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1 Bra-ket notation Quantum states representations Homo-nuclear diatomic molecule Hetero-nuclear diatomic molecule Bond energy The Diatomic Molecule MATS-535 Electronics and Photonics Materials Dr. Vladimir Gavrilenko Norfolk State University

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2 Bra and ket notation A wave function is a representation of the quantum state in real space. The is called a ‘ket’. At each point r in space the quantum state is represented by the function. The quantum state could be expanded in a set of ortho-normal basis states: Where C’s are called expansion coefficients

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3 Bra and ket notation

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4 (**) HW

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5 Wave Functions of Hydrogen Atom

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6 Atomic Wave Function Orthonormality (*)HW

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7 The Homonuclear Diatomic Molecule 1 2 Schrodinger equations for isolated H-atoms Full wave function of the H-molecule

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8 The Electronic Structure Schrodinger equation Projection onto basis set Orthogonality conditions:

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9 The Secular Equation Secular equation

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10 Solutions of the Secular Equation Solutions Bonding (b) and antibonding (a) molecular orbital energies Normalized eigen states

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11 Electron Energy Structure and Wave Functions of Hydrogen Molecule LUMO – Lowest Unoccupied Molecular Orbital HOMO – Highest Occupied Molecular Orbital

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12 Wave Functions Analysis

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13 Wave Functions Analysis

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14 Dependence on Time Time dependent Schrodinger equation Substitute:

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15 Dependence on Time First order differential equations with constant coefficients are solved by exponential functions: where Boundary conditions: at t=0 molecule is in state 1. Therefore: The probability that the molecule is in state 1 or 2:

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16 The Heteronuclear Diatomic Molecule A B Schrodinger equations for isolated H-atoms Assume: Full wave function of the H-molecule

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17 The Electronic Structure Schrodinger equation: Projection onto basis set

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18 The Secular Equation Secular equation

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19 The Secular Equation Substitution: Average on-site energy Solution:

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20 Charge Redistribution Insert Obtain for: For the bonding state For the antibonding state

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21 The Charge Transfer in Heteronuclear Diatomic Molecule AB 1. For: The homonuclear case: no charge transfer

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22 The Charge Transfer in Heteronuclear Diatomic Molecule AB 2. For: 1.Bonding state: charge is transferred to the B-molecule (lower on-site energy) 2.Antibonding state: charge is transferred to the A-molecule (higher on-site energy)

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23 The Ionic Bond Parameters Polarity: Covalency: Completely ionic limit Completely covalent limit

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24 Problems: 1.Using solutions of the secular equation for homonuclear diatomic molecule obtain orthonormal wave functions (see slide 10) 2.Show that wave functions of hydrogen atom are mutually orthogonal (problem marked by(*)) (slide 6). 3.Assuming mutual ortho-normality of atomic s- and p-functions show ortho-normality of the sp 3 hybrides (problem marked by(**)) (slide 4). 4.Obtain conditions for eigen function coefficients corresponding to bonding and antibonding states for heteronuclear diatomic molecule (slide 22).

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