Lecture 2: Hartree Fock

Question 1

What are operators (functions)?

Answer 1

They act upon the wave function and return observable properties of the system

Question 2

What is H in the Schrödinger equation?

Answer 2

Hamiltonian operator

Question 3

What is E in the Schrödinger equation?

Answer 3

The system energy

Question 4

What is the Hamiltonian operators made up of?

Answer 4

The sum of:
-electrons KE
-Nuclei KE
-electron-nuclei PE
-electron-electron PE
-nuclei-nuclei PE
- other terms

Question 5

What can the Hamiltonian operator be solved mathematically for?

Answer 5

A hydrogen atom (1 proton, 1 electron)
H2+ molecule (2 protons, 1 electron)

Question 6

Why do we approximate the Hamiltonian operator for larger systems?

Answer 6

Because we can’t solve it mathematically

Question 7

What does orthonormal mean?

Answer 7

The unit vectors are perpendicular to each other

Question 8

What does linear combination of atomic orbitals mean?

Answer 8

The bond orbitals are a sum of atomic orbital contributions
“Schrödinger atom”

Question 9

What are the two spins?

Answer 9

Alpha=+ ℏ/2 (up)
beta= -ℏ/2 (down)

Question 10

What is the Pauli exclusion principle?

Answer 10

No two electrons can have the same set of quantum numbers

Question 11

How many electrons does each spatial MO hold?

Answer 11

Two electrons

Question 12

For a system of 2N electrons how many MOs will there be?

Answer 12

N

Question 13

What can a wave function be split into?

Answer 13

The spatial and spin parts

Question 14

What is the Hartree product made up of?

Answer 14

The products of one-electron eigenfunctions

Question 15

In Hartree Fock, what are nuclei treated as?

Answer 15

Stationary points

Question 16

In Hartree Fock theory, what are electrons treated as?

Answer 16

Wave functions

Question 17

In Hartree Fock theory, what do you do to the electrons?

Answer 17

Integrate them over all space which accounts for interaction of electrons in an average way as a field

Question 18

How is the Hartree Fock Hamiltonian different to regular Hamiltonian?

Answer 18

-only considers electronic parts
-discards nucleus motion
-nucleus-nucleus interaction is constant

Question 19

What does the first integral for Hartree Fock represent?

Answer 19

Electron kinetic energy and interaction with nucleus

Question 20

What does the second integral for Hartree Fock represent?
What is it a combination of?

Answer 20

Electron-electron interactions
Combo of coulomb integral and exchange integral

Question 21

In Hartree Fock second integral, what is the coulomb integral?

Answer 21

The potential energy of electron i due to charge density or electron j
(Two electrons in same bonding orbital)

Question 22

In Hartree Fock second integral, what is the exchange integral?

Answer 22

Potential energy of electron i due to the non-indistinguishable nature of electron j
(2 Electrons in separate bonding orbitals)

Question 23

How do we use HF in comp chem?

Answer 23

-For each electron, treat rest as an average field (self consistent field theory)
-It’s an iterative procedure until it is converged

Question 24

Limitations of Hartree Fock

Answer 24

-Ignores all electron correlation due to 1-electron nature
-the accuracy of the HF is completely dependent on the basis set- how do we choose the right one?
-there’s lots of 2 electron/4 electron integrals

Question 25

What is a good analogy for HF?

Answer 25

-An orchestra all guessing the note R to play
-they all play their best guess, the sound is averaged, and then played back to them
-the musicians then adjust the note they play to be closer to the averaged note
-this continues until they converge to the same note