Given a set of k orthonormal spatial orbitals (MO) {i}, i=1,...k

RHF and UHF formalisms

2k spin-orbitals: i, i=1,...,2k

K,i)()r()x(

)()r()x(

ii

ii 12

12

unrestricted MOs unrestricted wave-function

Restricted wave-function for Li atom

sssRHF 211

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 sssUHF 211

ijδαj

αi ij

βj

βi δ αβ

ijβj

αi S usually, the two sets of spatial orbitals use the same basis

set

restricted MOs restricted wave-function

electrons with alpha and beta spin are constrained to be described by the same spatial wavefunction

UHF wave-functions are not eigenfunctions of S2 operator !!! spin contamination

-approximately a singlet - approximately a doublet

...642 26

24

22

2 ccc

|2> - exact doublet state

|4> - exact quartet state

|6> - exact sextet state

For an UHF wave-function, the expectation value of S2 is:

N

i

N

jijexactUHFSNSS

222

1

222

NNNNS

exact

exactUHFSS 22

where:

spin projection procedures (Gaussian)

generate singlet eigenstates from a UHF determinat by applying projection operators that interchange the spins of the electrons

ijδαj

αi ij

βj

βi δ αβ

ijβj

αi S

State S2 -> S(S+1)

singlet 0.00

doublet 0.75

triplet 3.75

Comparison of the R(O)HF and UHF formalisms

R(O)HF UHF

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

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):

N/2

1i

N/2

1jijij

N/2

1ii )K(2JHE 2

N/2

1jijijii )K(2JH

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 energyOr (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:

a) b) c) d)

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

1

2

3

e)

Hartree-Fock-Roothaan Equations

LCAO-MO

K

ii c1

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

)()()( 111 rcrcrf iii matrix equation for the cμi coefficients

Multiplying by μ*(r1) on the left and integrating we get:

111*

1111* )()()()()( drrrcdrrrfrc iii

1111* )()()( drrrfrF

- Fock matrix (KxK Hermitian matrix)

111* )()( drrrS

- overlap matrix (KxK Hermitian matrix)

KicScF iii ,...,2,1,

- Roothaan equations

FC=SC

KKKK

K

K

ccc

ccc

ccc

C

...

............

...

...

21

22221

11211

K

...00

............

0...0

0...0

2

1

More compactly:

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:

ijji Scc

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

Charge density

N/2

a

2a(r)Φ2ρ(r)

For a closed shell molecule, described by a single determinant wave-function

Nρ(r)dr

μν

*νμμν

μν

*νμ

(r)(r)P

(r)(r)

ρ(r)

2/*

2/**

2/*

2

)()(2)()(2

N

aaa

N

aaa

N

aaa

cc

rcrcrr

K

ii c1

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:

Where:

2/

*2N

aaaccP - elements of the density matrix

The integral of (r) is

NP(r)dr(r)P(r)(r)Pρ(r)drμν μν

μνμν*νμμν

μν

*νμμν Sdr

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.

μνμνP S -the electronic population of the atomic overlap distribution

-give an indication of contributions to chemical binding when and

centered on different atoms

Off-diagonal elements

Diagonal elements

- the net electronic charge residing in orbital μμμμSP

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

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:=>

εCSSS

εCSSCSSFS1/21/21/2

1/21/21/21/2

Notations:

C'CS

F'SFS1/2

1/21/2

εC'C'F'

Thus:

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)PS()( trSPNdrr

Basis set functions are normalized Sμμ=1

Pμμ - number of electrons associated with a particular BF

- net population of φμ

 Qμ = 2PμSμ (μ≠)      overlap population

- associated with two basis functions

Total electronic charge in a molecule consists of two parts:

K K K

NQP

first term is associated with individual BF

second term is associated with pairs of BF

(PS)μμ can be interpreted as the charge associated with the basis function φμ

AatomofchargeMulliken

)()PS( AqMA

SPPq - gross population for φμ NqK

A

AnetA PZq

the net charge associated with the atom A; P is the net

population of

A B

AB Qq

total overlap population between atoms A and B

where Qμ = 2PμSμ is the overlap population between two basis functions

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?

Formaldehyde (CH2O) (aqueous solution: formol)

Mulliken population analysis

Formaldehyde

#P RHF/STO-3G scf(conventional) Iop(3/33=6) Extralinks=L316 Noraff Symm=Noint Iop(3/33=1) pop(full)

Basis functions:

The summation is over occupied molecular orbitals

Example

)cc...ccc2(cP 18581252115151

μic

oc

1iνiμiμν cc2P

NSPρ(r)droc

μ

oc

νμνμν

μνμνSP

μνS

= 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

Atomic populations (AP)1 O 8.186789

2 C 5.926642

3 H 0.9432854 H 0.943285

Total atomic charges (Q=Z-

AP)1 O -0.186789

2 C 0.073358

3 H 0.0567154 H 0.056715

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

Excited state symmetry

Formaldehyde

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