Orbital approximation
|
|
- Meghan Kelley
- 5 years ago
- Views:
Transcription
1 Orbital approximation Assign the electrons to an atomic orbital and a spin Construct an antisymmetrized wave function using a Slater determinant evaluate the energy with the Hamiltonian that includes the electron-electron interactions Ψ ( r, r,, r ) = 1 N 1 N! φ ( r) φ ( r ) φ ( r ) Z Z Z 1s 1 1s 1s N Z Z Z 1s r1 1s r 1s rn φ ( ) φ ( ) φ ( ) φ ( r) φ ( r ) φ ( r ) Z Z Z N 1 N N N E = Ψ H ΨΨ Ψ
2 Lithium The antisymmetric 3 electron wavefunction can be written φ ( r) φ ( r ) φ ( r ) Ψ ( r, r, r ) =,, = ( r) ( r ) ( r ) Li Li Li 1s 1 1s 1s 3 Li Li Li 1 Li Li Li 1 3 φ1s φ1s φs φ1s 1 φ1s φ1s 3 3! Li Li Li φs r1 φs r φs r3 ( ) ( ) ( ) There are two possible configurations Li Li Li φ, φ, φ and φ1s, φ1s, φs Li Li Li 1s 1s s Construct the Hamiltonian matrix to see which has the lowest energy
3 Berylium The antisymmetric 4 electron wave function can be written, Ψ ( r, r, r, r ) = φ, φ, φ, φ Be Be Be Be s 1s s s = 1 4! φ ( r) φ ( r ) φ ( r ) φ ( r ) Be Be Be Be 1s 1 1s 1s 3 1s 4 Be Be Be Be 1s r1 1s r 1s r3 1s r4 Be Be Be Be s r1 s r s r3 s r4 Be Be Be Be s ( r1) φs ( r φs r 3 φs r 4 φ ( ) φ ( ) φ ( ) φ ( ) φ ( ) φ ( ) φ ( ) φ ( ) φ ) ( ) ( )
4 Gold The antisymmetric 79 electron wave function can be written, Ψ ( r, r,, r ) = φ ( r ), φ ( r ),, φ ( r ) s 1 1s 6s 79 The determinant of an N N matrix has N! terms. 79! = (more than the atoms in the observable universe) Start with the valence electrons. Stop adding electrons when the matrix gets too big.
5 Pauli exclusion The sign of the wave function must change when two electrons are exchanged Ψ 1 ( r) ( r) 1 ( r, r ) = φ φ = ( r) ( r ) ( r ) ( r) ( φ φ φ φ ) φ1( r) φ( r) If two columns are exchanged, the wave function changes sign. If two columns are the same, the determinant is zero. The Pauli exclusion principle only holds in the noninteracting electron approximation when the many electron wave function can be written as a product of single electron wave functions, only one electron can occupy each single electron state.
6 Atomic physics summary Solutions to the Schrödinger equation accurately describe the observed energy levels in atoms. We know the equation that needs to be solved but it is intractable. A common first approximation is the orbital approximation: Assign the electrons to an atomic orbital and a spin antisymmetrized product of spin orbitals. The energy is then evaluated including the electron-electron interactions. E = Ψ H ΨΨ Sometimes assumptions have to be made about the core wave functions and the wave functions of the valence electrons are determined numerically. Ψ
7 The full Hamiltonian of a molecule H Ze e = + + ZZe A A B i A i me A ma ia, 4πε 0riA i< j4πε 0rij A< B4πε 0rAB Everything you can know about the molecule is contained in the Hamiltonian. This explains life, the universe, and everything!
8 Born Oppenheimer approximation Fix the positions of the nuclei and consider the many electron Hamiltonian. H elec Ze e = + + ZZe A A B i i me ia, 4πε 0riA i< j4πε 0rij A< B4πε 0rAB This is still too difficult. Neglect the electron-electron interactions.
9 Separation of variables (Trennung der Veränderlichen) The Schrödinger equation can be solved by the separation of variables if the total Hamiltonian can be written as a sum of Hamiltonians each depending on only one variable. H t (r 1,r,...r n ) = H 1 (r 1 ) + H (r ) +... H n (r n ) H elec Ze = + A i i me ia, 4πε 0riA i< j4πε 0rij e H Ze A elec _ red = i HMO i m = e A 4πε 0r ia i
10 Molecular orbital Hamiltonian Fix the positions of the nuclei. Solve the Schrödinger equation for one electron. Find the ground state and the excited states. 1 - r 1C + r r1a 1B r AC C A + r AB + B r BC H MO Ze A = 1 m 4πε r r e A 0 1 A
11 Molecular orbitals Molecular orbitals of a molecule are like the atomic orbitals of an atom. ψ ψ ψ ψ MO1 MO MO3 MO4 ( r ) ( r ) ( r ) ( r ) φ1s ( r ) φs ( r ) φ p ( r ) x φ ( r ) You can put two electrons, spin up and spin down, in each molecular orbital. p y
12 Molecular orbitals The first approximation for the many electron wave function of a molecule is an antisymmetrized product of molecular orbitals. The energy of this wave function should be evaluated using the electronic Hamiltonian. Ψ ( r, r ) = 1 N 1 N! ψmo 1 ( r1) ψmo 1 ( r1 ) ψmo, N ( r1) ψ ( r ) ψ MO1 ( r ) ψ ( r ) MO1 N MO, N N
13 Molecular orbitals Calculate the energy of a molecular orbital using the electronic Hamiltonian. Ψ ( r, r ) = ψ, ψ, ψ, ψ, 1 N MO1 MO1 MO MO H elec Ze e = + + ZZe A A B i i me ia, 4πε 0riA i< j4πε 0rij A< B4πε 0rAB E = Ψ H elec ΨΨ Ψ
14 Bond potentials E = Ψ H elec ΨΨ Ψ Calculate the energies for different atomic distances. The minimum yields the bond length and bond strength.
15
16 XPS
17 Bond angles Find the angle that minimizes the energy. E = Ψ H elec ΨΨ Ψ
18 Shape of a molecule H elec Ze e = + + ZZe A A B i i me ia, 4πε 0riA i< j4πε 0rij A< B4πε 0rAB
19 Shape of a molecule In Jmol, double click to start and stop a measurement.
20 Chemical reactions Calculate the energy at every stage of the reaction. E = Ψ H elec ΨΨ Ψ To calculate the speed of a chemical reaction, solve the time-dependent Schrödinger equation.
21 Chemical reactions + + C 3 H O 3 CO + 4 H O + Energy Propane Oxygen Carbon Water dioxide It is possible to calculate if the reaction is endothermic or exothermic. E = Ψ H elec ΨΨ Ψ
22 The molecular orbital can be found by: Linear Combination of Atomic Orbitals Look for a solution to the molecular orbital Hamiltonian, of the form, Here φ n are atomic orbitals.
23 Molecular orbitals of H The molecular orbital Hamiltonian for H is, r A and r B are the positions of the protons. ψ = cφ + c φ + cφ + c φ + H H H H mo 1 1, sa 1, sb 3, sa 4, sb What about spin?
24 Molecular orbitals of H The time independent Schrödinger equation, Hψ mo = Eψ mo Multiply from the left by H φ1, sa ( ) c φ H φ + c φ H φ + = E c φ φ + c φ φ + H H H H H H H H 1 1, sa mo 1, sa 1, sa mo 1, sb 1 1, sa 1, sa 1, sa 1, sb H Multiply from the left by φ1, sb ( ) c φ H φ + c φ H φ + = E c φ φ + c φ φ + H H H H H H H H 1 1, sb mo 1, sa 1, sb mo 1, sb 1 1, sb 1, sa 1, sb 1, sb Two equations with two unknowns: c 1 and c
25 Molecular orbitals of H H H H H H H H H φ1, sa Hmo φ1, sa φ1, sa Hmo φ 1, sb c1 φ1, sa φ1, sa φ1, sa φ 1, sb c1 H H H H E c = H H H H φ1, sb Hmo φ1, sa φ1, sb Hmo φ1, sb φ c 1, sb φ1, sa φ1, sb φ 1, sb Roothaan equations: HAA HAB c1 SAA SAB c1 E * * HAB H BB c = SAB S BB c Hamiltonian matrix S Overlap matrix H AA = H BB
26 Molecular orbitals of H H H c c = E AA AB 1 1 * HAB H AA c c The eigenvalues and eigenfunctions are: E H H = ± ± AA AB 1 ψ± = φ ± φ H H ( r) ( r) ( r) ( ) 1sA 1sB Both H AA and H AB are negative E + < E -
27 Molecular orbitals of H In the ground state, both electrons occupy the lower energy symmetric orbital (bonding orbital). 1 ψ ( r) ψ ( r) 1 Ψ = = ψ ( r ) ψ ( r ) ( r ) ( ) 1, r ψ+ ( r1) ψ+ ( r) Ψ = H H H H H H H H ( r1, r) ( φ1sa( r1) φ1sa( r) φ1sa( r1) φ1sb ( r) φ1sa( r) φ1sb ( r1) φ1sb ( r1) φ1sb ( r) )( ) Use this wave function including the electron-electron interaction to calculate the bond potential.
28 Student project: Draw the Bond potential of H E = Ψ H elec ΨΨ Ψ
29 r 0 = A D 0 = 3.18 ev Student project Henögl / Pranter 014 exact values r 0 = 0.74 A D 0 = 4.5 ev
30 r 0 = 0.85 A D 0 = 3.63 ev student project: Lang / Röthel exact values r 0 = 0.74 A D 0 = 4.5 ev
31 Homonuclear diatomic molecules H, N, O,... All homonuclear diatomic molecules use the molecular orbitals of H. ψ = cφ + c φ + cφ + c φ + cφ + cφ + Z Z Z Z Z Z mo 1 1, sa 1, sb 3, sa 4, sb 5 p, A 6 p, B x x The Hamiltonian matrix is as large as the number of atomic orbitals in the molecular orbital sum.
32 Homonuclear diatomic molecules All homonuclear diatomic molecules use the molecular orbitals of H. 1σ g < 1σ u < σ g < σ u < 3σ g ~ 1π u < 1π g < 3σ u g inversion symmetry from: Blinder, Introduction to Quantum Mechanics
33 Separtion of variables number of electron pairs shared from: Blinder, Introduction to Quantum Mechanics
Lecture 10. Born-Oppenheimer approximation LCAO-MO application to H + The potential energy surface MOs for diatomic molecules. NC State University
Chemistry 431 Lecture 10 Diatomic molecules Born-Oppenheimer approximation LCAO-MO application to H + 2 The potential energy surface MOs for diatomic molecules NC State University Born-Oppenheimer approximation
More information3: Many electrons. Orbital symmetries. l =2 1. m l
3: Many electrons Orbital symmetries Atomic orbitals are labelled according to the principal quantum number, n, and the orbital angular momentum quantum number, l. Electrons in a diatomic molecule experience
More informationElectron States of Diatomic Molecules
IISER Pune March 2018 Hamiltonian for a Diatomic Molecule The hamiltonian for a diatomic molecule can be considered to be made up of three terms Ĥ = ˆT N + ˆT el + ˆV where ˆT N is the kinetic energy operator
More informationσ u * 1s g - gerade u - ungerade * - antibonding σ g 1s
One of these two states is a repulsive (dissociative) state. Other excited states can be constructed using linear combinations of other orbitals. Some will be binding and others will be repulsive. Thus
More informationwe have to deal simultaneously with the motion of the two heavy particles, the nuclei
157 Lecture 6 We now turn to the structure of molecules. Our first cases will be the e- quantum mechanics of the two simplest molecules, the hydrogen molecular ion, H +, a r A r B one electron molecule,
More informationQuantum mechanics can be used to calculate any property of a molecule. The energy E of a wavefunction Ψ evaluated for the Hamiltonian H is,
Chapter : Molecules Quantum mechanics can be used to calculate any property of a molecule The energy E of a wavefunction Ψ evaluated for the Hamiltonian H is, E = Ψ H Ψ Ψ Ψ 1) At first this seems like
More informationMolecular orbitals, potential energy surfaces and symmetry
Molecular orbitals, potential energy surfaces and symmetry mathematical presentation of molecular symmetry group theory spectroscopy valence theory molecular orbitals Wave functions Hamiltonian: electronic,
More informationKronig-Penney model. 2m dx. Solutions can be found in region I and region II Match boundary conditions
Kronig-Penney model I II d V( x) m dx E Solutions can be found in region I and region II Match boundary conditions Linear differential equations with periodic coefficients Have exponentially decaying solutions,
More informationCHEMISTRY Topic #1: Bonding What Holds Atoms Together? Spring 2012 Dr. Susan Lait
CHEMISTRY 2000 Topic #1: Bonding What Holds Atoms Together? Spring 2012 Dr. Susan Lait Why Do Bonds Form? An energy diagram shows that a bond forms between two atoms if the overall energy of the system
More informationLecture 9: Molecular Orbital theory for hydrogen molecule ion
Lecture 9: Molecular Orbital theory for hydrogen molecule ion Molecular Orbital Theory for Hydrogen Molecule Ion We have seen that the Schrödinger equation cannot be solved for many electron systems. The
More informationIn this lecture we will understand how the molecular orbitals are formed from the interaction of atomic orbitals.
Lecture 7 Title: Understanding of Molecular Orbital Page-1 In this lecture we will understand how the molecular orbitals are formed from the interaction of atomic orbitals. We will see how the electrons
More informationQuantum Mechanical Simulations
Quantum Mechanical Simulations Prof. Yan Wang Woodruff School of Mechanical Engineering Georgia Institute of Technology Atlanta, GA 30332, U.S.A. yan.wang@me.gatech.edu Topics Quantum Monte Carlo Hartree-Fock
More informationIntroduction to Computational Chemistry
Introduction to Computational Chemistry Vesa Hänninen Laboratory of Physical Chemistry room B430, Chemicum 4th floor vesa.hanninen@helsinki.fi September 3, 2013 Introduction and theoretical backround September
More informationCHAPTER 11 MOLECULAR ORBITAL THEORY
CHAPTER 11 MOLECULAR ORBITAL THEORY Molecular orbital theory is a conceptual extension of the orbital model, which was so successfully applied to atomic structure. As was once playfuly remarked, a molecue
More informationIdentical Particles in Quantum Mechanics
Identical Particles in Quantum Mechanics Chapter 20 P. J. Grandinetti Chem. 4300 Nov 17, 2017 P. J. Grandinetti (Chem. 4300) Identical Particles in Quantum Mechanics Nov 17, 2017 1 / 20 Wolfgang Pauli
More informationChem 442 Review for Exam 2. Exact separation of the Hamiltonian of a hydrogenic atom into center-of-mass (3D) and relative (3D) components.
Chem 44 Review for Exam Hydrogenic atoms: The Coulomb energy between two point charges Ze and e: V r Ze r Exact separation of the Hamiltonian of a hydrogenic atom into center-of-mass (3D) and relative
More informationMolecular Bonding. Molecular Schrödinger equation. r - nuclei s - electrons. M j = mass of j th nucleus m 0 = mass of electron
Molecular onding Molecular Schrödinger equation r - nuclei s - electrons 1 1 W V r s j i j1 M j m i1 M j = mass of j th nucleus m = mass of electron j i Laplace operator for nuclei Laplace operator for
More informationMolecular-Orbital Theory
Prof. Dr. I. Nasser atomic and molecular physics -551 (T-11) April 18, 01 Molecular-Orbital Theory You have to explain the following statements: 1- Helium is monatomic gas. - Oxygen molecule has a permanent
More informationChemistry 2. Lecture 1 Quantum Mechanics in Chemistry
Chemistry 2 Lecture 1 Quantum Mechanics in Chemistry Your lecturers 8am Assoc. Prof Timothy Schmidt Room 315 timothy.schmidt@sydney.edu.au 93512781 12pm Assoc. Prof. Adam J Bridgeman Room 222 adam.bridgeman@sydney.edu.au
More informationThe symmetry properties & relative energies of atomic orbitals determine how they react to form molecular orbitals. These molecular orbitals are then
1 The symmetry properties & relative energies of atomic orbitals determine how they react to form molecular orbitals. These molecular orbitals are then filled with the available electrons according to
More informationChemistry 2000 Lecture 1: Introduction to the molecular orbital theory
Chemistry 2000 Lecture 1: Introduction to the molecular orbital theory Marc R. Roussel January 5, 2018 Marc R. Roussel Introduction to molecular orbitals January 5, 2018 1 / 24 Review: quantum mechanics
More informationMolecular Orbitals in Inorganic Chemistry. Dr. P. Hunt Rm 167 (Chemistry)
Molecular Orbitals in Inorganic Chemistry Dr. P. Hunt p.hunt@imperial.ac.uk Rm 167 (Chemistry http://www.ch.ic.ac.uk/hunt/ Outline LCAO theory orbital size matters where to put the FO energy levels the
More informationNH 3 H 2 O N 2. Why do they make chemical bonds? Molecular Orbitals
N 2 NH 3 H 2 O Why do they make chemical bonds? 5 Molecular Orbitals Why do they make chemical bonds? Stabilization Bond energy Types of Chemical Bonds Metallic Bond Ionic Bond Covalent Bond Covalent Bond
More informationLecture 8: Radial Distribution Function, Electron Spin, Helium Atom
Lecture 8: Radial Distribution Function, Electron Spin, Helium Atom Radial Distribution Function The interpretation of the square of the wavefunction is the probability density at r, θ, φ. This function
More informationGeneral Physical Chemistry II
General Physical Chemistry II Lecture 10 Aleksey Kocherzhenko October 7, 2014" Last time " promotion" Promotion and hybridization" [He] 2s 2 2p x 1 2p y 1 2p z0 " 2 unpaired electrons" [He] 2s 1 2p x 1
More informationStructure of diatomic molecules
Structure of diatomic molecules January 8, 00 1 Nature of molecules; energies of molecular motions Molecules are of course atoms that are held together by shared valence electrons. That is, most of each
More informationVALENCE Hilary Term 2018
VALENCE Hilary Term 2018 8 Lectures Prof M. Brouard Valence is the theory of the chemical bond Outline plan 1. The Born-Oppenheimer approximation 2. Bonding in H + 2 the LCAO approximation 3. Many electron
More informationOn the Uniqueness of Molecular Orbitals and limitations of the MO-model.
On the Uniqueness of Molecular Orbitals and limitations of the MO-model. The purpose of these notes is to make clear that molecular orbitals are a particular way to represent many-electron wave functions.
More informationCh 125a Problem Set 1
Ch 5a Problem Set Due Monday, Oct 5, 05, am Problem : Bra-ket notation (Dirac notation) Bra-ket notation is a standard and convenient way to describe quantum state vectors For example, φ is an abstract
More informationQuantum Chemistry. NC State University. Lecture 5. The electronic structure of molecules Absorption spectroscopy Fluorescence spectroscopy
Quantum Chemistry Lecture 5 The electronic structure of molecules Absorption spectroscopy Fluorescence spectroscopy NC State University 3.5 Selective absorption and emission by atmospheric gases (source:
More informationChemistry 120A 2nd Midterm. 1. (36 pts) For this question, recall the energy levels of the Hydrogenic Hamiltonian (1-electron):
April 6th, 24 Chemistry 2A 2nd Midterm. (36 pts) For this question, recall the energy levels of the Hydrogenic Hamiltonian (-electron): E n = m e Z 2 e 4 /2 2 n 2 = E Z 2 /n 2, n =, 2, 3,... where Ze is
More informationIntroduction to Quantum Mechanics PVK - Solutions. Nicolas Lanzetti
Introduction to Quantum Mechanics PVK - Solutions Nicolas Lanzetti lnicolas@student.ethz.ch 1 Contents 1 The Wave Function and the Schrödinger Equation 3 1.1 Quick Checks......................................
More informationCHEM3023: Spins, Atoms and Molecules
CHEM3023: Spins, Atoms and Molecules Lecture 3 The Born-Oppenheimer approximation C.-K. Skylaris Learning outcomes Separate molecular Hamiltonians to electronic and nuclear parts according to the Born-Oppenheimer
More informationMOLECULAR STRUCTURE. The general molecular Schrödinger equation, apart from electron spin effects, is. nn ee en
MOLECULAR STRUCTURE The Born-Oppenheimer Approximation The general molecular Schrödinger equation, apart from electron spin effects, is ( ) + + V + V + V =E nn ee en T T ψ ψ n e where the operators in
More informationMolecular Orbitals. Based on Inorganic Chemistry, Miessler and Tarr, 4 th edition, 2011, Pearson Prentice Hall
Molecular Orbitals Based on Inorganic Chemistry, Miessler and Tarr, 4 th edition, 2011, Pearson Prentice Hall Images from Miessler and Tarr Inorganic Chemistry 2011 obtained from Pearson Education, Inc.
More information5.61 Physical Chemistry Exam III 11/29/12. MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Chemistry Chemistry Physical Chemistry.
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Chemistry Chemistry - 5.61 Physical Chemistry Exam III (1) PRINT your name on the cover page. (2) It is suggested that you READ THE ENTIRE EXAM before
More informationCHEM-UA 127: Advanced General Chemistry I
CHEM-UA 7: Advanced General Chemistry I I. LINEAR COMBINATION OF ATOMIC ORBITALS Linear combination of atomic orbitals (LCAO) is a simple method of quantum chemistry that yields a qualitative picture of
More informationLecture 15 From molecules to solids
Lecture 15 From molecules to solids Background In the last two lectures, we explored quantum mechanics of multi-electron atoms the subject of atomic physics. In this lecture, we will explore how these
More informationApplications of Quantum Theory to Some Simple Systems
Applications of Quantum Theory to Some Simple Systems Arbitrariness in the value of total energy. We will use classical mechanics, and for simplicity of the discussion, consider a particle of mass m moving
More information2 Electronic structure theory
Electronic structure theory. Generalities.. Born-Oppenheimer approximation revisited In Sec..3 (lecture 3) the Born-Oppenheimer approximation was introduced (see also, for instance, [Tannor.]). We are
More informationCHEM3023: Spins, Atoms and Molecules
CHEM3023: Spins, Atoms and Molecules Lecture 4 Molecular orbitals C.-K. Skylaris Learning outcomes Be able to manipulate expressions involving spin orbitals and molecular orbitals Be able to write down
More informationChemistry 6 (9 am section) Spring Covalent Bonding
Chemistry 6 (9 am section) Spring 000 Covalent Bonding The stability of the bond in molecules such as H, O, N and F is associated with a sharing (equal) of the VALENCE ELECTRONS between the BONDED ATOMS.
More informationMO theory is better for spectroscopy (Exited State Properties; Ionization)
CHEM 2060 Lecture 25: MO Theory L25-1 Molecular Orbital Theory (MO theory) VB theory treats bonds as electron pairs. o There is a real emphasis on this point (over-emphasis actually). VB theory is very
More informationThe Electronic Structure of Atoms
The Electronic Structure of Atoms Classical Hydrogen-like atoms: Atomic Scale: 10-10 m or 1 Å + - Proton mass : Electron mass 1836 : 1 Problems with classical interpretation: - Should not be stable (electron
More informationMolecular Orbital Theory
Molecular Orbital Theory While FMO theory allows prediction of reactions (by thermodynamics, regio or stereochemistry), all predictions seen so far have been qualitative We have predicted that HOMO or
More information( R) MOLECULES: VALENCE BOND DESCRIPTION VALENCE BOND APPROXIMATION METHOD. Example: H 2. at some fixed R, repeat for many different values
5.6 004 Lecture #9 page MOLECULES: VLENCE OND DESCRIPTION VLENCE OND PPROXIMTION METHOD Example: H H = + + r r r r r R Determine E ( R) at some fixed R, repeat for many different values Electronic Schrodinger
More informationThe Hartree-Fock approximation
Contents The Born-Oppenheimer approximation Literature Quantum mechanics 2 - Lecture 7 November 21, 2012 Contents The Born-Oppenheimer approximation Literature 1 The Born-Oppenheimer approximation 2 3
More informationIntroduction to Hartree-Fock Molecular Orbital Theory
Introduction to Hartree-Fock Molecular Orbital Theory C. David Sherrill School of Chemistry and Biochemistry Georgia Institute of Technology Origins of Mathematical Modeling in Chemistry Plato (ca. 428-347
More informationPAPER :8, PHYSICAL SPECTROSCOPY MODULE: 29, MOLECULAR TERM SYMBOLS AND SELECTION RULES FOR DIATOMIC MOLECULES
Subject Chemistry Paper No and Title Module No and Title Module Tag 8: Physical Spectroscopy 29: Molecular Term Symbols and Selection Rules for Diatomic Molecules. CHE_P8_M29 TLE OF CONTENTS 1. Learning
More informationChapter 9: Multi- Electron Atoms Ground States and X- ray Excitation
Chapter 9: Multi- Electron Atoms Ground States and X- ray Excitation Up to now we have considered one-electron atoms. Almost all atoms are multiple-electron atoms and their description is more complicated
More informatione L 2m e the Bohr magneton
e L μl = L = μb 2m with : μ B e e 2m e the Bohr magneton Classical interation of magnetic moment and B field: (Young and Freedman, Ch. 27) E = potential energy = μ i B = μbcosθ τ = torque = μ B, perpendicular
More informationLecture #30 page 1. Valence band approach: Molecular wavefunction described in terms of 1-electron atomic orbitals
5.6 4 Lecture #3 page MOLCUL: MOLCULAR ORBITAL DCRIPTION Valence band approach: Molecular wavefunction described in terms of -ectron atomic orbitals Molecular orbital approach: Construct new set of -ectron
More informationMO Calculation for a Diatomic Molecule. /4 0 ) i=1 j>i (1/r ij )
MO Calculation for a Diatomic Molecule Introduction The properties of any molecular system can in principle be found by looking at the solutions to the corresponding time independent Schrodinger equation
More informationExam 4 Review. Exam Review: A exam review sheet for exam 4 will be posted on the course webpage. Additionally, a practice exam will also be posted.
Chem 4502 Quantum Mechanics & Spectroscopy (Jason Goodpaster) Exam 4 Review Exam Review: A exam review sheet for exam 4 will be posted on the course webpage. Additionally, a practice exam will also be
More informationMOLECULES. ENERGY LEVELS electronic vibrational rotational
MOLECULES BONDS Ionic: closed shell (+) or open shell (-) Covalent: both open shells neutral ( share e) Other (skip): van der Waals (He-He) Hydrogen bonds (in DNA, proteins, etc) ENERGY LEVELS electronic
More informationChapter 14: Phenomena
Chapter 14: Phenomena p p Phenomena: Scientists knew that in order to form a bond, orbitals on two atoms must overlap. However, p x, p y, and p z orbitals are located 90 from each other and compounds like
More informationπ* orbitals do not Molecular Orbitals for Homonuclear Diatomics
Molecular Orbitals for Homonuclear Diatomics CHEM 2060 Lecture 26: MO theory contd L26-1 Molecular orbitals can be formed pictorially by looking at the way in which atomic orbitals overlap. Let s look
More informationUNIT III Chemical Bonding There are two basic approaches to chemical bonding based on the results of quantum mechanics. These are the Valence Bond
UNIT III Chemical Bonding There are two basic approaches to chemical bonding based on the results of quantum mechanics. These are the Valence Bond Theory (VB) and the Molecular Orbital theory (MO). 1)
More informationYingwei Wang Computational Quantum Chemistry 1 Hartree energy 2. 2 Many-body system 2. 3 Born-Oppenheimer approximation 2
Purdue University CHM 67300 Computational Quantum Chemistry REVIEW Yingwei Wang October 10, 2013 Review: Prof Slipchenko s class, Fall 2013 Contents 1 Hartree energy 2 2 Many-body system 2 3 Born-Oppenheimer
More informationDiatomic Molecules. 14th May Chemical Bonds in Diatomic Molecules: Overlaps and Delocalization of Electrons
Diatomic Molecules 14th May 2009 1 Chemical Bonds in Diatomic Molecules: Overlaps and Delocalization of Electrons 1.1 H + 2 Molecule Consider the process where 2 atomic nuclei and associated electron (1
More informationQUANTUM MECHANICS AND MOLECULAR STRUCTURE
6 QUANTUM MECHANICS AND MOLECULAR STRUCTURE 6.1 Quantum Picture of the Chemical Bond 6.2 Exact Molecular Orbital for the Simplest Molecule: H + 2 6.3 Molecular Orbital Theory and the Linear Combination
More informationPHYSICAL CHEMISTRY I. Chemical Bonds
PHYSICAL CHEMISTRY I Chemical Bonds Review The QM description of bonds is quite good Capable of correctly calculating bond energies and reaction enthalpies However it is quite complicated and sometime
More informationTopic 2. Structure and Bonding Models of Covalent Compounds of p-block Elements
Topic 2 2-1 Structure and Bonding Models of Covalent Compounds of p-block Elements Bonding 2-2 Many different approaches to describe bonding: Ionic Bonding: Elements with large electronegativity differences;
More informationQuantum Condensed Matter Physics Lecture 4
Quantum Condensed Matter Physics Lecture 4 David Ritchie QCMP Lent/Easter 2019 http://www.sp.phy.cam.ac.uk/drp2/home 4.1 Quantum Condensed Matter Physics 1. Classical and Semi-classical models for electrons
More informationBe H. Delocalized Bonding. Localized Bonding. σ 2. σ 1. Two (sp-1s) Be-H σ bonds. The two σ bonding MO s in BeH 2. MO diagram for BeH 2
The Delocalized Approach to Bonding: The localized models for bonding we have examined (Lewis and VBT) assume that all electrons are restricted to specific bonds between atoms or in lone pairs. In contrast,
More informationIntroduction to Electronic Structure Theory
Introduction to Electronic Structure Theory C. David Sherrill School of Chemistry and Biochemistry Georgia Institute of Technology June 2002 Last Revised: June 2003 1 Introduction The purpose of these
More informationLS coupling. 2 2 n + H s o + H h f + H B. (1) 2m
LS coupling 1 The big picture We start from the Hamiltonian of an atomic system: H = [ ] 2 2 n Ze2 1 + 1 e 2 1 + H s o + H h f + H B. (1) 2m n e 4πɛ 0 r n 2 4πɛ 0 r nm n,m Here n runs pver the electrons,
More informationonly two orbitals, and therefore only two combinations to worry about, but things get
131 Lecture 1 It is fairly easy to write down an antisymmetric wavefunction for helium since there are only two orbitals, and therefore only two combinations to worry about, but things get complicated
More informationMolecular Bond Theory
Molecular Bond Theory Short comings of the localized electron model: electrons are not really localized so the concept of resonance was added no direct information about bond energies Molecular Orbital
More information(1/2) M α 2 α, ˆTe = i. 1 r i r j, ˆV NN = α>β
Chemistry 26 Spectroscopy Week # The Born-Oppenheimer Approximation, H + 2. Born-Oppenheimer approximation As for atoms, all information about a molecule is contained in the wave function Ψ, which is the
More informationIntroduction to Density Functional Theory
Introduction to Density Functional Theory S. Sharma Institut für Physik Karl-Franzens-Universität Graz, Austria 19th October 2005 Synopsis Motivation 1 Motivation : where can one use DFT 2 : 1 Elementary
More informationShapes of Molecules. Lewis structures are useful but don t allow prediction of the shape of a molecule.
Shapes of Molecules Lewis structures are useful but don t allow prediction of the shape of a molecule. H O H H O H Can use a simple theory based on electron repulsion to predict structure (for non-transition
More informationMolecular Simulation I
Molecular Simulation I Quantum Chemistry Classical Mechanics E = Ψ H Ψ ΨΨ U = E bond +E angle +E torsion +E non-bond Jeffry D. Madura Department of Chemistry & Biochemistry Center for Computational Sciences
More informationThe general solution of Schrödinger equation in three dimensions (if V does not depend on time) are solutions of time-independent Schrödinger equation
Lecture 17 Page 1 Lecture 17 L17.P1 Review Schrödinger equation The general solution of Schrödinger equation in three dimensions (if V does not depend on time) is where functions are solutions of time-independent
More informationApproximate Molecular Orbital Calculations for H 2. George M. Shalhoub
Approximate Molecular Orbital Calculations for H + LA SALLE UNIVESITY 9 West Olney Ave. Philaelphia, PA 94 shalhoub@lasalle.eu Copyright. All rights reserve. You are welcome to use this ocument in your
More informationMolecular Orbital Theory. WX AP Chemistry Chapter 9 Adapted from: Luis Bonilla Abel Perez University of Texas at El Paso
Molecular Orbital Theory WX AP Chemistry Chapter 9 Adapted from: Luis Bonilla Abel Perez University of Texas at El Paso Molecular Orbital Theory The goal of molecular orbital theory is to describe molecules
More informationproblem very complex is applied to bonding in a molecule as a whole i.e., includes interaction of all nuclei & e s
CB VII Molecular Orbital (MO) Theory Ref 11: 5 14-1 General further improvement on Lewis, VSEPR & VB theory; resulting in better info on: bond energy bond order magnetic properties of molecules...... 14-2
More informationMOLECULAR ORBITAL THEORY Chapter 10.8, Morrison and Boyd
MOLECULAR ORBITAL THEORY Chapter 10.8, Morrison and Boyd more understanding: why oxygen is paramagnetic, why H2 + exists; explanation of excited electronic states (e.g., visible spectra) eliminates need
More informationSame idea for polyatomics, keep track of identical atom e.g. NH 3 consider only valence electrons F(2s,2p) H(1s)
XIII 63 Polyatomic bonding -09 -mod, Notes (13) Engel 16-17 Balance: nuclear repulsion, positive e-n attraction, neg. united atom AO ε i applies to all bonding, just more nuclei repulsion biggest at low
More informationChemistry 3502/4502. Final Exam Part I. May 14, 2005
Advocacy chit Chemistry 350/450 Final Exam Part I May 4, 005. For which of the below systems is = where H is the Hamiltonian operator and T is the kinetic-energy operator? (a) The free particle
More informationComputational Methods. Chem 561
Computational Methods Chem 561 Lecture Outline 1. Ab initio methods a) HF SCF b) Post-HF methods 2. Density Functional Theory 3. Semiempirical methods 4. Molecular Mechanics Computational Chemistry " Computational
More informationChapter 4 Symmetry and Chemical Bonding
Chapter 4 Symmetry and Chemical Bonding 4.1 Orbital Symmetries and Overlap 4.2 Valence Bond Theory and Hybrid Orbitals 4.3 Localized and Delocalized Molecular Orbitals 4.4 MX n Molecules with Pi-Bonding
More information4πε. me 1,2,3,... 1 n. H atom 4. in a.u. atomic units. energy: 1 a.u. = ev distance 1 a.u. = Å
H atom 4 E a me =, n=,,3,... 8ε 0 0 π me e e 0 hn ε h = = 0.59Å E = me (4 πε ) 4 e 0 n n in a.u. atomic units E = r = Z n nao Z = e = me = 4πε = 0 energy: a.u. = 7. ev distance a.u. = 0.59 Å General results
More informationVALENCE Hilary Term 2019
VALENCE Hilary Term 2019 8 Lectures Prof M. Brouard Valence is the theory of the chemical bond Outline plan 1. The Born-Oppenheimer approximation 2. Bonding in H + 2 the LCAO approximation 3. Many electron
More informationElectronic structure calculations: fundamentals George C. Schatz Northwestern University
Electronic structure calculations: fundamentals George C. Schatz Northwestern University Electronic Structure (often called Quantum Chemistry) calculations use quantum mechanics to determine the wavefunctions
More informationOn semiempirical treatment of hydrocarbons
On semiempirical treatment of hydrocarbons Debangshu Mukherjee B.Sc Physics, IInd Year Chennai Mathematical Institute 27..2009 Abstract As, we do in case of small systems like H + 2 or He atom, we write
More information0 belonging to the unperturbed Hamiltonian H 0 are known
Time Independent Perturbation Theory D Perturbation theory is used in two qualitatively different contexts in quantum chemistry. It allows one to estimate (because perturbation theory is usually employed
More information5.62 Physical Chemistry II Spring 2008
MIT OpenCourseWare http://ocw.mit.edu 5.6 Physical Chemistry II Spring 008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 5.6 Lecture #13: Nuclear Spin
More informationAb initio calculations for potential energy surfaces. D. Talbi GRAAL- Montpellier
Ab initio calculations for potential energy surfaces D. Talbi GRAAL- Montpellier A theoretical study of a reaction is a two step process I-Electronic calculations : techniques of quantum chemistry potential
More informationBe H. Delocalized Bonding. Localized Bonding. σ 2. σ 1. Two (sp-1s) Be-H σ bonds. The two σ bonding MO s in BeH 2. MO diagram for BeH 2
The Delocalized Approach to Bonding: The localized models for bonding we have examined (Lewis and VBT) assume that all electrons are restricted to specific bonds between atoms or in lone pairs. In contrast,
More informationHartree-Fock-Roothan Self-Consistent Field Method
Hartree-Fock-Roothan Self-Consistent Field Method 1. Helium Here is a summary of the derivation of the Hartree-Fock equations presented in class. First consider the ground state of He and start with with
More informationChapter 2 Approximation Methods Can be Used When Exact Solutions to the Schrödinger Equation Can Not be Found.
Chapter 2 Approximation Methods Can be Used When Exact Solutions to the Schrödinger Equation Can Not be Found. In applying quantum mechanics to 'real' chemical problems, one is usually faced with a Schrödinger
More informationBorn-Oppenheimer Approximation
Born-Oppenheimer Approximation Adiabatic Assumption: Nuclei move so much more slowly than electron that the electrons that the electrons are assumed to be obtained if the nuclear kinetic energy is ignored,
More information7.1 Variational Principle
7.1 Variational Principle Suppose that you want to determine the ground-state energy E g for a system described by H, but you are unable to solve the time-independent Schrödinger equation. It is possible
More informationFrom Last Time Important new Quantum Mechanical Concepts. Atoms and Molecules. Today. Symmetry. Simple molecules.
Today From Last Time Important new Quantum Mechanical Concepts Indistinguishability: Symmetries of the wavefunction: Symmetric and Antisymmetric Pauli exclusion principle: only one fermion per state Spin
More informationECE440 Nanoelectronics. Lecture 07 Atomic Orbitals
ECE44 Nanoelectronics Lecture 7 Atomic Orbitals Atoms and atomic orbitals It is instructive to compare the simple model of a spherically symmetrical potential for r R V ( r) for r R and the simplest hydrogen
More informationCHEM J-5 June 2014
CHEM1101 2014-J-5 June 2014 The molecular orbital energy level diagrams for H 2, H 2 +, H 2 and O 2 are shown below. Fill in the valence electrons for each species in its ground state and label the types
More informationAN INTRODUCTION TO QUANTUM CHEMISTRY. Mark S. Gordon Iowa State University
AN INTRODUCTION TO QUANTUM CHEMISTRY Mark S. Gordon Iowa State University 1 OUTLINE Theoretical Background in Quantum Chemistry Overview of GAMESS Program Applications 2 QUANTUM CHEMISTRY In principle,
More informationThe Hydrogen Molecule-Ion
Sign In Forgot Password Register ashwenchan username password Sign In If you like us, please share us on social media. The latest UCD Hyperlibrary newsletter is now complete, check it out. ChemWiki BioWiki
More information