advantages
Disadvantages
Computationally least
intensive - fast and
useful with limited
computer resources
Can be used for
molecules as large as
enzymes
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
Best for
Large systems
(thousands of atoms)
Systems or
processes with no
breaking or forming of
bonds
Molecular Mechanics
Package
Available versions
Characteristics
Amber
7/8/9/10
Bio. /Par. Dec./
DL_POLY
2.1x/3.0x
Bio&nonBio/Par.Dc
Gromacs
3.3.1
Bio./Par.Dec
2.5/2.6/2.7b1
Bio.
MM only
NAMD
General Purpose
Gaussian
Gamess (US)
Jaguar
NWChem
TURBOMOLE
G03C2/G03D2/G03E1/G09A2/G09B1
2004/2005/2006/2008/2009/2010
Bio./Serial Code
6.5/7.0/7.5
4.7/5.0/5.1.1/6.0
6.0
Availabe trough SDF
Macromodel
(Schrödinger 2009 suite)
Prime
(Schrödinger 2009 suite)
Bio: Designed mainly for biological systems; Par. Dec: parallelized using atom decomposition
Bio
Semiempirical
Uses quantum physics and experimentally derived empirical parameters
Uses approximation extensively
Advantages
Less demanding
computationally than ab
initio methods
Capable of calculating
transition states and
excited states
Disadvantages
Requires experimental
data (or data from ab initio)
for parameters
Less rigorous than ab
initio) methods
Best for
Medium-sized
systems (hundreds of
atoms)
Systems involving
electronic transitions
Semiempirical
Package
Available versions
Available methods
Semiempirical only
MOPAC
2007
AM1, PM3, MNDO,
MNDO/d, RM1 i PM6
General Purpose
Gaussian
Gamess (US)
G03C2/G03D2/G03E1/G09A2/G09B1
CNDO, INDO,
MINDO/3,MNDO,
AM1, PM3, PM6
2004/2005/2006/2008/2009/2010
MNDO, AM1,PM3
Ab initio
Uses quantum physics
Mathematically rigorous, no empirical parameters
Uses approximation extensively
Advantages
Useful for a broad
range of systems
Does not depend on
experimental data
Capable of calculating
transition states and
excited states
Disadvantages
Computationally
expensive
Best for
Small systems
Systems involving
electronic transitions
Molecules or systems
without available
experimental data
("new" chemistry)
Systems requiring
rigorous accuracy
HF and Post-HF
Package
Gamess (US)
Gaussian
Jaguar
NWChem
Available methods
Parallel
HF (RHF, ROHF, UHF, GVB),
SCF,MCSCF,CASSCF, CI, MRCI
Coupled cluster (closed shells)
MP2,ROMP2,UMP2, MCQDPT(CASSCF-MRMP2)
MPI
HF(RHF, UHF, ROHF), SCF,MCSCF,CASSCF,
RASSCF,MP2,MP3,MP4,MP5,CI,CID,CISD,CCD,
CCSD,CCISD,G1,G2,G2(MP2),G3,G3(MP2),
GVB-PP
OpenMP
HF(RHF, ROHF, UHF, GBV)
MP2, LMP2
GVB-RCI, GBV-LMP2, GVB-DFT
MPI
HF(RHF,ROHF,UHF),SCF,CASSCF,CCSD(T),CC
SDTQ, Plane wave, non-canonical MP2, MP3,MP4
MPI
Density Functional Theory
Package
Available versions
Parallel
20006.01/20007.01/2008.01/2009.01/2010.01
MPI
CPMD
3.11.1/3.9.1/3.13.2
MPI
Jaguar
6.5/7.0/7.5
MPI
SIESTA
2.0.1/2.0.2/3.0-rc2
MPI
G03C2/G03D2/G03E1/G09A2/G09B1
OpenMP
2004/2005/2006/2008/2009/2010
DDI/MPI
4.7/5.0/5.1.1
MPI
DFT only
ADF
General Purpose
Gaussian
Gamess (US)
NWChem
I.
II.
III.
IV.
V.
Introduction
Theoretical Chemistry
Computational Chemistry Methods
From theory to practice: Software at CESCA
Drug Design software
Drug Design software
7 laboratories and 9 academic groups
(2)
(2)
Drug Design software
Used hours
Licenses
Ending at
Tripos Sybyl
1 wildcard (acad. + ind.)
30-12-2010
2005
1.118
2006
2.242
2007
4.525
2008
5.776
2009
8.349
9.000
8.000
Schrödinger
7.000
32 academic tokens
30-04-2010
6.000
15 industrial tokens
31-12-2011
5.000
4.000
3.000
2.000
1.000
0
Sybyl
Schrödinger
Tripos software
Sybyl 7.3
Sybyl (1)
Unity Base (1)
Network (1-5)
Legion (1)
1 Wildcard
LeapFrog*
Selector*
DiverseSolution*
AdvancedCOMFA*
CombilibMaker*
Schrödinger software
Schrödinger (Suite 2010) 34 tokens acad.
Macromodel (2)
LigPrep (1)
Liason (4)
MINTA (1)
QickProp (2)
Epik (1)
ConfGen (1)
SiteMap (1)
Jaguar (2)
Strike (1)
Prime (8)
Glide (5)
Phase (1) (5)
CombiGlide (1) (5)
Qsite (1) (4)
XP Visualizer (1)
Database:
Phase database: Asinex, Bionet, Enamine, LifeChem,
Maybridge, Specs and TimTec
How to apply for the Drug Design service
Academic applications:
Submitted through the Vicerrectorat de Recerca of your own institution
Prices for 2010 : (allows unlimited access to all the software)
Initial fee: 227 € (only paid once)
Annual fee: 1.533 €
Additional information: [email protected]
Methods and applications in Quantum Chemistry,
Life Science and Drug Design software
Thank you for your attention!!!
QUESTIONS????
David Tur, PhD
Scientific Applications expert
[email protected]