Transcript Molekylfysik

The Born-Oppenheimer approximation
 The electrons are much lighter than the nuclei (me/mH1/1836)  their motion is much faster
than the vibrational and rotational motions of the nuclei within the molecule.
 The Schrödinger equation can then be divided into two equations:
1) One describes the motion of the nuclei.
2) The other one describes the motion of the electrons around the nuclei whose positions are
fixed.
The electronic Schrödinger equation
 The nuclear coordinate R appears as a parameter in the
expression of the electronic wave function.
R
 An electronic wave function elect(R,r) and an energy Eelect
are associated to each structure of the molecule (set of nuclei
coordinates R).
 For each variation of bond length in the molecule (each new
R), we can solve the electronic SE and evaluate the energy that
the molecule would have in this structure: the molecular
potential energy curve is obtained (see Figure).
D0
 The molecule is the most stable (minimum of energy) for
one specific position of the nuclei: the equilibrium position Re.
 The zero energy corresponds to the dissociated molecule.
 The depth of the minimum, De, gives the bond dissociation energy, D0, considering the fact
that vibrational energy is never zero, but 12  :
D0=De- 12 
Valence-bond theory
The hydrogen molecule
A(1) = H1sA(r1) is the wavefunction
of the electron 1 on the 1s orbital of
Hydrogen A.
+= A(1)B(2) + A(2)B(1)
When the atoms are close, it’s not
possible to know whether it is
electron 1 that is on A or electron 2.
 The system is described by a
superposition of the wavefunctions
for each possibility: A(1)B(2) and
A(2)B(1). Two linear combinations
are possible:
+= A(1)B(2) + A(2)B(1)  between the two nuclei: |+|2>0 
creation of a  bond described by a “bonding molecular orbital”.
-= A(1)B(2) - A(2)B(1)  between the two nuclei, - changes sign
 |-|2 =0  creation of a * bond described by a “antibonding
molecular orbital”.
-= A(1)B(2) - A(2)B(1)
Homonuclear diatomic molecules
N
1s2 2s2

2px1 2py1 2pz1
Valence electrons
y

z
In N2: 3 bonds are formed by combining the 3
different 2p orbitals of the 2 nitrogen atoms. This
is possible because of the symmetry and the
position of those 2p orbitals with respect to each
other.
 one  bond and 2  bonds (perpendicular to
each other) are formed by spin pairing.
x



Polyatomic molecules
O
x
1s2 2s2
H 1s1
y
z
H2O: This simple model suggests that the formed
angle between the (O-H) bonds is 90°, as well as
their position vs. the paired 2px electrons; whereas
the actual bond angle is 104.5°.
2px2 2py1 2pz1
H 1s1
Promotion
 C: 1s2 2s2 2px1 2py1. With the valence bond theory, we expect maximum 2 bonds.
 But, tetravalent carbon atoms are well known: e.g., CH4. This deficiency of the theory is
artificially overcome by allowing for promotion.
Promotion: the excitation of an electron to an orbital of higher energy. This is not what
happens physically during bond formation, but it allows to feel the energetics. Indeed,
following this artificial excitation, the atom is allowed to create bonds; and consequently, the
energy is stabilized… more than the cost of the excitation energy.
 C: 1s2 2s2 2px1 2py1  1s2 2s1 2px1 2py1 2pz1. CH4 should be composed of 3 bonds due to
the overlap between the 2p of C and the 1s of H, and another  bond coming from the 2s of C
and the 1s of H.
 But, it is known that CH4 has 4 similar bonds. This problem is overcome by realizing that
the wavefunction of the promoted atom can be described on different orthonormal basis sets:
1) either the orthogonal hydrogenoic atomic orbitals (AO): 1s, 2s, 2px, 2py, 2pz.
2) or equivalently, from another set of orthonormal functions: “the hybrid orbitals”
Energy of the states  and *
H-H+: One electron around 2 protons
2 2
H 
e  V
2me
e2  1 1 1 
   
V =4 0  rA rB R 
+
E = EH 1s 
H=E
=-
-
erA
A
 jk 


4 0 R  1  S 
e2
rB
R
E = EH 1s 
B
 j k 


4 0 R  1  S 
e2
=+
Structure of diatomic molecules
Now, we use the molecular orbitals (= + and *= -) found for the one-electron molecule
H-H+; in order to describe many-electron diatomic molecules.
The hydrogen and helium molecules
H2: 2 electrons  ground-state configuration: 12
E
He2: 4 electrons  ground-state configuration: 12 2*2
E
EE+
E+< E-
 He2 is not
stable and does not exist
Increase of electron density
Bond order
Bond order:
n= number of electrons in the bonding orbital
b(n-n*)
n*= number of electrons in the antibonding orbital
 The greater the bond order between atoms of a given pair of elements, the shorter is
the bond and the greater is the bond strength.
Period 2 diatomic molecules
According to molecular orbital theory,  orbitals are built from all orbitals that have the
appropriate symmetry. In homonuclear diatomic molecules of Period 2, that means that two 2s
and two 2pz orbitals should be used. From these four orbitals, four molecular orbitals can be
built: 1, 2*, 3, 4*.
1, 2*, 3, 4*.
With N atomic orbitals  the molecule will have N molecular orbitals, which are
combinations of the N atomic orbitals.
dioxygen O2: 12 valence electrons
The two last e- occupy both the
x* and the y* in order to
decrease their repulsion. The
more stable state for 2e- in
different orbitals is a triplet state.
O2 has total spin S=1
(paramagnetic)
The two 2px give one x and one x*
The two 2py give one y and one y*
Bond order = 2
Note: The  orbitals together give rise to an cylindrical
distribution of charge. Electrons can circulate around this torus
can create magnetic effect detected in NMR
Heteronuclear diatomic molecules
 A diatomic molecule with different atoms can lead to polar bond, a covalent bond in which
the electron pair is shared unequally by the 2 atoms.
Polar bonds
 2 electrons in an molecular orbital composed of one atomic orbital of each atom (A and B).
 = cA A + cB B
|ci|2= proportion of the atomic orbital “i” in the bond
 The situation of covalent polar bonds is between 2 limit cases:
1) The nonpolar bond (e.g.; the homonuclear diatomic
molecule): |cA|2= |cB|2
2) The ionic bond in A+B- : |cA|2= 0 and |cB|2=1
Example: HF
The H1s electron is at higher energy than the F2p orbital.
The bond formation is accompanied with a significant partial
negative charge transfer from H to F.
Electronegativity
 A measure of the power of an atom to attract electron to itself when it is part of a
compound.
There are different electronegativity definitions, e.g. the Mulliken electronegativity:
M=½ (IP + EA)
 IP is the ionization potential = the minimum energy to remove an electron from the ground
state of the molecule (Chap 3, p14).
 EA is the electron affinity = energy released when an electron is added to a molecule. EA>0
when the addition of the electron releases energy, i.e. when it stabilizes the molecule.
Electronic transitions
Vibrational transitions
Rotational
transitions
Selection rules in Raman spectroscopy
General principle
 Most of the scattered radiation has the same wavelength as the
incident beam. This radiation is called the Rayleigh radiation
and is the result of elastic scattering.
 About 1 in 107 of the incident photons collide with the
molecules, give up some energy, and emerge with a lower
energy. These inelastic scattered photons constitute the lowerfrequency Stokes radiation from the sample.
 Other incident photons may collect energy from the molecules
(if they are already excited), and emerge as higher-frequency
anti-Stokes radiation.
Scattered
radiation
Monochromatic
incident radiation
L = lens
Ei= hi
M = mirror
R - Rayleigh Scattering
AS - Anti-Stokes Raman Scattering
S - Stokes Raman Scattering
0
Ev = Ei - Es
The energy difference between the incident light (Ei) and the Raman scattered light (Es) is
equal to the energy involved in changing the molecule's vibrational state (i.e. getting the
molecule to vibrate, Ev). This energy difference is called the Raman shift: Ev = Ei - Es
Several different Raman shifted signals will often be observed; each being associated with
different vibrational (or rotational) motions of molecules in the sample. A plot of Raman
intensity vs. Raman shift is a Raman spectrum.
Virtual level X
hinc
Virtual level X
hinc
hinc- hvib
hinc+ hvib
hvib
lower-frequency lines: Stokes radiation
higher-frequency lines: anti-Stokes radiation
 At room temperature, few molecules are in the first excited vibrational levels.
Consequently, the anti-Stokes line have a low intensity.
The vibrational structure: Franck-Condon principle
E
Classical picture:
*s
Because the nuclei are heavier than electrons, an electronic
transition takes place much faster than the nuclei can
respond. This is represented by the vertical green arrow in the
graph: during the vertical electronic transition, the molecule
has the same geometry as before the excitation.
During the transition, the electron density is rapidly built up in
new regions of the molecule and removed from others, and the
nuclei experience suddenly a new force field, a new potential
(upper curve). They respond to this new force by beginning to
vibrate.
Re* > Regs because an excited state is characterized by an
electron in an anti-bonding molecular orbital, which gives rise
to an elongation of one or several bonds in the molecule.
gs
Reg Re*
s
Separation
distance between
atoms in the molecule
The fates of electronically excited states
 Nonradiative decay = the excitation energy
is transferred into the vibration, rotation,
translation of the surrounding molecules via
collisions.
Collisions
Molecule B
Molecule A
 Dissociation and chemical reaction
 Radiative decay = the excitation energy is
discarded as a photon (fluorescence, phosphorescence)
Fluorescence
Energy
*s
v=0 gs
v=2
v=1
v=0
Regs Re*
R
a max b
The excited molecule collides with the
surrounding molecules and steps down the
ladder of vibrational levels to v=0 of *s. The
surrounding molecules, however, might now be
unable to accept the larger energy difference
needed to lower the molecule to gs. It might
therefore survive long enough to undergo
spontaneous emission. As a consequence, the
transitions in the emission process have lower
energy compared to the absorption transition
In accord with the Franck-Condom principle,
the most probable transition occurs from *s to
the vibrational state of gs, for which the
molecule has the same inter-atomic separation
Re*. This vibrational state (v=1 in the Figure) is
characterized by a maximum intensity of its
vibrational wavefunction at Re*. This is the
origin of the maximum in the fluorescence or
emission spectrum.
Phosphorescence
*S
*T
Conditions:
1) The potential felt by the atoms when the molecule is in
its electronic singlet excited state () crosses the
potential for the molecule in its triplet excited state ( ).
In other words, the structure of the molecule in both
states is similar for specific vibrational levels of both
states.
gsS
2) If there is a mechanism for unpairing two electron
spins (and achieving the conversion of  to  ), the
molecule may undergo intersystem crossing and becomes
in *T. This is possible if the molecule contains heavy
atoms for which spin-orbit coupling is important.
When the molecule reaches the vibrational ground state of *T, it is trapped!
The solvent cannot absorb the final, large quantum of electronic energy, and the molecule
cannot radiate its energy because return to gsS is spin-forbidden….. However, it is not totally
spin-forbidden because the spin-orbit coupling mixed the S and T states, such that the
transition becomes weakly allowed.
 weak intensity and slow radiative decay (can reach hours!!).
Note: Phosphorescence more efficient for the solid phase
Dissociation
A dissociation is characterized by an absorption spectrum
composed of two parts:
(i) a vibrational progression
(ii) a contiuum absorption
For some molecules, the potential surface of the excited
state is strongly shifted to the right compared to the
potential of the ground state.
As a consequence, lot of vibrational states of the electronic
excited state are accessible (vibrational progression
described by the Franck-Condon principle), and the
dissociation limit can be reached.
Beyond this dissociation limit, the absorption is continuous
because the molecule is broken into two parts. The energy
of the photon is used to break a bond and the rest in
transformed in the unquantized translational energy of the
two parts of the molecule.