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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 methodApproximation 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 methodVariation 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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