Tasir Project Updated

8/11/2019 Tasir Project Updated

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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The Hckel model has been largely superseded by more accurate MO calculations.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



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


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


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.



1 -0.851 0.526 0.553

2 0.526 0.851 1.447

METHYLIMINE: H2C NH

a =

0. 1.

1. 0.5



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


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.



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



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


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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Tasir Project Updated