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Chemistry 700

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Chemistry 700

Lectures

- 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.

- 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

- 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

- 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

- 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:

- 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

- 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

- 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

- Classical treatment of the nuclei (e,g. classical trajectories)
- Quantum treatment of the nuclei (e.g. molecular vibrations)

- 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

- 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

- 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

- 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

- 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

- kinetic energy operator
- nuclear-electron attraction 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)

- 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.

- 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)

- 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

- 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)

- ’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

- 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

- 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
- overlap 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)

- 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.

- 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

- start with the Schrödinger equation
- use the variational energy
- Born-Oppenheimer approximation
- Hartree-Fock approximation
- LCAO approximation

1. The Hartree-Fock method (HF)

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)

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 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

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

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 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”

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

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 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

Post-HF methods

Hartree-Fock

Møller-Plesset

Perturbation theory

(MP2, MP3, MP4,…)

Multiconfigurational

SCF (MCSCF)

Coupled Cluster

(CCSD, CCSDT, …)

Configuration

Interaction (CI)