Download Molecules Molecular Bonding

Molecules
Atoms can combine in various ways to form stable molecules. Atoms pair up into
molecules when there is an energy gain (I.e. their energy is lowered; the atoms
become more stable).
Two views:
1. A molecule is a stable arrangements of a set of nuclei and electrons. The exact
arrangement is determined by the electromagnetic interactions and quantum
mechanics.
“Set up the Schrödinger equation for all particles (nuclei + electrons) and solve.”
2. A molecule is a stable structure formed by two or more atoms.
“Atoms in a molecule mostly preserve their identity.”
Molecular bonding affects the outermost shell of electrons only (“the valence
electrons”); often in quite dramatic ways. Here we typically speak of molecular
orbitals instead of distinct atomic orbitals.
Lower lying core-orbitals/subshells are almost exactly like those in the atom.
Molecular Bonding - Overview
A balance between attractive and repulsive forces between atoms.
Attractive Forces:
•Coulomb forces between oppositely charged ions.
•Covalent bonds (courtesy of quantum mechanics).
•Coulomb interactions between dipoles.
•Induced dipole-dipole interactions.
RAB
Repulsive Forces:
RAB
•Coulomb forces between non-opposite charged ions.
•Covalent antibonds (“antibonding molecular orbitals”).
•Repulsion between nuclei.
•Exchange Force/Pauli Exclusion Principle.
The balance between attractive and repulsive
potential often creates a net-potential with an energy
minimum.
The energy minimum defines the classical (!)
equilibrium separation or bond length. The depth of
the potential well defines the classical (!) molecular
binding energy or dissociation energy.
(! see vibrational quantization, zero-point energy for the
more accurate quantum mechanical definitions).
Molecules - Ionic Bonding
Ionic bonding results, when the Coulomb attraction between oppositely charged ions is larger
than the energy required to form ions from the neutral elements.
This occurs when small ionization energies in one atom are paired with large electron affinities in
another atom.
Ionization:
Na + 5.1 eV # Na+ + eCl + e- # Cl- + 3.8 eV
" Ionized elements.
+1.3 eV
" Neutral elements.
0 eV
Repulsive forces at short distances due to:
(1) Repulsion between the nuclear charges;
less shielding of the ZNa and ZCl with
decreasing R. (2) Exchange forces - more
electrons compete for less space.
-3.6 eV
The Simplest Molecule - H2+
Two hydrogen nuclei (Z=1) and one electron.
Time independent Schrödinger Equation:
The Born-Oppenheimer Approximation - H2+
“Electrons move much faster than the nuclei. From the
timescale perspective of the electrons, the nuclei remain
essentially fixed. From the timescale perspective of the nuclei,
the electronic motion is averaged out.”
This allows us to separate electronic and nuclear motion
into two separate quantum problems:
Starting with the time-independent Schrödinger equation from the previous slide:
Assuming static nuclei, separation gives rise to an electronic Schrödinger equation:
Can solve this with nuclear
positions
given.
Obviously,
different solutions for different
nuclear positions.
Molecular potential energy curve/surface:
… and a nuclear Schrödinger equation:
Describes vibrational
and rotational motion
of the molecule
H2+ Correlation Diagram
Interpolation between two extremes:
RHH=0: He+
combined atom limit (Z=2).
RHH=$: H+ + H
separated atom limit (Z=1).
RHH
Energies at limits given by Bohr-equation for
one-electron atoms:
2 2
n=4: -3.4 eV
n=3: -6.0 eV
En= -13.6 eV (Z /n )
n=2: -3.4 eV (8%)
n=2: -13.6 eV (4%)
n=1: -13.6 eV (2%)
Note: This diagram
only
shows
the
electronic energy. The
Coulomb
repulsion
between the two nuclei
is not included.
n=1: -54.4 eV (1%)
RHH / bohrs
Combined Atom Limit
He+
Towards
separated Atom Limit
H+ + H
H2+ Potential Energy Curves
Potential energy curve for the two lowest
electronic states of H2+
with nuclear repulsion included.
Correlation Diagram
without nuclear repulsion
&-
&+
&&+
H2+ - Molecular Orbitals
&+ = &1s,A + &1s,B
&- = &1s,A - &1s,B
&+
Higher electron
density between atoms.
Lower electron
density between atoms.
Lower energy (more stable), because
the electrons are more located where
the nuclear potential is attractive
(negative).
Higher energy
(less stable)
Bonding molecular orbital
Antibonding molecular
orbital.
H2-Molecule - Two electrons !
We need to make sure that our two-electron eigenfunction is anti-symmetric w.r.t.
electron exchange. This is achieved by combining a symmetric space-eigenfunction
with an antisymmetric spin-eigenfunction ' or vice versa. Remember from earlier
lecture that triplet’s (“parallel spins”) have symmetric and singlets “(antiparallel spins”)
have antisymmetric spin functions.
[ &+(1)&-(2) + &-(1)&+(2) ]S 'A(1,2) ! singlet
[ &+(1)&-(2) - &-(1)&+(2) ]A 'S(1,2) ! triplet
[ &+(1)&+(2) + &+(1)&+(2) ]S 'A(1,2) ! singlet
With both electrons in the same one-electron eigenfunction &+,
an antisymmetric space function can not be constructed; so a
triplet state is not possible for this configuration.