1. Introduction to Molecular Orbitals
Chemistry 6440 / 7440
Introduction to
Molecular Orbitals

2. Resources Grant and Richards, Chapter 2
Foresman and Frisch, Exploring Chemistry with Electronic Structure Methods (Gaussian Inc., 1996)
Cramer, Chapter 4
Jensen, Chapter 3
Leach, Chapter 2
Ostlund and Szabo, Modern Quantum Chemistry (McGraw-Hill, 1982)

3. Potential Energy Surfaces
molecular mechanics uses empirical functions for the interaction of atoms in molecules to calculate potential energy surfaces
these interactions are due to the behavior of the electrons and nuclei
electrons are too small and too light to be described by classical mechanics
electrons need to be described by quantum mechanics
accurate potential energy surfaces for molecules can be calculated using modern electronic structure methods

4. 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 (as probability distribution, ||2 needs to be continuous, single valued and integrate to 1)

5. 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

6. 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

7. 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:

8. 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

9. 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

10. 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

11. 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)

12. 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

13. Hartree-Fock Approximation
the Pauli principle requires that a wavefunction for electrons must change sign when any two electrons are permuted
since |(1,2)|2=|(2,1)|2, (1,2)=(2,1) (minus sign for fermions)
the Hartree-product wavefunction must be antisymmetrized
can be done by writing the wavefunction as a determinant
determinants change sign when any two columns are switched

14. 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 down)
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

15. Fock Equation take the Hartree-Fock wavefunction
put it into the variational energy expression
minimize the energy with respect to changes in the orbitals while keeping the orbitals orthonormal
yields the Fock equation

16. 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

17. Fock Operator kinetic energy operator
nuclear-electron attraction operator

18. 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)

19. 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.

20. 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)

21. 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

22. 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)

23. 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

24. 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

25. Roothaan-Hall Equations
basis set expansion leads to a matrix form of the Fock equations
F Ci = i S Ci
F – Fock matrix
Ci – column vector of the molecular orbital coefficients
I – orbital energy
S – overlap matrix

26. Fock matrix and Overlap matrix

27. 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)

28. 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 F Ci = i S Ci for a new set of Ci
if the new Ci are different from the old Ci, go back to step 4.

29. 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

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