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Molecules
Chapter 10
November 09
Modern Physics
Outline
Bonding mechanisms
Molecular Rotation and Vibration
Molecular Spectra
Electron Sharing and the Covalent Bond
Molecular orbital picture
November 09
Modern Physics
Force between atoms
The attraction and bonding of atoms is
electrostatic in original - electrons and protons
attract each other.
The attraction is a result of a dynamical
correlation of the positions of electrons relative to
the nuclei and is subtle - different models are
used to understand the effect in different
systems.
November 09
Modern Physics
Attraction of electrons
p+
e-
e-
The far electron is closer to the repelling electron
than to the attracting proton so repelled.
e- p+
e-
The far electron is farther from the repelling
electron than from the attracting proton so
attracted.
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Modern Physics
Induced dipole moment
p+
e-
e-
If the near electron is in a fast circular orbit, the
time average force vanishes.
e-
p+
e-
If the near electron is on average pushed away
from the proton, the time average force is attractive
and the hydrogen atom has an induced (dipole)
polarization.
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Induced dipole interactions
p+
e-
e-
p+
Two neutral atoms can co-induce dipole resulting in
a net attraction.
The net force is in this case called the van der
Waals force.
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Field of an electric dipole
e-
p+
r
At distance r along the x-axis of a charge -e at r=0
and charge +e at r=d (dipole moment p=ed), the
electric field strength is inversely proportional to
the third power of distance from the dipole
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Van der Waals potential
e-
p+
r
If the dipole moment p induced on an atom is
proportional to the inducing electric field strength
(p=constant* E), the interaction energy between
two dipoles is inversely proportional to the 6th
power of distance - effectively rather short ranged.
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Interatomic effective potential
The total potential
energy includes the
repulsive energy of
interaction
(1/r2)between the
protons and attractive
induced dipole-dipole
energy (~1/r6) yielding
a minimum at a finite
“equilibrium
separation.”
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Modern Physics
Ionic dissociation
Bring a neutral sodium and a neutral chlorine atom
into contact. Then pull them apart.
The sodium may lose its relatively loosely bound 3s
outer electron to the chlorine which when neutral has
a closed shell minus one 2p5 outer electron
configuration. (The energy to remove an electron
from Na is 5.1 eV and the attachment energy or
electron affinity of chlorine is 3.7 eV.) The up-shot is
the pair of Na+ and Cl - ions.
The attractive force between such ions ~1/r2 at large
distances.
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Na-Cl
Total energy versus the nuclear separation for Na+ and Cl- ions.
The energy required to separate the NaCl molecule into neutral
atoms of Na and Cl is the dissociation energy, 4.2 eV.
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Electron Sharing
An ionic bond is formed when one atom essentially
steals an electron from a different atom, the two ions
then attracting each other.
A covalent bond is formed when electrons involved in
bonding are more equally shared.
In H2, the two electrons on
equal footing orbit both
protons and both tend to be
between the protons - more
of this later.
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Modern Physics
Carbon
The electron configuration in carbon (1s22s22p2)
favors formation of 4 covalent bonds with say
hydrogen to form tetrahedral methane.
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Hydrogen bonding
In some circumstances, hydrogen can donate an electron
to two high electron affinity atoms.
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Quantum excitations of
molecules
In its lowest energy state, a molecule is
comprised of nuclei and core electrons at “fixed”
equilibrium positions and a cloud of shared
electrons.
Electronic excitations involve promoting electrons
to excited states of the complex 3-d potential
associated with the nuclei . The minimum
electronic excitation energy is a few eV. At lower
available energies the electrons are frozen in the
ground state.
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Non-electronic excitations
Assuming the binding electrons are frozen in the
lowest energy state for nuclei of fixed relative position,
the entire molecule may still absorb energy internally
in collective (rigid) rotational motion.
With or without rotation present, the relative positions
of the nuclei may change with the binding electrons
constantly adjusting adiabatically to remain in the
instantaneous electronic ground state - nuclear
motion is slow compared to electronic motion.
Vibrational motions of the nuclear positions are
approximately harmonic about the (stable)
equilibrium positions.
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External and internal kinetic
energy
We can separate the kinetic energy associated with
the motion of the entire molecule from the internal
motion:
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Rotational kinetic energy
For rigid body rotational motion at fixed angular
frequency about direction n, the internal kinetic
energy may be written in terms of the moment of
inertia I and angular momentum L:
I and L
refer to n.
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Quantization of rotational
energy
The orientation of a free rigid body is described by
an angular wave function similar to that which
describes the angular position of an electron in a
spherically symmetric potential.
The angular momentum and energy a rigid body are
quantized.
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Diatomic molecule
Note that the mass in a molecule is dominated by the
pointlike nuclear masses and the rotational nuclear
energy is small compared to electronic rotational
energy:
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Example
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Diatomic molecule rotations
Allowed rotational energies of a diatomic
molecule
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Rotational transitions
Electromagnetic transitions between rotational states
correspond to emission and absorption of microwave
(IR) frequencies, not optical frequencies.
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Diatomic vibration
The potential energy of a diatomic molecule versus
atomic separation. Oscillation about the equilibrium
separation R0 has effective spring constant:
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Recall quantum oscillator
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Diatom vibrational states
Allowed vibrational energies
of a diatomic molecule,
where ω is the fundamental
frequency of vibration given
by ω = √K/ µ .
Note that the spacings
between adjacent vibrational
levels are equal.
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Vibrational data
Note that (ground state) vibrational frequencies are
larger than (low excitation) rotational frequencies.
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Nonlinear vibrations
Potential energy U(r) around R0 is harmonic, but rises sharply as
the atoms are brought closer together. The separation between
adjacent levels decreases with increasing energy.
Morse
potential
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Combined rotational-vibrational
spectrum
The rotation– vibration
levels for a typical
molecule. Note that the
vibrational levels are
separated by much larger
energies so that a
complete rotational
spectrum can be
associated with each
vibrational level.
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Transitions
(a) Absorptive transitions between the v =
0 and v = 1 vibrational states of a
diatomic molecule obey the selection rule
Δ l = ± 1 and fall into two sequences:
those for which Δ l = + 1 and those for
which Δ l = - 1. The transition energies
are given by Equation 11.14. (b)
Expected lines in the optical absorption
spectrum of a molecule. The lines on the
right side of center correspond to
transitions in which l changes by +1, and
the lines to the left of center correspond
to transitions for which l changes by -1.
These same lines appear in the emission
spectrum.
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HCL example
The absorption spectrum of the HCl molecule. Each line
is split into a doublet because chlorine has two isotopes,
35Cl and 37Cl, which have different nuclear masses.
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Electronic-rot-vibration spectrum
If an electron is excited out of the (molecular)
state (requires optical frequency excitation), the
spring constant, the equilibrium separation, and
the moment of inertia are slightly changed.
The excited molecule exhibits a slightly different
rotational-vibrational spectrum.
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Raman scattering
An incoming photon with energy E scatters from a molecule and
emerges with reduced energy E ‘. The energy lost by the photon
increases the rotational energy of the molecule in accordance with the
selection rule Δ l= 2. The energy loss translates into a change in
photon frequency, the Raman shift, that can be used to probe
molecular structure.
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Modern Physics
Flourescence
A photon with energy E is absorbed, in the process exciting a higher
vibrational state of the molecule. This excess energy is lost to
collisions with neighboring molecules. The molecule returns to its
original state in step 3 by emitting a photon with reduced energy E’.
In phosphorescence, the final transition is forbidden by selection
rules, resulting in delayed photon emission.
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Modern Physics
Simplest molecule
The hydrogen
molecular ion H2+. The
lone electron is
attracted to both
protons by the
electrostatic force
between opposite
charges. The
equilibrium separation
|R| of the protons in
H2+ is about 0.1 nm.
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H2+
For R=0, the electron is bound in a
hydrogen like orbital for Z=2. The
ground state electronic binding
energy is Z2=4 times that of
hydrogen or -54 eV.
For R=infinity, the electron is on
one or the other proton and the
energy is -13.6 eV.
The total energy is the sum of the
electronic energy and the repulsive
energy of interaction between the
protons.
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Electronic energy spectrum
versus separation
[1 bohr (a0) = 0.529 Å.]
At R = 0, the levels are those of the
united atom (ion) He+, 1s, 2s,2p,..
At R = ∞ the levels are those of
neutral H. The degeneracy of the
various levels (excluding spin) given
by the numbers in parentheses is
twice that of hydrogen as there are
two protons.
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Electronic wave functions
We expect if the electron is on one atom, it will tunnel
to the other and back. The two atomic states will mix
to produce two energy eigenstates, one symmetric,
one antisymmetric. At large R, we describe the
electron approximately as a combination of localized
atomic orbital waves.
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Symmetric wavefunction
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Symmetric probability density
Note large
probability
for
electron
between
the
protons.
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Antisymmetric wavefunction
The antisymmetric wavefunction vanishes
between the protons and the energy is different.
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Antisymmetric probability
density
Note a) small
probability for
electron between the
protons, and b)
higher derivatives
and more
confinement implies
higher energy.
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Estimation of electronic energy
If the wave function is only approximately a solution to
Schrodinger’s equation, the final expression
approximates the energy.
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Estimation of electronic energy
With hydrogenic wave solutions, the result for the integral is:
where R is units of the Bohr radius.
For large R
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H2+ energy
The antisymmetric
wavefunction yields
positive molecular energy
(not bonding even
excluding pp repulsion).
The symmetric
wavefunction yields
negative molecular energy
(bonding). The predicted
bond length occurs at the
point of stable equilibrium,
around R = 2.5 bohrs. The
predicted bond energy is
about09 1.77 eV.
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Modern Physics
H2+ energy compared to data
Predict R = 2.5 bohrs= 0.132 nm. Observe 0.106 nm.
Predict bond energy (energy to separate H and H+ )=
1.77 eV. Observe 2.65 eV.
Refined calculation technique: Assume wave is
symmetric but not simply a sum of atomic orbitals but
distorted. Characterize distortion by analytical factor
with parameters. Calculate E in terms of these
parameters. Minimize result as a function of the
parameters and separation simultaneously.
It can be proved this procedure will produce better
results and it does.
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Modern Physics
Now on to H2
The diatomic neutral hydrogen molecule contains
two electrons.
First approximation: Treat the two electrons as
independent and assume both electrons occupy the
symmetric bonding molecular orbital with opposite
spins to be consistent with the exclusion principle.
The case of one in bonding one in antibonding with
parallel spins is less bound.
2nd approximation: Include e-e repulsion and
recalculate the energy and minimum energy
equilibrium separation.
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Molecular Hydrogen: H2
Total molecular energy for
the bonding and antibonding
orbitals of H2. The bond
energy for H2 is 4.5 eV (not
quite twice that of H2+)and
the bond length is 0.074 nm
(less than 0.1 nm of H2+).
Since the energy of the
antibonding orbital exceeds
that of the isolated H atoms,
no stable molecule can be
formed in this state.
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Modern Physics
Sigma bonds
The electronic configuration of N is 1s22s22p3 and the
three outer p-state (L=1) electrons bond. We can think
of them in pairs. The axially symmetric states can
overlap to form an axially symmetric (sigma type) bond.
Formation of
a sigma
bond in N2
from the
overlap of
the 2pz
orbitals on
adjacent N
atoms.
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Sigma bond
The symmetric
bonding wave for
two electrons
formed from the pz
states concentrates
the electron pair
between the
atoms.
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Pi bond
Formation of a pi bond by overlap of the 2px orbitals
on adjacent N atoms. A similar bond is formed by
overlap of the 2py orbitals.
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Complexity
In heteronuclear diatoms (eg HF), the bonds are
not symmetric and can be ionic in character if
one atom is significantly more “attractive.”
In complex molecules, electrons are distributed
over more than two atoms and concepts from
condensed matter (solids) are applicable.
Condensed matter is the subject of the next
chapter.
November 09
Modern Physics
Important lessons
Molecular orbitals can be crudely understood as
linear combinations of atomic orbitals.
Weak bonds may be qualitatively understood as
a case of tunneling between atomic bound states.
In the simplest bond, an electron is localized and
shared equally between two atoms in an
antisymmetric wavefunction. Two electrons may
share that wave state provided their spins are
antiparallel. As in atoms, the Coulombic energies in
junction with the exclusion principle favor spin pair
formation and the absence of paramagnetism.
November 09
Modern Physics