Quantum and Statistical Mechanics of Molecules and Solids

What is the Schrodinger equation?

• Psi = Wavefunction

• E = Energy

• H = Hamiltonian

• T = Kinetic Energy

• V = Potential Energy

What information does the wavefunction contain?

ALL information about the system

What are the conditions about the wavefunction?

• Normalised

• Finite

• Single-valued

• Continious

What is wavefunction² proportional to?

Probability of finding particle at x

→ BORN INTERPRETATION

What atom do you need to use to get solution of Schrodinger equation?

Hydrogen atom

What are the ways to show electron spin?

What quantum numbers are needed to specify state of an electron?

n, l , m_l , m_s

What is the Pauli principle?

“If the labels of two electrons are exchanged, the total wavefunction must change sign”

How do you satisfy the Pauli principle?

Why are atomic units (au) used?

Calculations become clumsy so set physical calculations = 1

What are Born-Oppenheimer approximations?

Allows us to separate nuclear and electronic motion

What does the full wavefunction depend on for Born-Oppenheimer approximations?

Depends on nuclear and electronic coordinates, R and r

→ CAN BE SEPARATED TO A GOOD APPROXIMATIONS (see picture)

What is the electronic Schrodinger equation?

V_NN is a constant for fixed nuclear geometry

→ Can now find internal energy,U, for any arrangement of nuclei: Potential energy surface

How do you work out the Hamiltonian for a H₂ molecule with nuclei A & B and electrons 1 & 2 in Born-Oppenheimer approx?

How do you express the full wavefunction that describes a molecule is a function of coordinates of all electrons?

• Psi_e has a value for every possible arrangement of electrons

→ High amplitude for likely arrangements

→ Low amplitude for unlikely arrangements

→ Zero amplitude for impossible arrangements

What is the 3-body problem so what do you use instead?

When you cannot exactly solve the molecular Schrodinger equation with many-electron atoms

→ Use orbital approximation instead

What is the orbital approximation?

• Each molecular orbital (psi_i) describes motion of ONE electron

What is the Hartree-Fock method?

Standard approach for calculating molecular orbitals

• Set of 1-electron Schrodinger equations, where each electron feels average repulsion from all others

How do you use Hartree-Fock method equation effectively?

Effective potential each electron feels depends on all others, so solve equations iteratively until they are “Self-consistent”

→ Self consistent field method

What is the Variation theorem?

If psi_approx is an approximate wavefunction for the ground state of a molecule, then E_approx is greater or exact to E_exact

→ E_approx is the energy corresponding to psi_approx

What is the linear combination of atomic orbital approach?

• c_ij = Determines how much atomic orbital (kai_j) contributes to the molecular orbital (psi_i)

How do you minimise the Hartree-Fock energy?

Find best choice of “MO coefficients” c_ij

• Solve the equation by solving Roothan-Hall equations

What happens if you use the LCAO approach to H₂?

Forms MOs as linear combinations of s-orbitals on each nucleus X_A and X_B

What does bonding MO create?

Constructive interference

What does anti-bonding MO create?

Destructive interference

What does the variation theorem tell us about psi₊ and psi_-?

That psi₊ is better description of ground state of H₂

How can you calculate HF energy for any bond length?

Draw out diagram

→ HF is qualitatively correct, but quantitatively wrong

• Potential energy curve

What is a basis set?

Set of known functions used to build unknown functions in quantum calculation

What is the basis set for the pi-system of conjugated buatiene p_z orbitals describing all the pi-MOs

• In phase orbitals show positive numbers

How do you show the radial part of a Slater-type orbital (STO)

• Zeta ζ = Controls how quickly function decays: Higher values for higher effective nuclear charge

• r = Radius

What are Slater-type orbitals?

Family of cuntions used as atomic-orbital-like building blocks in LCAO basis set

Why is the exponential part important here?

• Matches true hydrogen-like orbtial, forms a cusp at the nucleus

• Tricky to wortk with and awkward

Why is Gaussian-type orbital (GTO) used instead of Slater-type orbitals (STO)

• Has no cusp at nucleus

• Easy to integrate

How do you improve the shape of Gaussians at small and large r?

Use several Gaussians linked together

What is STO-3G?

• Using 3 Gaussian type orbitals to mimic one slater type orbital

  

• Acts as one Slater basis function when used in fixed combination

What is a minimal basis set?

Smallest possible basis set uses single basis function for each occupied orbital on each free atom

What is the minimal basis set for hydrogen?

1s orbital

What is the minimal basis set for carbon?

1s², 2s², 3 × 2p

• Or 1s orbital, 2s orbital and full set of three 2p orbitals

What is a disadvantage of using minimal basis sets such as STO-3G?

Lacks flexibility to describe bonding in real molecules

How do you make minimal basis sets “more flexible”?

Use 2 or 3 basis functions per occupied orbital

• Termed as double zeta (DZ) or triple zeta (TZ)

• e.g = 2 × 1s or 2 × (1s, 2s, 3×2p)

  

• Basis functions of different sizes allow MOs to expand or contract according to their environment

What are the disadvantages of using double zeta (DZ) or triple zeta (TZ) basis sets?

• More coefficients to evalulate = Slower calculations and more integrals/RAM requirements

• Chemistry takes place mainly in valence orbitals so:

→ Core electrons do not change in response to environment

→ Adding flexibility for core electrons is inefficient

What are split-valence basis sets?

• Gives more flexibility to valence electrons than core

• Solution to using DZ and TZ

What do DZ and split-valence basis sets allow MOs to do?

• Allow MOs to expand or contract, but not to change shape from atomic orbitals

What function do you use to describe multiple bonding, lone-pairs,hypervalency

Polarisation functions

What are polarisation functions?

Functions with higher angular momentum than required in free atom

What do polarisation functions allow?

Allow proper description of electrons in molecular environment

What are the common polarised split-valence basis sets?

What is the variation theorem for the ground state?

Lower energy = Better wavefunction