1.  2k spin-orbitals: i, i=1,...,2k
RHF and UHF formalisms
Given a set of k orthonormal spatial orbitals (MO) {i}, i=1,...k
 2k spin-orbitals: i, i=1,...,2k
unrestricted MOs  unrestricted wave-function
restricted MOs  restricted wave-function
electrons with alpha and beta spin are constrained to be described by the same spatial wavefunction
Restricted wave-function for Li atom
But: K1s()2s( )≠0 and K1s()2s()=0
1s() and 1s() electrons will experience different potentials so that it will be more convenient to describe the two kind of electrons by different wave-functions
Unrestricted wave-function for Li atom
usually, the two sets of spatial orbitals use the same basis set

2. UHF wave-functions are not eigenfunctions of S2 operator !!!
 spin contamination
|2> - exact doublet state
|4> - exact quartet state
|6> - exact sextet state
-approximately a singlet
- approximately a doublet
For an UHF wave-function, the expectation value of S2 is:
State
S2 -> S(S+1)
singlet
0.00
doublet
0.75
triplet
3.75
where:
spin projection procedures (Gaussian)
generate singlet eigenstates from a UHF determinat by applying projection operators that interchange the spins of the electrons

3. Comparison of the R(O)HF and UHF formalisms
Spin-orbitals for pairs of electrons with α and β spin are constrained to have the same spatial dependence
Spin-orbitals for electrons with α and β spins have different spatial parts
Wavefunction is an eigenfuction of the S2 operator
Wavefunction is not an eigenfuction of the S2 operator;
Spin-contamination
Not suitable for the calculation of spin-dependent properties
Yields qualitatively correct spin densities
EUHF ≤ ER(O)HF
Different density matrices for the two sets of electrons; their sum gives the electronic density, while their difference gives the spin density

4. For a closed-shell system in RHF formalism, the total energy and molecular orbital energies are given by (see Szabo and Ostlund, pag.83 and chapter 4 in D.B. Cook, Handbook of Computational Quantum Chemistry):
Each occupied spin-orbital i contributes a term Hi to the energy
 Each unique pair of electrons (irrespective of their spins) in spatial orbitals i and j contributes the term Jij to the energy
Each unique pair of electrons with parallel spins in spatial orbitals i and j contributes the term –Kij to the energy
Or (over the spin orbitals):
Each occupied spin-orbital i contributes a term Hi to the energy and every unique pair of occupied spin orbitals i and j contributes a term Jij-Kij to the energy
Examples: 3 2 1 a) b) c) d) e)
a) E=2H1+J11
b) E=2H1+H2+J11+2J12-K12
c) E=H1+H2+J12-K12
d) E=H1+H2+J12
e) E=H1+2H2+H3+2J12+J22+J13+2J23-K12-K13-K23

5. Hartree-Fock-Roothaan Equations
LCAO-MO
i=1,2,...,K
{μ} – a set of known functions
The more complete set {μ}, the more accurate representation of the exact MO, the more exact the eigenfunctions of the Fock operator
The problem of calculating HF MO  the problem of calculating the set cμi LCAO coefficients
matrix equation for the cμi coefficients
Multiplying by μ*(r1) on the left and integrating we get:
- Fock matrix (KxK Hermitian matrix)
- overlap matrix (KxK Hermitian matrix) 
- Roothaan equations

6. the matrix of the expansion coefficients
More compactly:
FC=SC
where
the matrix of the expansion coefficients
(its columns describe the molecular orbitals)
The requirement that the molecular orbitals be orthonormal in the LCAO
approximation demands that:
The problem of finding the molecular orbitals {i} and orbital energies i involves solving the matrix equation FC=SC.
For this, we need an explicit expression for the Fock matrix

7. Charge density
For a closed shell molecule, described by a single determinant wave-function
The integral of this charge density is just the total number of electrons:
Inserting the molecular orbital expansion
into the expression for the charge density we get:

8. Where:
- elements of the density matrix
The integral of (r) is
By means of the last equation, the electronic charge distribution may be decomposed into contributions associated with the various basis functions of the LCAO expansion.
Off-diagonal elements
the electronic population of the atomic overlap distribution 
give an indication of contributions to chemical binding when  and 
centered on different atoms
Diagonal elements
- the net electronic charge residing in orbital 

9. FC=SC.
S is Hermitian and positive definite => exist the S1/2 and S-1/2 matrices with the properties: S-1/2S1/2=1 and S1/2S1/2=S
Trick: multiply the HFR matrix equation from the left by S-1/2, put S-1/2 . S1/2 in front of C from the left-hand side and write S in the right-hand side as S1/2S1/2:
=>
Notations:
Thus:
Computational effort
Time nedeed for solving the SCF equations scakes as M4 (M- # of basis functions)
Accuracy
Greather M → more accurate MOs and MO’s energies
Complete basis set limit (HF limit): M→∞ (never reached in practice
the best result that can be obtained based on a single determinantal wavefunction

10. Population analysis
- allocate the electrons in the molecule in a fractional manner, among the various parts of the molecule (atoms, bonds, basis functions) 
→ partial atomic charges, spin density distribution, bond orders, localized MOs                
-  Mulliken population analysis (MPA) – strongly depends on the particular basis set used
Substituting the basis set expansion we get:
(PS)μμ can be interpreted as the charge associated with the basis function φμ
Basis set functions are normalized  Sμμ=1
Pμμ - number of electrons associated with a particular BF
- net population of φμ
Qμ = 2PμSμ (μ≠)      overlap population
- associated with two basis functions
 Total electronic charge in a molecule consists of two parts:
first term is associated with individual BF
second term is associated with pairs of BF

11. - gross population for φμ
the net charge associated with the atom A; P is the net population of 
total overlap population between atoms A and B
where Qμ = 2PμSμ is the overlap population between two basis functions

12. Formaldehyde (CH2O) (aqueous solution: formol)
an important precursor to many chemical compounds, especially for polymers.
gas at room temperature which converts readily to a variety of derivatives.
annual world production: more than 21 million tonnes.
intermediate in the oxidation (or combustion) of methane as well as other carbon compounds
(forest fires, automobile exhaust, tobacco smoke).
can be produced in the atmosphere by the action of sunlight and oxygen on atmospheric methane
and other hydrocarbons (part of smog).
the first polyatomic organic molecule detected in the interstellar medium
(Zuckerman, B.; Buhl, D.; Palmer, P.; Snyder, L. E., Observation of interstellar formaldehyde,
Astrophys. J. 160 (1970) 485) → used to map out kinematic features of dark clouds
mechanism of formation: hydrogenation of CO ice:
H + CO → HCO HCO + H → H2CO (low reactivity in gas phase)
Due to its widespread use, toxicity and volatility, exposure to formaldehyde is very important for human health. It is used to make the hard pill coatings that dissolve slowly and deliver a more complete dosage.
Is it carcinogen? 12

13. Formaldehyde Basis functions: Mulliken population analysis
#P RHF/STO-3G scf(conventional) Iop(3/33=6) Extralinks=L316 Noraff Symm=Noint Iop(3/33=1) pop(full)
Basis functions: 13

14. The summation is over occupied molecular orbitals
Example 14

15. 15

16. = sum over the line (or column) corresponding to the C(1s) basis function
= sum over the line (or column) corresponding to the O(2px) basis function 16

17. Atomic populations (AP) Total atomic charges (Q=Z-AP)

19. Molecular orbitals of formaldehyde (RHF/STO-3G)

20. Excited state symmetry
Formaldehyde
The symmetry of the first excited state
of formaldehyde (as a result of HOMO-LUMO transition)