Download Introduction to molecular structure – Part I

Introduction to molecular structure – Part I
Contents:
atomic orbitals - generalities
models used in developing the molecular structure
molecular orbital theory
application for hydrogen-ion molecule
diatomic molecules
examples of molecular orbital diagrams
Mathematical description of electrons
Schrödinger equation which for one electron system has the form:
 2   2   2  8 2m
 2  2  2 ( E  V )  0
2
x
y
z
h
The solution of this equation are themselves mathematical formulas, but
not differential equations. They describe the electrons as waves can be
represented graphically. These graphs are 3D pictures that show
electron density and are called orbitals or electron clouds.
e
Atomic orbitals
Waves added togehter can occur in two ways:
•constructive interference
When waves interact, they
don't reflect off each other-they combine. If the
amplitudes of the waves are
both positive or both negative,
the combined wave
will have a larger amplitude.
• destructive interference
If the waves have opposite
amplitudes, the resulting wave
will have a smaller amplitude.
first-wave
second-wave
sum of the waves
Types of atomic orbitals
• atomic orbitals: s, p, d, f,g
For example, in a simple lowest-energy state of hydrogen atom, the electrons
are most likely to be found within a sphere around the nucleus of an atom. In a
higher energy state, the shapes become lobes and rings, due to the interaction of
the quantum effects between the different atomic particles
z
z
+
+
x
1s orbital
Each p orbital has a node. The
orbitals lobe are labeled (+) and (-)
which referes to the signs of the
wave function .
y
x
2s orbital
y
+
-
-
+
2p orbitals
+
Again to Shrödinger equation ....
H   E
where  represents the many-electron wave-function
Possible to solve for one electron system, but not possible for molecules containing
two or more electrons  ...need for approximation
Principals models of molecular structure
1. Molecular orbital
theory
2. Valence bond theory
Formation of
chemical bonding
Valence Bond Theory
• Explains the structures of covalently bonded molecules
 ‘how’ bonding occurs
• Principles of VB Theory:
 Bonds form from overlapping atomic orbitals and electron pairs are
shared between two atoms
 Lone pairs of electrons are localized on one atom
Molecular Orbital (MO) Theory
• was developed in 1920’s
• explains the distributions and energy of electrons in molecules
• useful for describing properties of compounds
– Bond energies, electron cloud distribution and magnetic properties
• basic principles of MO Theory
– Atomic orbitals combine to form molecular orbitals
– Molecular orbitals have different energies depending on overlaping type :
• Bonding orbitals (lower energy than corresponding atomic orbitals AO)
• Nonbonding orbitals (same energy as corresponding AO)
• Antibonding orbitals (higher energy than corresponding AO)
Note:
Both theories are limited case, but Chapter 9 treat a more exact theory
Molecular orbital theory - hydrogen molecule–ion H2+
Hydrogen atom has only 1 electron situated on the s orbital
atomic orbitals
atomic orbitals
1s
1s
2 antibond
1 bond
and
molecular orbitals
molecular orbitals
An application: the hydrogen molecule-ion H2+
• the simplest system: two nuclei and one electron
• the equation which characterize this system is given by:
2
2
2
2

e
e
e
2
H 
 


2me
40rA 40rB 40 R
kinetic energy
of the electron
attraction of
the electron e
by A
attraction of
the electron
e by B
e
rA
A
Coordinates used to
specify the hamiltonian
for the hydrogen
molecule-ion
rB
R
repulsion
between
A and B
B
.... and the equation set of solution
the set of solution lead to
construction of the molecular
potential energy
the molecular potential energy
curves vary with internuclear
distance, R.
Bonding: the electronic density
transfer into the internuclear region:
lowering of energy
Antibonding: the electronic density
undergo to a small expansion around
nuclei increasing the energy
Bonding and antibonding of atomic orbitals molecular orbital formation
Atomic orbitals of the H2+ are added in the same way as the waves. The in-phase
addition of two 1s orbitals will form a molecular orbital with electron density
between two nuclei  bonding orbital. The out-of-phase addition of the same
type of orbitals will produce a molecular orbital with no electron density between
two nucleii, called node  antibonding orbital.
Bonding and antibonding of atomic orbitals for H2+ - interpretation
In an antibonding orbital,
the nuclei are attracted to
an accumulation of electron
density outside the
internuclear region
2 antibonding orbital
.
1 bonding orbital
In a bonding orbital, the
nuclei are attracted to the
accumulation of electron
density in the inter nuclear
region.
Mathematical system in bond formation
Theoretically, molecular orbitals can be formed by overlapping the wave function of
the atomic orbitals. This overlaping procedure is constructed by forming linear
combination of atomic orbitals, LCAO. Adition of the atomic orbitals give the
bonding MO:
e
A  c AA  cBB
bonding
B  c AA  cBB
antibonding
rA
A
rB
R
A, B denote molecular orbitals
Atkins notations
A, B wave function for the atomic orbital of atom
having the nucleus in A, B respectively;
cA, cB electrons coeficients which show what part of
the electron was involved in the bond formation
B
Liniar combination of atomic orbitals LCAO
For species with many atoms, LCAO will have the form:
   crr
fomation of basis set
r
In principle, we should use an infinite basis set for a precise recreation of the
molecular orbital, but in practice only a finite basis set is used. In Atkins is assume
that the members of the basis set are real and each one is normalize to 1.
The optimum value of the coefficients are found by applying the variation principle,
which means to solve the secular equation:
c (H
r
rs
 ESrs )  0
r
See section 6.10 from Atkins.
Hrs – matrix element of the hamiltonian
Srs – overlap matrix element
The energies for LCAO-MO:
The energy expression are derived in Atkins, chapter 8.3
j'  k '
E  E1s 

40 R 1  S
e2
j'  k '
E  E1s 

40 R 1  S
e2
Positive integral which is
attributed to the total
Coloumbic potential energy
arising from the interaction of
electron clound around A with
nucelus in B and B with
nucelus in A, respectively
j’
Interaction of the overlap
electronic density with nucleus A
k’
A
A B
B
A
B
The molecular orbital energy level diagram of
H+2 in LCAO approximation
Energy
E-
2
A
B
E1s
E1s
E+
1
atomic orbitals
molecular orbitals
The bonding orbital is of lower
energy than its atomic orbitals
and the antibonding orbital is of
higher energy
Diagram asymmetry: an antibonding
orbital is more antibonding than a
bonding orbital is bonding
Cal. with LCAO: R=2.5a0
E=170kJ/mol
Exp: R=2.0a0 ; E=255kJ/mol
How we will use this theory?
1. we used the atomic orbitals from the hydrogen molecule-ion to construct
the wave –functions for many electrons atoms, by adding them one by one into the
calculations
2. we will use the orbitals of the hydrogen molecule-ion to build the MO for diatomic
molecules
Diatomic molecules – angular momentum
• the spatial symmetries of atomic orbitals and the number of each symmetry type
are determined by the angular momentum of the electron.
•the angular momentum vector for molecule case will lie along the bond axis. The
quantum number in this case is denoted by  and it is analogous to the quantum
number m in the atomic case. The possible values for  are: 0, 1, 2, 3...
correspondingly, the allowed values of the angular momentum about the
internuclear axis are 0, ±1 (h/2p), ±2(h/2p), etc., or in general, ±l(h/2p). Thus when 
is different from zero, each energy level is doubly degenerate corresponding to the
two possible directions for the component  along the bond axis.
The molecular orbitals are labelled according to the values of the quantum
number l . When  = 0, they are called s orbitals; when  = 1, p orbitals; when  =
2, d orbitals, etc.
the molecule as a whole rotates in
space and the nuclei contribute to the
total angular momentum of the system.
The nuclei and the electrons of a
diatomic molecule can rotate around
both axes which are perpendicular to
the bond axis
Overlaping of 2p orbitals
2px, 2py, 2pz: the direction of orbitals along the axes.
1. Overlaping of 2px orbitals:
Constructive interference from the 2px orbitals
Destructive interference for the 2px orbitals
2. Overlaping of 2py /2pz orbitals
Constructive interference from the 2py orbitals
Destructive interference from the 2py orbitals
Diatomic molecules
Basic principle of molecular orbital formation:
-Linear combination of atomic orbitals that have the same symetry species, i.e.span
the same irreducible representation, within the molecular point group. As was
shown in chapter 5.16 ( see M.H. Lecture), only orbitals of the same symetry
species may have a nonzero overlap (S0) and hence contribute to the bonding
Criteria for selecting the correct bonding
Group theory provides techniques for selecting the atomic
orbitals that may contribute to bonding, but others types of
arguments must be used to decide whether these orbitals do in
fact contribute and to what extent two important criteria:
1. Atomic orbitals must be neither too diffuse or too compact
(their overlaping is too diffuse to have a high significance)
2. The energies of the orbitals should be similar.
Filling of Orbitals
The valence eletctrons of the atomic orbitals are used to fill in the molecular orbitals.
The rules of filling in the MO are the same as filling in the AO
Diatomic molecules – periodic table where only s and p orbitals are important
1. Homonuclear molecules – the same atomic species:ex: Li2, O2, N2...
2. Heteronuclear molecules – different atomic species: ex: CO, HF...
The HOMO and LUMO state
HOMO and LUMO are acronyms for Highest Occupied Molecular
Orbitals and Lowest Unoccupied Molecular Orbitals respectively.
The energy level difference of the two (HOMO-LUMO) can (sometimes)
serve as a measure of the excitability of the molecule: the smaller the
energy, the easier it will be excited.
The MO diagram for Li2
The MO diagram for O2
2 valence electrons
12 valence eletrons
*2p
*2p
2p
2p
LUMO
2p
*2s
2s
Energy
Energy
*2p
2p
AO
2p
2p
*2s
HOMO
2s
MO
2p
2p
2s
2s
2s
2s
AO
LUMO
HOMO
*2p
AO
MO
AO
MO for CO molecule
C has 4 valence electrons
O has 6 valence electrons
*2p
The CO molecule
has 10 valence
eletrons
LUMO
*2p
2p
2p
2p
2p
*2s
2s
2s
Carbon AO
MO
2s
Oxygen AO
HOMO
..... To be continued by Staffan in two weeks