5th Set of Notes
Molecules!!!!!!
Simplest Molecule Possible?
For H atom separate Schrodinger eqn. into a part that deals with translation of the whole atom
and one dealing with electrons!
Similar situation - but molecule has additional degrees of freedom. vibrations, rotations - poses
problem!
Born-Oppenheimer Approximation
The nuclei, being so much heavier than the electron, move relatively slowly, and we treat them
as being fixed with respect to the Electron motion, at least for H2+
For heavier molecules?
(Exception is ground state of some cations.
With Born Oppenheimer - Schrodinger eqn. becomes single particle Schrodinger eqn. for an
electron in the field of two stationary protons at some specified separation.
R = distance between protons
So write Schrodinger eqn.
H= E ................................................... in terms of Laplacian
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Exact solution to this is obtainable at any given value of R!
However!
The total energy of the molecule at separation R is
...................... E + e2/(4o) 1/R
Use this to generate the energy of the H2+ molecule by:
1) choosing the smallest value of R (R1) then calculate Eschrodinger and add Vnuc
2) Increment R (R2) calculate Eschrodinger for protons at R2 distance and electron - Add Vnuc
3) increment R again recalculate (keep doing this at many other bond lengths to generate the
electronic energy curve as a function of internuclear distance (bond distance)
The minimum value on the curve is the
How would the energy of vibrations fit on this curve - Rotation?
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TWO BASIC APPROACHES TO QUANTUM THEORY
Valence Bond Theory versus Molecular Orbital Theory
VB treatment starts with individual atoms and considers the interaction between them
For two atoms a and b, with two electrons 1 and 2 a possible wavefunction is
............................. 1 = a(1)b(2)
..................... or ... 2 = a(2)b(1) .......... since electrons are indistinguishable
VB = a(1) b(2) + a(2) b(1)
The M.O. treatment starts with two nuclei. If a(1) is wavefunction for electron 1 on nucleus a nd
b(1) is a wavefunction for electron (1) on nucleus b.
The wavefunction for single electron in a field of two nuclei can be written as an LCAO
........... 1 = c1 a(1) + c2b(1)
for 2nd electron 2 = c1a(2) + c2b(2)
mo = 12 = c12a(1)a(2) + c22b(1)b(2) + c1c2[a(1)b(2) + a(2)b(1)]
Comparing VB with mo: mo gives large weight to configurations that place both electrons on
the same nucleus. In molecule AB, these are the ionic structures A+-B- = AB


VB neglects ionic terms. mo considerable overestimates the ionic terms.
VB base on chemical concept that in some sense atoms exist within molecules - structure of
molecule can be interpreted in terms of its constituent atoms and bonds between them.
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MO method seeks to discard the idea of atoms within molecules, and start with bare positive
nuclei arrayed at definite positions in space. Total # of electrons fed one-by-one into this field MO theory more physical then chemical, sees molecule as electronic pudding of varying density
with some positive nuclear plums.
Molecular Orbitals
One electron wave functions obtained from Schrodinger eqn. - Molecular Orbitals, M.O.s

2 gives distribution of the electron in the molecule as a whole
WAYS TO GET MOs
H2+ - could solve the Schrodinger eqn. analytically.
Another way - use approximate orbitals. Make from a linear combination of Atomic Orbitals
Consider H2+
If Electron is:
1. Close to HA - MO resembles an A.O.! ....... 1s A.O. on HA. Schrodinger is the same as that for
an isolated H atom. Lowest energy solution is H 1s orbital 1s(A)
2. Close to B - MO resembles an A.O. on B. Solution to Schrodinger is H 1s orbital 1s(B)
Thus an Approximate Overall Molecular Wavefunction is:
1 = N1 {1s(A) + 1s(B)} ...... or .......... 2 = N2 {1s(A) - 1s(B)}
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N1, N2 are defined as the normalization factors!
Approximate Molecular Orbitals formed from Linear Combination of Atomic Orbitals is called
an LCAO-MO.
General Rule:
# of LCAO-MO’s Formed is equal to the number of AOs used in the linear combination
Could Normalize the LCAO MO’s - Normalize 2
1=
Since 1s is normalized and REAL, 1st two terms are:
Define S as the overlap integral. It tells if (A) and (B) are simultaneously large in some
region of the defined space.
In the limit that (A) = , then
1=


) identical with (B)
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For two hydrogenic 1s orbital on nuclei separated by a distance R, OVERLAP is given by:
S = {1 + ZR/a0 + 1/3 (ZR/a0)2 exp(-ZR/a0) ...
a0 defined as the Bohr radius = ................................... = 52.9 pm
Z = nuclear charge
S = 0.59 for two 1s orbitals at equilibrium bond length in H2+. Typical values with n=2 are in the
range 0.2 to 0.3!
So
N2 = 1.10 for 2 LCAO-MO wave function in the case of H2+
S, the overlap integral also arises in the expression for the probability density of the electron in
H2+


*= ...............................................................
Look at the individual terms:
a) 1s(A)2 = probability density if electron is confined to an orbital on atom B.
b) 1s(B)2 =
c) 2 1s(A)2s(B) is extra contribution to the density overlap
For one LCAO-MO of H2+ - = N1 {1s(A) + 1s(B)}
see above ................
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the overlap is constructive since there is extra probability that the electron is between the nuclei Bonding MO
For other LCAO-MO of H2+ = N2 {1s(A) - 1s(B)} Overlap is destructive - get node place
where the electron probability is zero between the two nuclei - Antibonding MO.
* = ...............................................................
Relative to separated H atoms - “reduces probability”
Electron in antibond destabilizes molecule
1) Electron is excluded from internuclear region so distributed outside bonding region.
2) In a sense electron pulls nuclei apart!
Structures of DIATOMICS USING MO Theory
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Use Aufbau Principle as with many electron atoms - built up Atomic Electronic Configuration
From Hydrogen Atom orbitals
COULD!
Similarly - use the H2+ molecular orbitals to build up molecular orbitals - Apply Hunds Rule in
the case of degenerate molecular orbitals.
Procedure to Make Molecular MO
“Get the MO’s from A.O.s on atoms bonding:
1) Homonuclear Diatomics
Each Atom has:
a) Core Electrons - inner closed shell
b) Valence Electrons - orbitals of the valence shell
c) Virtual orbitals - orbitals of the atom that are unoccupied in grnd state.
Here:
Valence shell electrons most important in bonding - build Mos from these
A) Overall - All orbitals of the appropriate symmetry contribute to a molecular orbital.
( made form s and pz orbitals) from px,py)
B) Orbitals with similar symmetry and similar energy may combine to form MO s.
C) Ordering of MO s is:
1) 12*, 1, 3, 2*, 4* .........
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For diatomics through N2, where * indicates antibonding MO. (note type orbitals alternate
bonding, antibonding, bonding .... etc.
2) 12*, 3, 1

Overhead
Consider H2 and He2 (M.O. Energy level Diagram)
Energy Level of Separated Atoms
Note in case of H2 - bothe electrons go into M.O. which is stable relative to the separated atoms.
(bonding orbital 1)
In He2 - two electrons are in bonding and two are in an antibonding orbital - Energy of
antibonding orbitals is more positive than the bonding orbital is negative ........ so the atoms
separate.
Consider C2
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 orbitals are a result of the fact that the 2px and 2py orbitals of each atom (which are
perpendicular to the internuclear axis) overlap broadside.
Thus there is off-axis electron overlap giving rise to angular momentum.
Paramagnetic vs Diamagnetic If there are unpaired electrons then the molecule is said to be paramagnetic
If all spins are paired the molecule is diamagnetic!
Bond Order
Measures the net bonding in a diatomic molecule.
Defined by:
b = ½ (n-n*) where
n = # of electrons in bonding orbital
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n* = # of electrons in .................
b = ? .............................................. for H2
...................................................... for He2?
..................................................... for C2, C2+, or C2-
Molecular Electronic Configuration (like Atomic Electronic Configuration, but lists the MO for
valence electrons)
H2
He2
C2
Example:
Give the Molecular Orbital Energy Diagram
bond order, and Molecular Electronic Diagram for the valence electrons, and if paramagnetic of
N2
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F2
Back to Valence Bond
Valence - bond wavefunctions for H2
 = H1SA(1)H1SB(2) +/- H1SA(2)H1SB(1)
H=
 = H1SA(1)H1SB(2) + H1SA(2)H1SB(1) is the lower energy state!
Formation of the H2 molecule from atoms involves electrons participating in atom swapping
Electrons exist in internuclear regions in a positive way and form bonds.
Overlap of wavefunctions tells how electrons are enhanced in this region.
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If distribution is cylindrically symmetric around the internuclear axis - then sigma - bond.
No orbital angular momentum about the internuclear axis.
Calculate the energy curve for H2 - .......... change R
HOMO Nuclear Diatomics and Valence Bond
N2
N(valence) 2s2 2px 2py 2pz ............................. Convention - z axis is the internuclear axis.


 =
() =
HETERONUCLEAR Diatomics
Describe Bonding in ground state HBr in valence bond terms.
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POLYATOMICS
H2O
H - 1s1
O - 2s2 2px2 2py 2pz
Suggest unpaired electrons in O, 2p orbitals can each pair withan electron in the H 1s orbitals or 

What should the molecule look like
How about NH3?
What about CH4
Hybridization
Discrepancy for carbon being tetravalent can be understood by promotion of 2s electron.
Bonds would not be _______________ in CH4 Need hybridization.
Hybrid Orbitalsallow all four bonds in CH4 to be EQUIVALENT!
Hybrid orbitals for methane are formed by combining the C2s and C2p orbitals
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Four atomic orbitals - Four equivalent hybrids
h1 =
h2 =
h3 =
h4 =
Constructive and destructive interferences between AO give hybrid consisting of large lobe
pointing in direction of regular tetrahedron.
Each hybrid contains a single unpaired electrons which

Hybrids have enhanced amplitude in internuclear region
Ethene?
Bonding?
h1 = s + (2)½ py ........ h2 = s + (3/2)½ py + (½)½ py ............h3 =
lie in plane pointing towards corners of equilateral triangle.
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Structure of Ethene:
Three sp2 hybrid orbitals form:
Unhybridized p orbitals can pair and form the .............. bond.

 bond has angular momentum about .....................
Ethyne - ............. hybridization!
h1 = s + pz ............ h2 = s - pz
Why triple bond?
Explain NH3 bonding in terms of hybrids?
Explain H2O bonding in terms of hybrids!
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ARE Valence Bond and MO Theory totally separate or can we use a combination to
explain the structure of molecules?
ACCORDING to MO Theory we can form an MO from a linear combination of AOs.
Example MO (HF) = cHH1s + cF 2pF OR
Could use a hybrid orbital as an AO and us it in a linear combination for instance in Ethylene:
MO() =
MO() =
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Another Approximation Technique of Quantum Mechanics
Variation Principle
Given any Approximate wavefunction satisfying the boundary conditions of a problem, the
expectation value of the energy calculated from this function will always be HIGHER than the
true energy of the ground state.
Suggest Procedure For Solving Quantum Problems:
1) Guess several functions - trial functions or have one in mind.
2) Calculate expectation value for each one.
3) Choose the one with the lowest energy and conclude it is the best function that can be
obtained from the original guesses
In practice - start with the trial wavefunction containing arbitrary parameters and minimize the
expectation energy with respect to these parameters!
Present application is the building Molecular Orbitals from atomic orbitals:
LCAO-MO method.
Trial Function  = u Cuu
where u are the appropriate atomic orbitals and cu are the parameters to be chosen to minimize
the energy.
Use Variation Principle to find best and Energies for H2+ molecule
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Huckel Theory:
Have backbone made out of sigma MO. Consider only conjugated pi orbitals
Make some assumptions about the integrals.
Assumptions include:
1) All overlap integrals are set to zero.
2) All resonance integrals between non-neighbors are set equal to zero.
3) All remaining resonance integrals are set equal to b.
Approximations are very severe but allow calculation of a general picture of
the molecular orbital electronic energy levels.
Assumptions result in simplifications to the secular determinant as follows:
1) All diagonal elements become:
2) Off-diagonal elements become:
3) All other elements become:
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Ethylene
Butadiene
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Benzene
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Extended Huckel Theory
EHT included both  and  orbitals, not confined to planar or conjugated hydrocarbons
SCF Calculation
Difficulties associated with EHT are overcome by more sophisticated theories that calculate
shapes and energies of molecular orbitals and reactivity and structures
Hartree Fock
Semiempirical and Ab-initio
Continue calculation using semiempirical or Ab-initio methods
Semi-empirical - many integrals are estimated by appealing to spectroscopic data or physical
properties like ionization energy.
Ab-initio - attempt is made to calculate all of the integrals that appear in the secular determinant
from first principles.
DFT - Density Functional Theory - central focus of DFT is electron density rather than the
wavefunction i Functional comes from the fact that the energy of the molecule is a function of
the electron density.
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