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Huckle Molecular orbital theory

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الكلية كلية العلوم     القسم قسم الكيمياء     المرحلة 4
أستاذ المادة عباس عبد علي دريع الصالحي       18/12/2017 17:24:29
University of Babylon Undergraduate Studies
College of Science
Department of Chemistry
Course No. Chsc 424 Physical chemistry
Fourth year - Semester 1
Credit Hour: 3 hrs.

Lectures of Quantum mechanics
Scholar year 2017-2018
Prof. Dr. Abbas A-Ali Draea
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Lecture No. Fifteen: Hückl Molecular Orbital theory
1-Introudction.
2-Formalim of Hamilton operator.
3-Approximation of Hückl molecular orbital theory.
4-Application of HMOT.

1-Introudction.
As an application of quantum mechanics theories on real applicable systems, is very useful tool for organ chemist. Theory have been achieved to be as treatment for unsaturated organic molecules (consisting conjugated double bonds), likes butadiene, Benzene, pyrimidine, and pyrenes (essential component of DNA). Valance electrons in unsaturated organic molecules have been classified into two main separated groups, the first one is ?- electrons( loosely bonded to atoms)that’s form double bonds and the second one is ?- electrons(strongly bonded to atoms)that’s form strong single bonds (inactive electrons relatively). The ?- electrons are more active than other electrons in the same molecules due their delocalization properties into significance atom, therefore its play important role in chemical reactions.
2-Formalsim of Hamilton operator.
The treatment of this theory involved two difficulties, the first, is the ?- electrons at the chemical reactions depend on ?- electrons. The second, Hamilton operator of ?- electrons system is very complicated and it s not give up precisely solution of equations that’s give up energies of ?- electrons. For this reasons, they supposed that Hamilton operator of ?- electrons system is consisted from summation terms of unpaired electrons, so that:
?=?_i??h_effective (i)? ----1
Since,i refers to electrons, and h_effective, refers to mathematic operator for one electron as a function for dimensions and momentums of electron i. the mathematic operator of electron i is involved potential energy that’s effected by other rest electrons. Due that the mathematic operator of electron i is a function for dimensions and momentums, so can be used the Schrodinger equation separation method of variables for ?- electrons system, that’s produced by eq.1, this generate N equations of unpaired electrons, since N represent the number of these electrons and each equation have following formula:
h_effective (i).?_i=E_i ?_i----2
Since ?_iis represented the molecular orbitals for single electron, E_i is energy of orbitals. The total energy of ?- electrons is:
E=?_i?E_i ----3
The filling electrons in their orbital occurs according to Pauli principles to build up the molecular orbitals, also the method of linear combination of atomic orbitals.
?_i=?_???a_?i.?_? ?----4
Since ?_? is represented the wave function orbitals, that’s contributed ?- system and the total represent all atoms in the system. By using Hamiltonian operator of eq. 1 and variation theory, can be get on a secular equations determinate, that’s likes eq. 14 and 15 of previous situation as follow:
|?(H_11-ES_11&H_12-ES_12&H_1n-ES_1n@H_21-ES_21&H_22-ES_22&H_2n-ES_2n@H_n1-ES_n1&H_n2-ES_n2&H_nn-ES_nn )|=0—5
Also can be write in summarized formula as follow;
|H_??-ES_?? |=0 ----6
The atomic orbitals are essential orbitals for ?- system, they found some approximation for this system to be more achievable.
3-Approximation of Hückl molecular orbital theory.
The approximation involved some rules to build up ?- system as follow:
The integral H_?? is the same for every atom in the system and symbolized by (?).
If that µ-atom bonded with ?-atom, so that integral H_?? is constant quantity in the system and symbolized by (?).
If that µ-atom did not bonded with ?-atom, so that integral H_?? is equal to zero quantity in the system.
If that µ-atom is equal to ?-atom (µ= ?), so that integral S_?? is equal to one quantity in the system.
If that µ-atom did not equal to ?-atom (µ??), so that integral S_?? is equal to zero quantity in the system (no overlap between them).
Hückl molecular orbital is represented the results by (?) and (?) without any experimental calculus. The bond length of different single bonds of C—C in the molecules have effect on values of ?-values, therefore must be using different value of ? for single and double bonds respectively, to be the calculus more precisely. They found another difficulties are comes out from hetero atoms in unsaturated hydrocarbon compounds, like pyridines, therefore must be using a different values of ? and ? for hetero atoms differ than for that carbon atoms. At the same time can be optimized HMO- theory by introduced interacted of electrons interactions.
4-Application of HMOT.
Some molecular examples can determined to find their structures and stable energies as follow:
4-1 Ethylene molecule.
Ethylene molecule is consisted from two carbon atom with hybridization of SP2, that’s mean they found sixteen electron at electronic system. According to MOHT, only two electrons are treated as ?- system. Two atomic non hybridized orbitals of 2p_z^1at carbon atoms are overlapped linearly to form ? bond. Following figure is represent the molecular structure of ? orbital.


Figure 1. Calculation of HMOT at Ethylene molecule.
By application the linear combination of atomic orbitals for ?-electrons to get on:
?_1=a_1 ?_1+a_2 ?_2------7
Orbitals of ?_1and ?_2 are represent the atomic orbitals of 2P_1^1 Z and2P_2^1 Z
Respectively. To form the determinate of ethylene molecule according to eq.5 from type (2x2) as follow:
|?(H_11-ES_11&H_12-ES_12@H_21-ES_21&H_22-ES_22 )|=0 ---8
By applying approximation of HMOT, to get on
|?(?-E&?@?&?-E)|=0 ----9
If we divided the element of determinant on ? and supposed that:
x=(?-E)/?----10
Therefore
|?(x&1@1&x)|=0-----11 or x^2-1=0 And x=±1
So that: E=?±? by substituted these values in the secular equations and normalized the functions to get on following functions:
?_1=1/?2 ?(??_1+?_2)------12
?_2=1/?2 ?(??_1-?_2)------13
The association energy value of eq.12 equal to ?+? , and The association energy value of eq.13 equal to ?-? , due that both ? and ? have negative values, therefore level energy of ?_1is lowest than level energy of ?_2 . Conclusion is involved that’s ?-electrons of ethylene molecule are stated in ground state at level of energy?_1. The total energy of ?-electrons in ethylene molecule equal to2(?+?).
Following figure represent the diagram of energy levels of ?-electrons in ethylene molecule. According to calculus of HMOT, ? is energetic value of P-electron at non bonded carbon atom that’s equal to integral:
(????_? ??_? ??)? .


Figure 2. Diagram of energy levels of ?-electrons in ethylene molecule.
4-2 Butadiene molecule.
They found four electrons refers to four carbon atoms respected to Sp2 hybridization, following figure represent the four linear combination of atomic orbitals 2Pz to form ?-electronic system as conjugated 1,3 butadiene.


Figure 3. Molecular orbital of ?-electronic system of Butadiene.

By application the linear combination of atomic orbitals for ?-electrons to get on:
?_1=a_1 ?_1+a_2 ?_(2 )+a_3 ?_(3 )+a_3 ?_(3 )------14
Orbitals of?_1,?_2, ?_3, and ?_4 are represent the atomic orbitals of 2P_1^1 Z , 2P_2^1 Z , 2P_3^1 Z, and2P_4^1 Z Respectively. To form the determinate of Butadiene molecule (open chain) according to eq.14 from type (4x4) as follow:

H_11-ES_(11 ) H_12-ES_12 ? H?_13-ES_13 ? H?_14-ES_14
H_21-ES_21 ? H?_22-ES_22 ? H?_23-ES_23 ? H?_24-ES_24
H_31-ES_31 H_32-ES_(32 ) H_33-ES_(33 ) H_34-ES_34
H_41-ES_(41 ) H_42-ES_(42 ) H_43-ES_(43 ) H_44-ES_44
---------15
By applying approximation of HMOT, to get on
|?(?-E ? 0 0@? ?-E ? 0@0 ? ?-E ?@0 0 ? ?-E)|=0 ----16
If they divided the element of determinant on ? and supposed that:
x=(?-E)/?----17
Therefore
|?(x 1 0 0@1 x 1 0@0 1 x 1@0 0 1 x)|=0-----18
x|?(x 1 0 @1 x 1 @0 1 x @0 0 1 )|-|?(@1 1 0@0 x 1@0 1 x)|=0

Or x^4-3x^2+1=0 And x=±1.62,x=±0.62
By applying this values for butadiene into eq.14
?_1=a_1 ?_1+a_2 ?_(2 )+a_3 ?_(3 )+a_3 ?_(3 )------14
To get on the results of Hückl calculus as follow:
?_i Ei a_i1 a_i2 a_i3 a_i4
?_1 ?+1.62? 0.37 0.60 0.60 0.37
?_2 ?+0.62? 0.60 0.37 -0.37 -0.60
?_3 ?-0.62? 0.60 -0.37 -0.37 0.60
?_4 ?-1.62? 0.37 -0.60 0.60 -0.37
Following figure represent the energy diagram distribution of ?- electron system butadiene.



Energy

E_4=?-1.62?

E_3=?-0.62?

E=?
??
E_2=?+0.62?
??
E_1=?+1.62?
Energy levels distribution of ?- electron system butadiene according to HMOT.
The function of bonding orbitals are ?_1 and ?_2and the function of antibonding orbitals are?_3 and ?_4. At ground state, the four electrons are stay in orbitals?_1 and ?_2, therefore the total energy of ?-electrons in open chain butadiene molecule.
E_1=?+1.62?+?+1.62? for? two electrons of ??_1
E_2=?+1.62?+?+1.62? forwo electrons of?_2
-------------------------------------------------------------------------------------
?E=E?_1+E_2=4?+4.48?
E_Butadiene=4?+4.48?-----19
By comparative the energy of ?-electrons in ethylene for two molecules with energy of ?-electrons in one open chain butadiene molecule as follow:
?E=E?_1+E_2=?+?+?+?
E_Ethylene=2?+2?-----20
As they concluded that the total energy of two molecules equal to: 2E_Ethylene=4?+4? for two double bond, that’s mean the difference equal to 0.48? , this additive energy value is called delocalization energy electrons in butadiene molecule causes more stability to the molecular structure, since electrons have a longer distance for motion on the molecule.
Cyclic Butadiene molecule is consisted from four carbon atoms with Sp2 hybridization. These carbon atoms are bonded with each other as a ring, sins C1 is bonded with C4, that’s mean two double bonds is delocalized between four atoms in 90o with lowest stability than open chain butadiene.


Figure 4. Molecular structure of ?-electron system at cyclic butadiene.
By applying approximation of HMOT, to get on
|?(?-E ? 0 ?@? ?-E ? 0@0 ? ?-E ?@? 0 ? ?-E)|=0 ----21
If we divided the element of determinant on ? and supposed that:
x=(?-E)/?----22
Therefore
|?(x 1 0 1@1 x 1 0@0 1 x 1@1 0 1 x)|=0-----23



4-3 Benzene molecule.
System of benzene molecule is consisted from three double bonds (three ?-bonds), that’s provided through 2Pz orbitals of six carbon atoms with SP2 hybridization. HMOT calculus gives large stability energy through delocalization energy of six electrons at the benzene molecule, comparative to butadiene and ethylene. Benzene ring have some probable structures, that’s called resonance phenomena like Kekule structures and Dewar structures.






Kekule structures Dewar structures

The required determinant of this system as follow:
|?(?-E ? 0 0 0 ?@? ?-E ? 0 0 0@0 ? ?-E ? 0 0@0 0 ? ?-E ? 0@0 0 0 ? ?-E ?@ ? 0 0 0 ? ?-E )|=0 ----24
If we divided the element of determinant on ? and supposed that:
x=(?-E)/?----25
Therefore
|?(x 1 0 0 0 1@1 x 1 0 0 0@0 1 x 1 0 0@0 0 1 x 1 0@0 0 0 1 x 1@ 1 0 0 0 1 x )|=0-----26
Expansion eq.26 to get on six degree equation as follow:
x^6-6x^4+9x^2-4=0 -----27
According to eq. 27, they found six values for x of molecular orbitals as follow:
?_i Ei value
?_1 E1 ?+2?
?_2,?_3 E2=E3 ?+?
?_4,?_5
E4=E5 ?-?
?_6 E6 ?-2?
-------28
Bonding molecular orbitals are ?_1,?_2 and ?_3 and antibonding molecular orbitals are?_4,?_5 and ?_6. T ground state two electrons occurs in orbital of?_1, also four electrons occurs in ?_2 and ?_3because these orbitals are double degenerates.
Total energy of benzene ring equal to
E=2E_2+4E_2
E=2(?+2?)+4(?+?)
E=6?+8?----29
The comparative between benzene ring and three double bonds of ethylene molecules that’s equal to
E=3x(2?+2?)=6?+6?
The delocalization energy of benzene, can be calculated as follow:
E_delocazation=6?+8?-6?+6?=2?-----30
The stabilization energy of benzene ring equal to2? , all six electrons are moving around all the six carbon atoms.
Because arguments based on atomic orbitals focus on the bonds formed between valence electrons on an atom, they are often said to involve a valence-bond theory.
The valence-bond model can t adequately explain the fact that some molecules contains two equivalent bonds with a bond order between that of a single bond and a double bond. The best it can do is suggest that these molecules are mixtures, or hybrids, of the two Lewis structures that can be written for these molecules.
Molecular orbital theory is more powerful than valence-bond theory because the orbitals reflect the geometry of the molecule to which they are applied. But this power carries a significant cost in terms of the ease with which the model can be visualized.


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