Download Quantum Chemical Simulations and Descriptors

Quantum Chemical Simulations
and Descriptors
Dr. Antonio Chana, Dr. Mosè Casalegno
Classical Mechanics: basics
• It models real-world objects as point particles, objects with negligible size.
• The motion of a point particle is characterized by a small number of
parameters: its position, mass, and the forces applied to it.
• The 3 Newton's laws (1687) provide the relationships between the forces
acting on a body and its motion. They are the basis of classical mechanics.
At the macroscopic level, energy varies continuously:
motion is a continuous sequence of stationary states.
For instance, the turning points of the pendulum are
determined by its energy.
Limits of Classical Mechanics
Classical mechanics can describe the motion of
macroscopic objects with great accuracy – for
example, the analysis of projectile motion –,
but...
...does not to explain some physical phenomena
observed since the 19th century, such as:
-1838: the cathode ray emissions,
-1839: the photoelectric effect,
-1859: the black body radiation.
Example 1: photoelectric effect
Simple experiment: shining a metal surface with ultraviolet light trigger the emission of electrons.
1) the emission did not occur below a specific threshold
frequency
2) increasing intensity had no effect at all on the
average amount of energy that each ejected electron
carried away
Einstein realized that light itself is quantized: all the light
of a particular frequency comes in little bullets of the
same energy, called photons.
1_only photons with a certain amount of energy
(frequency) set the electrons free.
2_increasing intensity only increases the number of
electrons. Increasing frequency also increases the kinetic
energy of free electrons
Example 2: atomic spectra
When a chemical element is irradiated over a range of frequencies, an atomic spectra
is obtained:
The spectral lines are not continuous: every chemical element only absorbs
electromagnetic radiations at particular wavelengths.
This can only be explained considering that at the atomic level, energy is quantized
Example 3: electron diffraction
A simple question: suppose one electron is emitted
from a source, directed toward a double-slit gate. What
happens ?
Classical expectation: since the electron is a particle it
should “choose” one path and hit the screen.
Former experiments showed a diffraction pattern,
like that generated by a light beam...
...electrons, like photons, exhibit wave-like as well
as particle-like properties
The birth of Quantum Mechanics
Quantum mechanics was developed with the aim to describe non classical
phenomena, such as:
z
Quantization of energy
z
Wave-particle duality
z
Uncertainty principle (Heisenberg, 1926)‫‏‬
In quantum mechanics:
Measurable properties are called observables (energy, position,
momentum, etc.)‫‏‬
Observables have no definite value, but each value is associated to
a probability.
The quantum state of atoms and molecules is represented by a
wave function.
The Schrödinger Equation
The wave function is a scalar function that describes the state of the quantum system.
Starting from the wave function the “expectation” value of the observable F is given
by:
Here, the integration is performed over the set of variables, “<..>” indicates the
expectation value, while “^” refers to the operator associated to F.
Similarly, the Schrödinger equation allows the calculation of the energy:
where:
This example refers to a particle of mass m, subject to a potential V, defined in the
coordinate space (r)‫‏‬
Solutions for the Harmonic Oscillator
The harmonic oscillator provides a simple example of energy quantization: a diatomic
molecule vibrates somewhat like two masses connected by a spring.
The Schrödinger equation is:
where
Solutions are Hermite polynomials:
where
Hydrogen wave functions: atomic
orbitals
On going from the harmonic oscillator to single atoms, the solutions to the Schrödinger
equation become more complex. So complex that the exact solution is known only for
hydrogen-like atoms (polar coordinates):
The integer numbers: n, l, ml are called quantum numbers.
Molecular Hamiltonian
If we want to calculate molecular properties, we should explicitly consider that the
total molecular energy is the sum of different contributions:
Since the energies associated to each term differ by orders of magnitude, we can
consider them separately.
As far as electronic properties are considered, the Hamiltonian is:
This refers to a molecule composed by N electrons and M nuclei.
Since analytical solutions are not known, numerical methods have to be used.
Approaches in Quantum Chemistry
Different computational methods are available to solve the Schrödinger equation.
The goal of such methods is to determine the ground-state wave function and the
ground-state energy.
The Hartree-Fock method
Hartree-Fock is an iterative, fixed-point method: the positions of the nuclei in the
space are fixed:
Assumption: the exact wave function can be expressed as a Slater determinant:
The wave function is a LCAO: linear combination of atomic orbitals, whose
coefficients are recursively optimized, until convergence is reached.
Basis Sets
Quantum chemical calculations are typically performed
within a finite set of basis functions.
Wavefunctions under consideration are all represented as
vectors, the components of which correspond to coefficients
in a linear combination of the basis functions in the basis set
used
We select a finite number of orbitals to calculate which will
be starting point for the linear combinations.
Slater-type orbitals STO:
Denoted by STO-nG (n is the number of primitive gaussians to combine)
Gaussian-type orbitals (GTO)
Denoted as X-YZg (X primitive gaussians upon each core atomic orbital basis function,
Y and Z are also combinations for the valence orbitals in Y and Z).
* Polarization (adding a p orbital over an s)
** Polarization over light atoms (H)
+ Diffuse functions (tail functions to big distances from the orbital center)
++ Diffuse functions for light atoms
HF LCAO for hydrogen: an example
The hydrogen molecule is one of the simplest to illustrate how LCAO works.
The LCAO combine giving 2 molecular orbitals: one occupied, one unoccupied
Drawbacks of HF method
The HF energy is an upper bound to the exact energy (variational principle):
In fact, the Coulomb electron-electron repulsion, known as electronic correlation term,
is not explicitly taken into account.
So called Post HF methods attempt to account for this effect,
like Møller-Plesset, where perturbation theory is used to add
correlation.
Density Funcitonal Theory: DFT
The main objective of density functional theory is to replace the many-body
electronic wave function with the electronic density as the basic quantity.
Whereas the many-body wavefunction is dependent on 3N variables, three spatial
variables for each of the N electrons, the density is only a function of three
variables and is a simpler quantity to deal with both conceptually and practically.
In 1964, Hohenberg and Sham demonstrated that the electronic
energy is a functional of the density.
The exact functional is not known, several have been proposed
over the years.
The Kohn-Sham equations (1965) can be solved iteratively, like the Hartree -Fock
ones. The solution is the density.
Advantages: reduced computational cost. DFT is often applied to solids, or large
molecules.
Quantum Monte-Carlo method
The Monte Carlo method is a computational algorithm that relies on repeated
random sampling.
Consider the square area on the right, and the circle
inscribed within. Now, scatter some small objects (for
example, grains of rice or sand) throughout the square.
If the objects are scattered uniformly, then, the
proportion of objects within the circle is proportional to
its area....
...Monte Carlo provides a simple way to compute
areas, or integrals !
Since computing the molecular electronic energy requires the estimation of integrals,
Monte Carlo can be used to this purpose.
VMC: variational Monte-Carlo, in this case we sample a wave function by randomly
changing the electrons coordinates.
DMC: diffusion Monte-Carlo attempts find an exact solution to the Schrödinger
equation.
Limits of the ab initio methods
There are limits to the size of the system for
calculating the electronic structure
Post-HF
HF
DFT
<10 heavy atoms
<30 heavy atoms
Calculation time ~N4
The higher the complexity the more difficult to obtain a proper solution
Schroedinger equation can be approximated by means the
use experimental parameters: SEMIEMPIRICAL
METHODS
Semi-empirical methods: introductory
remarks
Semi-empirical methods are the methods which treat the electron system
quantum mechanically (we solve the Schroedinger equation), but the true
Hamiltonian is replaced with a model one; the parameters of the model
Hamiltonian are fitted to reproduce the reference data (usually experiments) of
Let’s call the “real semi-empirical methods” (as opposed to the tight-binding
methods) the methods which are close in spirit to the Hartree-Fock formalism
but have fitted parameters, such as MINDO, MNDO, AM1, PM3, etc. (MOPAC)
the experiments)
Appeared in the 60th-70th as an attempt to find a compromise between
the computational efficiency and accuracy
Nowadays not that much in use (sometimes applied to big organic
molecules and similar problems in organic chemistry)
The localized orbitals are used as basis functions
The methods normally do not make use of the Slater-Koster approximation;
the one-electron integrals are assumed to be proportional to the overlap integrals
which are evaluated through the recursion formulas.
Approximations used in semiempirical
methods
Only valence electrons are taken into account;
In many versions the overlap integrals in the secular (Hartree-Fock Roothaan)
equation are ignored. Thus, we solve the ordinary eigen value problem
instead of the generalized eigenvalue problem:
HΨ = ESΨ
HΨ = EΨ
We completely neglect all three- and four-center
integrals
Practically all matrix elements are approximated by
analytical functions of interatomic separation and
atom environment. Parameters are chosen to
reproduce the characteristics of reference systems.
Core-core (nucleus-nucleus) Coulombic repulsion is
replaced with a parametized function.
How well these methods work?
Calculations of the cohesive (binding) energy for a carbon dimer by various
semiempirical methods incorporated into the MOPAC package
E b = E ( 2 ) − E (1)
E exp = − 6 . 0 eV
Energy
AM1(R)
AM1(U)
MINDO (3R)
MINDO (3U)
PM3 (R)
PM3 (U)
C(1)
-118.96
-120.20
-117.61
-118.85
-108.60
-110.35
C(2)
-244.85
-247.08
-247.23
-247.44
-226.08
-227.85
Eb
-6.94
-6.68
-12.01
-9.74
-8.89
-7.15
R stands for (RHF) U for (UHF)
Which approximation should we trust? Note that we have up to 50 fitted parameters.
Quality of a model:
Quality=
predictive power (transferability)
complexity
Alternative: tight-binding models which have much less parameters, physically are
more transparent and possess more predictive power
Quantum Chemical descriptors in
QSAR
Quantum chemical descriptors include some observables that can be computed by
means of QM approaches.
The most important are:
•Energy
•Dipole moment
•Molecular polarizability
•Molecular electron density
•HOMO-LUMO energies
•Atomic charges
The QM level of theory is the highest...
...at the same time, computing these descriptors is computationally expansive.
Dipole moment
The dipole moment is a measure of the bond polarity in a molecule.
The distribution of the electron density is not uniform in case different elements are
combined.
The dipole moment is defined as:
Where Q is the charge magnitude and R the separation distance.
The dipole moment is expressed in Debye.
Molecular polarizability
Polarizability is the relative tendency of a charge distribution, like the electron cloud
of an atom or molecule to be distorted from its normal shape by an external electric
field.
Application of an external electric filed “induces” a dipole moment:
The dipole moment is proportional to the intensity of the electric field applied, and
the polarizability, α. The polarizability is a tensor:
Polarizability values have been shown to be related to the hydrophobicity and other
biological activities.
Molecular electron density
The electron density is the measure of the probability of an electron being present
at a specific location.
The image shows the electrostatic potential for benzyne.
The more red an area is, the higher the electron density and
the more blue an area is, the lower the electron density. Here
we see the triple bond as a region of high electron density
(red). As a result of the non-linear triple bond, benzyne is
highly reactive.
•The electron density provides indication about the most reactive sites in a
compound.
•It also provides useful means for the detailed characterization of donor-acceptor
interactions.
•This descriptor has been employed in QSAR studies to describe drug-receptor
interaction sites.
Molecular Electrostatic Potential
Derived from the molecular electron density appears the Molecular Electostatic
Potential MEP
ZA
p(r ' )
−∫
V (r ) = ∑
dr '
r − rA
r − r'
Nuclear Charge
1 n
V S = ∑ VS (ri )
n i =1
+
S
V =
−
S
V =
1
α
1
β
α
∑V
j =1
β
+
S
∑V
k =1
(rj )
−
S
(rk )
electronic density
1 n
Π = ∑ VS (ri ) − V S
n i =1
σ
2
tot
= σ +σ =
2
+
2
−
[
V
∑
α
1
α
j =1
σ +2σ −2
ν= 2 2
[σ tot ]
+
S
(rj ) − V
] + β ∑ [V
+ 2
S
1
β
k =1
−
S
(rk ) − V
]
− 2
S
HOMO-LUMO Energies
HOMO-LUMO energy is the difference between the energies of HOMO and LUMO
molecular orbitals.
LUMO: Lowest Unoccupied Molecular Orbital
HOMO: Highest Occupied Molecular Orbital
z
The transition between HOMO and LUMO is observed in some chemical reactions.
• The smaller the gap, the higher the reactivity of the compound.
• HOMO-LUMO generically describes the “reactivity” and can be responsible for high
toxicity.
Atomic charges
The partial atomic charges are due to the asymmetric distributions of electron in
chemical bonds
z
Charge based descriptors are widely
employed as chemical reactivity indices or
as a measure of weak intermolecular
interactions.
• The calculation of partial charges relies on
the Mulliken population analysis, and is
carried out also by means of semi-empirical
quantum methods.
Software for QM descriptors
Several software packages are available for quantum chemical simulations and the
calculation of the related properties:
z
Hartree-Fock: Gaussian, Gamess, Jaguar, MOLPro, Qchem, Turbomole
• DFT: Gaussian, Gamess, DFT++, ADF, Spartan
• Monte-Carlo: Casino, CHAMP, QMCmol, QWalk
The computational cost follows this hierarchy: DFT < Hartree-Fock << Monte-Carlo
Thank you !