Chemistry 700
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Chemistry 700. Lectures. Resources. Grant and Richards, Foresman and Frisch, Exploring Chemistry with Electronic Structure Methods (Gaussian Inc., 1996) Cramer, Jensen, Ostlund and Szabo, Modern Quantum Chemistry (McGraw-Hill, 1982). Why is one interested in computational chemistry?.

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Chemistry 700

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Chemistry 700

Lectures


Resources

  • Grant and Richards,

  • Foresman and Frisch, Exploring Chemistry with Electronic Structure Methods (Gaussian Inc., 1996)

  • Cramer,

  • Jensen,

  • Ostlund and Szabo, Modern Quantum Chemistry (McGraw-Hill, 1982)


Why is one interested in computational chemistry?

  • • experiments are expensive, and often indirect.

    • structure and property prediction is of great value.

  • We need to know where the electrons are to be able to predict how eg. light will affect them or whether they are ready to create bonds with other atoms/molecules.

  • Possible applications include:

  • 1. drug design.

  • 2. development of new materials.


  • There exist various approximation levels, the major being:

  • Molecular Dynamics – called also Molecular Mechanics – treats atoms as classical

  • objects, with interactions described by predetermined potentials, usually fitted

  • to some analytical functions. Major applications are to perform geometry

  • optimization and eg. study docking (how a small drug molecule can bind to a

  • molecular macromolecule). This approach allows to study systems consisting

  • of thousands of atoms but its quality is limited by the choice of the force

  • field/potential.

  • Ab-initio Theory starts from fundamental equations of quantum theory and works

  • is up from there. Since strict analytical formula exists for energies and other

  • system properties, many various properties can be computed, including MD

  • potentials and interaction with light or magnetic field. However, full quantum-

  • mechanical treatment is expensive and the systems studied are severely limited

  • in the size to tens or hundreds of atoms. This course is focused mostly on this

  • method.


Schrödinger Equation

  • H is the quantum mechanical Hamiltonian for the system (an operator containing derivatives)

  • E is the energy of the system

  •  is the wavefunction (contains everything we are allowed to know about the system)

  • ||2 is the probability distribution of the particles


Hamiltonian for a Molecule

  • Kinetic energy of the electrons

  • Kinetic energy of the nuclei

  • Electrostatic interaction between the electrons and the nuclei

  • Electrostatic interaction between the electrons

  • Electrostatic interaction between the nuclei


Solving the Schrödinger Equation

  • Analytic solutions can be obtained only for very simple systems

  • Particle in a box, harmonic oscillator, hydrogen atom can be solved exactly

  • Need to make approximations so that molecules can be treated

  • Approximations are a trade off between ease of computation and accuracy of the result


Expectation Values

  • for every measurable property, we can construct an operator

  • repeated measurements will give an average value of the operator

  • the average value or expectation value of an operator can be calculated by:


Variational Theorem

  • the expectation value of the Hamiltonian is the variational energy

  • the variational energy is an upper bound to the lowest energy of the system

  • any approximate wavefunction will yield an energy higher than the ground state energy

  • parameters in an approximate wavefunction can be varied to minimize the Evar

  • this yields a better estimate of the ground state energy and a better approximation to the wavefunction


Born-Oppenheimer Approximation

  • The nuclei are much heavier than the electrons and move more slowly than the electrons

  • In the Born-Oppenheimer approximation, we freeze the nuclear positions, Rnuc, and calculate the electronic wavefunction, el(rel;Rnuc) and energy E(Rnuc)

  • E(Rnuc) is the potential energy surface of the molecule (i.e. the energy as a function of the geometry)

  • on this potential energy surface, we can treat the motion of the nuclei classically or quantum mechanically


Born-Oppenheimer Approximation

  • freeze the nuclear positions (nuclear kinetic energy is zero in the electronic Hamiltonian)

  • calculate the electronic wavefunction and energy

  • E depends on the nuclear positions through the nuclear-electron attraction and nuclear-nuclear repulsion terms

  • E = 0 corresponds to all particles at infinite separation


Nuclear motion on the Born-Oppenheimer surface

  • Classical treatment of the nuclei (e,g. classical trajectories)

  • Quantum treatment of the nuclei (e.g. molecular vibrations)


Hartree Approximation

  • Assume that a many electron wavefunction can be written as a product of one electron functions

  • If we use the variational energy, solving the many electron Schrödinger equation is reduced to solving a series of one electron Schrödinger equations

  • each electron interacts with the average distribution of the other electrons


Hartree-Fock Approximation

  • the Pauli principle requires that a wavefunction for electrons must change sign when any two electrons are permuted

  • the Hartree-product wavefunction must be antisymmetrized

  • can be done by writing the wavefunction as a determinant


Spin Orbitals

  • each spin orbital I describes the distribution of one electron

  • in a Hartree-Fock wavefunction, each electron must be in a different spin orbital (or else the determinant is zero)

  • an electron has both space and spin coordinates

  • an electron can be alpha spin (, , spin up) or beta spin (, , spin up)

  • each spatial orbital can be combined with an alpha or beta spin component to form a spin orbital

  • thus, at most two electrons can be in each spatial orbital


Fock Equation

  • take the Hartree-Fock wavefunction

  • put it into the variational energy expression

  • minimize the energy with respect to changes in the orbitals

  • yields the Fock equation


Fock Equation

  • the Fock operator is an effective one electron Hamiltonian for an orbital 

  •  is the orbital energy

  • each orbital  sees the average distribution of all the other electrons

  • finding a many electron wavefunction is reduced to finding a series of one electron orbitals


Fock Operator

  • kinetic energy operator

  • nuclear-electron attraction operator


Fock Operator

  • Coulomb operator (electron-electron repulsion)

  • exchange operator (purely quantum mechanical -arises from the fact that the wavefunction must switch sign when you exchange to electrons)


Solving the Fock Equations

  • obtain an initial guess for all the orbitals i

  • use the current I to construct a new Fock operator

  • solve the Fock equations for a new set of I

  • if the new I are different from the old I, go back to step 2.


Hartree-Fock Orbitals

  • for atoms, the Hartree-Fock orbitals can be computed numerically

  • the ‘s resemble the shapes of the hydrogen orbitals

  • s, p, d orbitals

  • radial part somewhat different, because of interaction with the other electrons (e.g. electrostatic repulsion and exchange interaction with other electrons)


Hartree-Fock Orbitals

  • for homonuclear diatomic molecules, the Hartree-Fock orbitals can also be computed numerically (but with much more difficulty)

  • the  ‘s resemble the shapes of the H2+ orbitals

  • , , bonding and anti-bonding orbitals


LCAO Approximation

  • numerical solutions for the Hartree-Fock orbitals only practical for atoms and diatomics

  • diatomic orbitals resemble linear combinations of atomic orbitals

  • e.g. sigma bond in H2

     1sA + 1sB

  • for polyatomics, approximate the molecular orbital by a linear combination of atomic orbitals (LCAO)


Basis Functions

  • ’s are called basis functions

  • usually centered on atoms

  • can be more general and more flexible than atomic orbitals

  • larger number of well chosen basis functions yields more accurate approximations to the molecular orbitals


Roothaan-Hall Equations

  • choose a suitable set of basis functions

  • plug into the variational expression for the energy

  • find the coefficients for each orbital that minimizes the variational energy


Roothaan-Hall Equations

  • basis set expansion leads to a matrix form of the Fock equations

    FCi = iSCi

  • F – Fock matrix

  • Ci – column vector of the molecular orbital coefficients

  • I – orbital energy

  • S – overlap matrix


Fock matrix and Overlap matrix

  • Fock matrix

  • overlap matrix


Intergrals for the Fock matrix

  • Fock matrix involves one electron integrals of kinetic and nuclear-electron attraction operators and two electron integrals of 1/r

  • one electron integrals are fairly easy and few in number (only N2)

  • two electron integrals are much harder and much more numerous (N4)


Solving the Roothaan-Hall Equations

  • choose a basis set

  • calculate all the one and two electron integrals

  • obtain an initial guess for all the molecular orbital coefficients Ci

  • use the current Ci to construct a new Fock matrix

  • solve FCi = iSCi for a new set of Ci

  • if the new Ci are different from the old Ci, go back to step 4.


Solving the Roothaan-Hall Equations

  • also known as the self consistent field (SCF) equations, since each orbital depends on all the other orbitals, and they are adjusted until they are all converged

  • calculating all two electron integrals is a major bottleneck, because they are difficult (6 dimensional integrals) and very numerous (formally N4)

  • iterative solution may be difficult to converge

  • formation of the Fock matrix in each cycle is costly, since it involves all N4 two electron integrals


Summary

  • start with the Schrödinger equation

  • use the variational energy

  • Born-Oppenheimer approximation

  • Hartree-Fock approximation

  • LCAO approximation


Ab initio methods

1. The Hartree-Fock method (HF)


The Hartree-Fock method

We want to solve the electronic Schrödinger equation:

For this, we need to make some approximations

These will lead to the Hartree-Fock method (which is

the simplest ab initio method)


The Hartree-Fock method

Approximation 1:

Decompose Y into a combination of molecular orbitals (MOs)

MO: one-electron wavefunction (fn)

However, this is not a good wavefunction, as

wavefunctions need to be antisymmetric: swapping the

coordinates of two electrons should lead to sign change

Good wavefunction:


The Hartree-Fock method

The antisymmetry of the wavefunction can be achieved by

constructing the wavefunction as a Slater Determinant:

fi is a “spinorbital”: contains also the spin of the electron


The Hartree-Fock method

Approximation 2:

The Hartree-Fock wavefunction consists

of a single Slater Determinant

(This implies that the electron-electron repulsion

is only included as an average effect => the

Hartree-Fock method neglects electron correlation)


LCAO: Linear Combination of Atomic Orbitals

AO or basis function

MO

expansion coefficients

The Hartree-Fock method

Approximation 3:

The MOs fi are written as linear combinations of

pre-defined one-electron functions (basis functions or AOs)


How to obtain the optimal coefficients cmi?

The Hartree-Fock method

  • The Hartree-Fock wavefunction is a single Slater Determinant

  • The MOs in the Slater Determinant are expressed as

    linear combinations of atomic orbitals

  • The exact form of the wavefunction depends on the

    coefficients cmi

  • The Hartree-Fock method aims to find the

    optimal wavefunction


“The energy calculated from an approximation to the true

wavefunction will always be greater than the true energy”

The Hartree-Fock method

Variation Principle

So, just find the coefficients cmi that give the lowest energy!

This leads to the Hartree-Fock equations, which can be

solved by the Self-Consistent Field (SCF) method


The Hartree-Fock method

Approximations leading to the Hartree-Fock method

  • Start with the electronic Schrödinger equation

  • Decompose Y into a combination of MOs => antisymmetry

    imposed by using Slater Determinant wavefunctions

  • Wavefunction consists of a single Slater Determinant

  • MOs are linear combinations of AOs (which are predefined)

  • Variation principle to find optimal coefficients


The Hartree-Fock method

The main weakness of Hartree Fock is that it neglects

electron correlation

In HF theory: each electron moves in an average field

of all the other electrons. Instantaneous electron-electron

repulsions are ignored

Electron correlation: correlation between the spatial

positions of electrons due to Coulomb repulsion

- always attractive!

Post-HF methods include electron correlation


Ab initio methods

Post-HF methods

Hartree-Fock

Møller-Plesset

Perturbation theory

(MP2, MP3, MP4,…)

Multiconfigurational

SCF (MCSCF)

Coupled Cluster

(CCSD, CCSDT, …)

Configuration

Interaction (CI)


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