Post on 02-Jun-2018
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Submitted by: Tashir AwanM.Sc Chemistry (Final)
Jamia M il lia I slamiaNew Delhi.
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To determine energies of molecular orbital ofpi electron in conjugated hydrocarbonsystem such as ethylene, butadiene, benzene,cyclobutadiene etc using electronicspreadsheet.
To determine Charge densities for each atomin the molecule and bond orders for bonded
pairs of atoms using electronic spreadsheet.
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Hckel Theory (Erich Hckel) E. Hckel, Z. Physik, 1931, 70, 204.
Illustration of LCAO approach formulated in the early 1930s
Used to describe unsaturated/aromatic hydrocarbons
The Hckel model has been largely superseded by more accurate MO calculations.
However, it is still useful to obtain qualitative predictions of bonding and reactivity in
conjugated systems.
Hckel Molecular Orbital Theory
Variation Theorem :
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The significance of the variation theorem is that the trial function giving the lowest
Rayleigh ratio is the optimum function of that form. Moreover, because the Rayleigh ratio
is not less than the true ground-state energy of the system, we have a way of calculating
an upper bound to the true energy of the system. Typically, the trial function is expressed
in terms of one or more parameters that are varied until the Rayleigh ratio is minimized.
If trial= p1 1+ p2 2where 1 are 2 are suitable
arbitary orthonormal trial functions
The variation principle seeks the
values of the parameters (two are
shown here) that minimize theenergy. The resulting wavefunction
is the optimum wavefunction of the
selected form.
0
1
p
E
02
p
E
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Huckel theory begins with 2 Structural Assumptions:
1. The electrons of interest initially occupy a system of carbon 2p orbitals having a
common nodal plane; that is, with their long axes parallel; they interact to form -type
molecular orbitals;
2. The rest of the electrons in the molecule occupy a -orbital framework that is
orthogonal to the 2p orbitals and therefore does not interact with them.
The description of Hckel theory as an LCAO method means that it assumes that
-molecular orbitals, , can be represented as a linear combination of atomic orbitals,
(basis set):
where
j is an index over molecular orbitals (MOs)
n is an index over atomic orbitals (Aos)
c is a set of coefficients weighting the contributions of the atomic orbitals to the
molecular orbitals
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The wave functions, , are called one-electron wave functions:
They represent the motion of a single electron in the electric field provided
by the nuclei and the averaged distribution of the other electrons.
Because they take account of the other electrons only in an average, they
do not account properly for electron correlation, the tendency of electrons to
avoid each other insofar as possible.
This problem is particularly severe for paired electrons in the same
orbital.
more sophisticated MO methods attempt to overcome this difficulty.
The energy of the electron for which the wave functions, is given by
Mother Nature always builds systems in states of lowest energy.
Energy calculated from the above equation always will be greater than
Eo, the true minimum energy, unless we have chosen the correct set of
coefficients, crj,So, if we can find the minimum of the energy with respect to the coefficients,
we must have the right answer for both the energy and the coefficients.
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The molecular wave function can be written as the linear combination of the two
carbon 2p atomic wave functions:
Substituting the right side of this expression into the Schroedinger equation
Multiply through
Extract the coefficients from the integrals (which we can do because they're
constants)
Integrals of the type iH jwill be replaced by HijIntegrals of the type ijwill be replaced by Sij.
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With these substitutions the energy expression becomes:
To make differentiation easier, multiply through by the denominator:
Now take the partial derivative with respect to c1, leading to:
Apply the variation principle and set d/c1= 0, and we obtain:
which can be rearranged to
Likewise, partial differentiation with respect to c2leads to
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This is a completely general result: taking a linear combination of n AOs of any
kind (s, p, d....) leads to a set of n simultaneous equations:
Now, we make a set of additional assumptions to reduce the complexity of the
linear equations.
1. The integrals of type Hij, where i = j, are related to the energy of an electron inan isolated carbon 2p orbital. They are called Coulombintegrals.
We assume they all are equal, regardless of the molecular environment
of the particular carbon
We use for all of them the symbol .
2. The integrals of the form Hij, where I j, are related to the energy lowering
that occurs upon allowing an electron to occupy both orbitals. This energy is dependent upon the distance between the orbitals.
Hckel theory assumes that if i and j are on adjacent atoms, the
interaction will be the same.
These integrals are represented by ; they are the resonance integrals.
If i and j are not adjacent, we assume there is no energy gain, and set
these integrals equal to zero.
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3. The S type integrals are called overlap integrals. They are related to the
energy of interaction between electrons in i and j. We assume they can be
divided into two groups.
If i = j, we set the integrals = 1
If i does not equal j, we set them = 0 This trick, of ignoring differences in interaction between orbitals, is
calledneglect of differential overlap, or NDO.
We can summarize the assumptions in the form of a table:
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Substituting the values into the set of simultaneous equations yields:
This determinant is called the secular equation.
Returning to ethylene, we find that the secular equation comes out to be:
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Some facts about the roots of polynomial equations from Secular
determinants:
1. Since no carbon atom in a -lattice can be bounded to more than 3 other
carbons, no root can equal or exceed three in magnitude; i.e. |xj| < 3.
2. The algebraic sum of all roots vanishes; i.e. xj=0.
3. The hydrocarbons can be classified into Alternant hydrocarbon (AH)
and Non-Alternant hydrocarbon (Non-AH).
4. Alternant hydrocarbons are planar conjugated hydrocarbons having no
odd-membered rings, in which the carbons can be divided into two sets,
s(starred) and u(unstarred), such that each s-carbon has only u-
neighbours and vice versa.
Even AH: no. of s-carbon = no. of u-carbon positions, the roots take
the form of xj= x1, x2, x3, *
* *
*
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In some cases no. of starred positions is larger than no. of unstarred
positions (ns > nu). In such cases (ns-nu) xs= 0.
Odd-AH, The starred set exceeds the unstarred set by one; the
starred carbons are referred to as active positions. For these systems,
the roots also occurs in pairs and extra root has the value zero.
*
*
*
*
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12
34
5
6
78
9
10
1.173
1.173
1.047
0.8550.986
0.870
0.855 0.986
1.027
1.027
NucleophilicSubstitution
ElectrophilicSubstitution
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Atom
Bond Type - electrons
for atomhX kXY
C -C=C- 1 0 1.0
N -C=N-(Pyridine) 1 0.5 1.0
N =C-N
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FRMALDEHYDE:
a =
0. 1.1. 1.
H2C O
Energy() 0.618034 -1.618034 -electron density
Atom No. Coefficients
1 -0.851 0.526 0.553
2 0.526 0.851 1.447
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METHYLIMINE:
0. 1.
1. 0.5
H2C NH
Energy() 0.781 -1.281 -electron density
Atom No. Coefficients
1 -0.788 0.615 0.757
2 0.615 0.788 1.243
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Computation of HMO Coefficients using
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Computation of HMO Coefficients using
Electronic Spreadsheet
Idea behind finding eigenvalues and eigenvectors of real, symmetric
matrices using MS Excel.
Excel can find eigenvalues and eigenvectors of real, symmetric matrices
(encountered Hckel Molecular Orbital Theory),as follows. If the
eigenvalues of the nth order real, symmetric matrix H are arranged in
increasing order: 1 2.n, an extension of a theorem due toRayleigh and Ritz sates that: 1= min (x
T Hx/xTx), where xis an nth order
non zero column vector whose elements are varied to minimize the
quantity in parentheses; also 2 =min (yTHy/yTy) if yTc1= 0, where c1is
the eigenvector corresponding to 1: 3= min (zTHz/zTz) If zTc1= 0 and
zTc2= 0 where c2 eigenvector corresponding to 2:etc. Use this theorem to
have excel find the eigenvalues and nomalized eigenvectors of the matrix.
Ho to implement Ra leigh and Rit theorem in MS E cel
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How to implement Rayleigh and Ritz theorem in MS Excel.
Assign name in excel to the various matrices involved. To multiply matrices A and
B, select an appropriately sized rectangular array of cells where we want the
matrix product to appear; then type =MMULT(A,B) and press the Control, Shift
and Enter keys simultaneously. The transpose of matrix C is found similarly using
the formula =TRANSPOSE(C). To find 1, start with guess for x and use the
Solver to vary x so as to minimize xTHx subject to the constraint that xTx= 1.After you find the first eigenvalue and eigenvector add the relevant orthogonality
constraint to the Solver and find the next eigenvalue eigenvector; and so on.
The calculation for the simplest framework of adjacent sp2 hybridised
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The calculation for the simplest framework of adjacent sp hybridised
carbon atoms using EXCEL is illustrated below.ETHENE: H2C CH2
We start with the 77 matrix, named as H, given in Cells D5:J11. Here we
will use the Excel Solver, if the Solver Add-In is missing then you can add
the Solver Add-In in the following steps: Click the Microsoft Office
Button then click Excel Options Click Add-Ins then in the
Manage box select Excel Add-ins
Click Go In the Add-Insavailable box select the Solver Add-in check box then click
OK.After you load the Solver Add-in, the Solver command is available in
the Analysis group on the Data tab.
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We now define the matrix x in cellsD16:D22 by naming it
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y g
asA. We define xTby using the formula=TRANSPOSE(A)
in cellsF16:L16 and pressing the Control, Shift and Enter
keys simultaneously. We name this matrix as AT.
To find 1, start with guess for A and use the Solver to vary A
so as to minimize ATH(A) in CellF17subject to the constraint
that AT(A)= 1 in Cell F18 & I19, The formula
=MMULT(AT,(MMULT(H,A))) was used in F17 where as
=MMULT(AT,A) formula was used in F18. and I19.
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Now we call the Solver dialog box as:
And set the Target Cell F17 to minimum by Changing cells
D16:D17(as ethylene has 2 sp2hybridized carbon atoms) subject
to the constraints that F18 = 1. Then Click Solve button to get the
following dialog box.
We press Ok button to get the first eigenvalue in Cell F19 orF17 and eigenvectors in cellsD16:D17 as shown below:
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p g g g
After we find the first eigenvalue and eigenvector, we call the
Solver Dialog box again and add the relevant orthogonality
constraint to find the next eigenvalue and eigenvector;
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To find 2, start with guess for y (named asB) in the cells D26:D32and yT
(named asBT) in the cellsF36:L36.
Use the Solver to vary Bso as to minimize BTH(B) in CellF27subject to
the constraint that BT(B)= 1 in Cell F28 & I29, The formula
=MMULT(BT,(MMULT(H,B))) was used in F27where as =MMULT(BT,B)formula was used in F28. and I29. The additonal constraint I30 = 0was
entered to the Solver dialog box by inserting the formula =MMULT(BT,A) in
cell I30. The Solver Dialog box looked like:
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When the Solve button is clicked, we get second eigenvalue in Cell
F29 and eigenvectors in cellsD26:27 as shown below
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Results of Ethene are Summarised as :
Energy() 1 -1 -electron density
Atom No. Coefficients
1 -0.707107
0.707106
8 1.0000001
2
0.707106
8
0.707106
8 1.0000001
By using the same procedure for other molecules, we get
the Eigenvalues and Eigenvectors
2. ALLYL RADICAL
Energy() 1.414 0.000 -1.414 -electron density
Atom No. Coefficients Allyl Radical Allyl Cation Allyl Anion
1 0.500 -0.707 0.500 1.000 0.500 1.500
2 -0.707 0.000 0.707 1.000 1.000 1.000
3 0.500 0.707 0.500 1.000 0.500 1.500
1,3 BUTADIENE :
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,
Energy() 1.618 0.618 -0.618 -1.618 -electron density
Atom No. Coefficients
1 0.3717 -0.6015 -0.6015 0.3717 1.0000
2 -0.6015 0.3717 -0.3717 0.6015 1.0000
3 0.6015 0.3717 0.3717 0.6015 1.0000
4 -0.3717 -0.6015 0.6015 0.3717 1.0000
4. CYCLOBUTADIENE:
Energy() 2 0 0 -2 -electron densityAtom No. Coefficients
1 0.5 0.7071 0.0000 -0.5000 0.500
2 -0.5 0.0000 -0.7071 -0.5000 1.500
3 0.5 -0.7071 0.0000 -0.5000 0.500
4 -0.5 0 0.7071 -0.500 1.500
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5. BENZENE :
Energy(
) 2 1 1 -1 -1 -2 -electron densityAtom No. Coefficients
1 0.408 -0.289 -0.500 0.500 0.289 -0.408 1.000
2 -0.408 -0.289 0.500 0.500 -0.289 -0.408 1.000
3 0.408 0.577 0.000 0.000 -0.577 -0.408 1.000
4 -0.408 -0.289 -0.500 -0.500 -0.289 -0.408 1.000
5 0.408 -0.289 0.500 -0.500 0.289 -0.408 1.000
6 -0.408 0.577 0.000 0.000 0.577 -0.408 1.000
When discussing pericyclic reactions, it is useful to know a little bit more about
the HMOs of conjugated system HMO schemes for conjugated chains involving
2-6 carbon atoms. There are no nodes in the most stable bonding MO. One
additional node is added. The more nodes, the higher the orbital energy.
For example, HMO theory can be used to explain the mechanism and
stereochemistry of the thermal as well as photochemical eletrocyclic ring
closure of trans,trans-2,4-hexadiene to trans-3,4-dimethylcyclobutene and cis-
3,4-dimethylcyclobutene, respectively.
2.4 Computations for Conjugated Systems with Heteroatoms
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Example:
FORMALDEHYDE:
a =
0. 1.1. 1.
Energy() 0.618034 -1.618034 -electron density
Atom No. Coefficients
1 -0.851 0.526 0.553
2 0.526 0.851 1.447
METHYLIMINE: H2C NH
a =
0. 1.
1. 0.5
Energy() 0.781 -1.281 -electron density
Atom No. Coefficients
1 -0.788 0.615 0.757
2 0.615 0.788 1.243
ENAMINE:NH2
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NH2
Energy() 1.129 -0.682 -1.947 -electron density
Atom No. Coefficients
1 0.222 0.483 -0.847 1.901
2 -0.730 -0.494 -0.473 0.934
3 0.646 -0.723 -0.243 1.164
a =
1.5 0.8 0.
0.8 0. 1.
0. 1. 0.
Acetaldehyde Radical:H2C O
a =
1. 1. 0.
1. 0. 1.
0. 1. 0.
Energy() 1.247 -0.445 -1.802 -electron density
Atom No. Coefficients
1 0.328 0.591 -0.737 1.436
2 -0.737 -0.328 -0.591 0.806
3 0.591 -0.737 -0.328 0.758
Vinyl Fluoride:
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Vinyl Fluoride:F
Energy() 1.062 -0.891 -3.172 -electron density
Atom No. Coefficients
1 0.124 0.216 -0.969 1.969
2 -0.722 -0.649 -0.237 0.956
3 0.680 -0.729 -0.075 1.075
a =
3. 0.7 0.
0.7 0. 1.
0. 1. 0.
Acrolein : O
Energy() 1.532 0.347 -1.000 -1.879 -electron density
Atom No. Coefficients
1 -0.228 0.429 0.577 -0.657 1.529
2 0.577 -0.577 0.000 -0.577 0.667
3 -0.657 -0.228 -0.577 -0.429 1.034
4 0.429 0.657 -0.577 -0.228 0.771
a =
1. 1. 0. 0.
1. 0. 1. 0.
0. 1. 0. 1.
0. 0. 1. 0.
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2.5 STRENGTHS AND WEAKNESSES OF THE HMO
Strengths
The HMO has been extensively used to correlate, rationalize, and predict
many chemical phenomena, having been applied with surprising success to
dipole moments, esr spectra, bond lengths, redox potentials, ionization
potentials, UV and IR spectra, aromaticity, acidity/basicity, and reactivity,
The method will probably give some insight into any phenomenon that
involves predominantly the n electron systems of conjugated molecules.The HMO may have been underrated and reports of its death are probably
exaggerated.
Because of the phenomenal success of all-valence-electron SE methods,
which are applicable to quite large molecules, and of the increasing power
of all-electron ab initio and DFT methods.
WeaknessesThe defects of the HMO arise from the fact that it treats onl n electrons
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The defects of the HMO arise from the fact that it treats only n electrons
This approximation, as explained earlier, reduces the matrix form of the secular
equations to standard Eigen value form HC = Ce, so that the Fock matrix can (after
giving its elements numerical values) be diagonalized without further ado.
In the older determinant, as opposed to matrix, treatment, the approximation greatlysimplifies the determinants.
The neglect of electron spin and the deficient treatment of interelectronic repulsion is
obvious. In the usual derivation, the integration is carried out with respect to only spatial
coordinates (ignoring spin coordinates; contrast ab initio, we simply took the sum of the
number of electrons in each occupied MO times the energy level of the MO.
If we calculate the total electronic energy by simply summing MO energies timesoccupancy numbers, we are assuming, wrongly, that the electron energies are
independent of one another. The resonance energies calculated by the HMO can thus be
only very rough, unless the errors tend to cancel in the subtraction step, which in fact
probably occurs to some extent.
The neglect of electron repulsion and spin in the usual derivation of the HMO.
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2.7 CONCLUSIONS:One reason that expanding the characteristic determinant and
solving the characteristic equation is not a good way to find the
eigenvalue of large matrices is that, for large matrices, a verysmall change in a coefficient in the characteristic polynomial may
produce a large change in the eigenvalue. Hence we might have
to calculate the coefficient in the characteristic polynomial to
hundreds or thousands of decimal places in order to get
eigenvalue accurate to a few decimal places. Hence, search for aunitary matrix C(matrix diagonalization) using MS EXCELsuch
that CTHC is a diagonal matrix. The diagonal elements of CTHC
are the eigenvalues of H, and the columns of C are the
orthonormal eigenvectors of H is computationally much faster.
The HMO method can be widely used to rationalize the effect of
heteroatoms and predict the properties and reactivities of
conjugated compounds.
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