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 ultra-violet 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:

Quantization of energy

Wave-particle duality

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:

Similarly, the Schrödinger equation allows the calculation of the energy:

Here, the integration is performed over the set of variables, “<..>” indicates the expectation value, while “^” refers to the operator associated to F.

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.

Solutions are Hermite polynomials:

The Schrödinger equation is:

where

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 SetsQuantum 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

In fact, the Coulomb electron-electron repulsion, known as electronic correlation term, is not explicitly taken into account.

The HF energy is an upper bound to the exact energy (variational principle):

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.

There are limits to the size of the system for calculating the electronic structure

Post-HFHFDFT

<10 heavy atoms

<30 heavy atoms

Calculation time ~N4

Limits of the ab initio methods

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 are the methods which treat the electron systemquantum mechanically (we solve the Schroedinger equation), but the trueHamiltonian is replaced with a model one; the parameters of the modelHamiltonian are fitted to reproduce the reference data (usually experiments) of

Let’s call the “real semi-empirical methods” (as opposed to the tight-bindingmethods) the methods which are close in spirit to the Hartree-Fock formalismbut 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 betweenthe computational efficiency and accuracy

Nowadays not that much in use (sometimes applied to big organicmolecules 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.

Semi-empirical methods: introductory remarks

Approximations used in semiempiricalmethods

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:

Ψ=Ψ SH E Ψ=Ψ EH

We completely neglect all three- and four-center integralsPractically all matrix elements are approximated byanalytical functions of interatomic separation and atom environment. Parameters are chosen to reproduce the characteristics of reference systems.Core-core (nucleus-nucleus) Coulombic repulsion isreplaced 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

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

)1()2( EEE b −= eV0.6exp −=E 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:

Alternative: tight-binding models which have much less parameters, physically aremore transparent and possess more predictive power

predictive power (transferability)complexityQuality=

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:

Polarizability values have been shown to be related to the hydrophobicity and other biological activities.

The dipole moment is proportional to the intensity of the electric field applied, and the polarizability, α. The polarizability is a tensor:

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 PotentialDerived from the molecular electron density appears the Molecular Electostatic Potential MEP

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Nuclear Charge

∑=

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electronic density

∑=

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α 1)(1

jjSS rVV

∑=

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∑=

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HOMO-LUMO Energies

HOMO-LUMO energy is the difference between the energies of HOMO and LUMO molecular orbitals.

HOMO: Highest Occupied Molecular Orbital

LUMO: Lowest Unoccupied Molecular Orbital

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

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:

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 !