Molecular orbitals, potential energy surfaces and symmetry
|
|
- Clement Long
- 6 years ago
- Views:
Transcription
1 Molecular orbitals, potential energy surfaces and symmetry mathematical presentation of molecular symmetry group theory spectroscopy valence theory molecular orbitals Wave functions Hamiltonian: electronic, nuclear wave functions H = Hel + Hvib + Hrot Energy scales: ev, 0.1 ev, 0.01 ev
2 Molecular orbitals: constructed from atomic orbitals Atomic orbitals have spherical symmetry basis set is spherical harmonics Yl m Molecular orbitals have symmetry determined by molecular geometry basis set is atomic orbitals, ϕ A ϕ B Pauli principle applies Effective bonding energies of atomic orbitals are similar overlap between orbitals strong same symmetry with respect to rotation about molecular bond axis
3 Molecular bonds: bonding and antibonding Antibonding orbital Node (zero density) between nuclei DESTRUCTIVE INTERFERENCE + Bonding orbital nonzero electron density between nuclei CONSTRUCTIVE INTERFERENCE
4 Energy level diagrams for homonuclear diatomic molecules Li2 to N2 O2 and F2
5 Energy level diagram for heteronuclear molecules Atomic orbitals ϕ A ϕ B s-orbitals different atomic energy form σ molecular orbitals
6 Example: hydrogen molecular ion Calculate orbitals for the hydrogen molecular ion, H2+ Calculate potential energies for bonding and antibonding sigma states Use atomic 1s orbitals as a basis set Effective charge described by ζ (= 1 for hydrogenic case) The LCAO wave function in it s simplest form
7 Basic strategy and equations Solve the SE using the Hamiltonian for a 3-body system (H1, H2, e-) Two wave functions, two equations use symmetry to simplify solve for energy eigenvalues use atomic units...
8 Symmetry... For a homonuclear molecule the basis wave functions are identical, and the coefficients in the LCAO wave function are equal. H11=H22 Two solutions: c1 =
9 Symmetry.. i gerade i ungerade
10 Calculation of wave functions, potentials and other quantities... A brief MATLAB interlude...
11 What does it mean?
12 Polyatomics: Symmetry operations and elements I, identity Cn, n-fold rotation σ Reflection (in a plane) i, inversion (through a center of symmetry) Sn, improper rotation (n-fold rotation in a plane followed by reflection through a plane perpendicular to rot plane)
13 Rotation Reflection Improper rotation Inversion S = C n. σ"
14 Symmetry groups Lowest symmetry C1, Cs, Ci Cn: I, n-fold rotation (about principal axis) Cnv: I, n-fold rotation, vertical reflections Cnh: I, n-fold rotation, horizontal reflection Dn: I, n-fold rotation, about principal and perpendicular axis Dnh: Dn and horiz reflection (homonuclear) Sn: I, n-fold improper rotation C2v
15 Calculus of symmetry elements Symmetry elements generate other elements: C3 2 = C3 x C3 All symmetry elements are generated from others Matrix representation Group theory applies to molecular symmetries
16 Character tables Symmetry classification of properties of molecules (wave functions, orbitals, states) Degeneracy Symmetry elements Basis for constructing orbitals, vibrational motion, electronic transitions
17 Example: C2v point group character table (water, SO2, etc) Generating elements Irreducible representation of symmetry species I C 2 σ v σ v A A B B
18 Heteronuclear diatomics Σ+ Σ Π Δ I C... σ v A A E 1 2 2cosφ 0 E 2 2 2cos2 φ$... 0
19 Molecular bonds: σ symmetry
20 Molecular bonds: π symmetry
21 Potential energy function Neutral oxygen molecule
22 Symmetry adapted Linear Combination AO Example: water C2v symmetry I C2 σ v(xz) σ v(yz) A A B B
23 Construct wave functionsfollowing the rules I C2 σ v(xz) σ v(yz) A B O 2s O 2px O 2py O 2pz A1 B2 B1 A1 H 1sA 1 H 1sB 1 H 1sB H 1sB 1 H 1sA 1 H 1sA 1/2 1sA + 1sB 1/2 1sA - 1sB A1 B2
24 Symmetry adapted wave functions Ψ(A1) = c1 O 2s + c2 O 2pz + c3 ϕ (A1) Ψ(B1) = O 2py Ψ(B2) = c4 O 2px + c5 ϕ (B2) where the symmetry adapted molecular orbitals are ϕ (A1) = 1/2 1sA + 1sB ϕ (B2) = 1/2 1sA - 1sB
25 Molecular orbitals for water
26 Molecular orbital labels Water molecule: 8 oxygen electrons 2 hydrogen 1s Walsh diagram Ground-state configuration and term bent XH 2 molecules linear O1s 2 1a1 2 1b2 2 2a1 2 1b1 2 1 A1 4a 1 3 g 1b1 1 u 1b2 A1... a1 B1...b1 B2... b2 3a 1 2a 1 1 u 2 g Bond angle 2
27 Vibrational modes A1 B1 Symmetric antisymmetric
28 Quantum mechanical treatment of vibrations in molecules The full wave function is a product of electronic and nuclear wave functions (Born-Oppenheimer approx) electrons (r) immediately follow the nuclear motion (R) Break down matrix element to electronic and nuclear transition elements
29 Quantum mechanical treatment of vibrations in molecules Attractive restoring force (approximate harmonic osc)... solve the Schrödinger equation:! With eigenfunctions based upon Hermite polynomials v = 1 1/2 H 2 v v! 1/2 v (y) exp( y 2 /2) and eigenfrequencies: = 1 2 k µ! 1/2. Table 2.1: Hermite polynomials. v H v (y) v H v (y) y 3 12y 1 2y 4 16y 4 48y y y 5 160y y! 1/2 y = 4 2 µ (r r e ) h
30 Vibrational progressions E ν = E 0 + ω 0 (ν +1/ 2)+ ω 0 x e (ν +1/ 2) 2 Adiabatic binding energy Harmonic term Anharmonic term
31 Molecular potential function, vibrations and dissociation
32 Franck-Condon principle Absorption of radiation-vibronic transitions Electronic transition from state with energy E el and vibrational quantum number v =0 to an excited state E el The electronic state transition is independent of the vibrational transition Electronic state transition: DIPOLE selection rules Vibronic transition, no strict selection rules E el(v ) --> E el(v )
33 Frank-Condon principle: intuitive picture Franck-Condon region is defined by the ground-state wave function Absorption from the ground state is strongest to the wave function in the excited state that has a maximum in the center of the Franck-Condon region
34 Franck-Condon principle: quantum mechanical treatment Transition moment for vibrational transitions S = Ψ 0 (x)ψ' o (x)dx general form of vibrational wave functions Hv are Hermite polynomials, and α = 2π/h mhv S 0 0 = e α 4 2 ' ( R e R e ) 2 Δq = 2 S µω gr 1/ 2 ω gr ω fin
The 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 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 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 informationwbt Λ = 0, 1, 2, 3, Eq. (7.63)
7.2.2 Classification of Electronic States For all diatomic molecules the coupling approximation which best describes electronic states is analogous to the Russell- Saunders approximation in atoms The orbital
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 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 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 informationBrief introduction to molecular symmetry
Chapter 1 Brief introduction to molecular symmetry It is possible to understand the electronic structure of diatomic molecules and their interaction with light without the theory of molecular symmetry.
More informationChapter IV: Electronic Spectroscopy of diatomic molecules
Chapter IV: Electronic Spectroscopy of diatomic molecules IV.2.1 Molecular orbitals IV.2.1.1. Homonuclear diatomic molecules The molecular orbital (MO) approach to the electronic structure of diatomic
More informationLecture 9 Electronic Spectroscopy
Lecture 9 Electronic Spectroscopy Molecular Orbital Theory: A Review - LCAO approximaton & AO overlap - Variation Principle & Secular Determinant - Homonuclear Diatomic MOs - Energy Levels, Bond Order
More informationSymmetry and Molecular Orbitals (I)
Symmetry and Molecular Orbitals (I) Simple Bonding Model http://chiuserv.ac.nctu.edu.tw/~htchiu/chemistry/fall-2005/chemical-bonds.htm Lewis Structures Octet Rule Resonance Formal Charge Oxidation Number
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 informationLecture 14 Chemistry 362 M. Darensbourg 2017 Spring term. Molecular orbitals for diatomics
Lecture 14 Chemistry 362 M. Darensbourg 2017 Spring term Molecular orbitals for diatomics Molecular Orbital Theory of the Chemical Bond Simplest example - H 2 : two H atoms H A and H B Only two a.o.'s
More informationMolecular Orbital Theory
Molecular Orbital Theory 1. MO theory suggests that atomic orbitals of different atoms combine to create MOLECULAR ORBITALS 2. Electrons in these MOLECULAR ORBITALS belong to the molecule as whole 3. This
More informationNPTEL/IITM. Molecular Spectroscopy Lectures 1 & 2. Prof.K. Mangala Sunder Page 1 of 15. Topics. Part I : Introductory concepts Topics
Molecular Spectroscopy Lectures 1 & 2 Part I : Introductory concepts Topics Why spectroscopy? Introduction to electromagnetic radiation Interaction of radiation with matter What are spectra? Beer-Lambert
More informationConcept of a basis. Based on this treatment we can assign the basis to one of the irreducible representations of the point group.
Concept of a basis A basis refers to a type of function that is transformed by the symmetry operations of a point group. Examples include the spherical harmonics, vectors, internal coordinates (e..g bonds,
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 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 informationMolecular spectroscopy Multispectral imaging (FAFF 020, FYST29) fall 2017
Molecular spectroscopy Multispectral imaging (FAFF 00, FYST9) fall 017 Lecture prepared by Joakim Bood joakim.bood@forbrf.lth.se Molecular structure Electronic structure Rotational structure Vibrational
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 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 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 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 informationLearning Objectives and Worksheet VIII. Chemistry 1B-AL Fall Lectures (13-14) Molecular Orbital Theory of Covalent Bonding
Learning Objectives and Worksheet VIII Chemistry 1B-AL Fall 2016 Lectures (13-14) Molecular Orbital Theory of Covalent Bonding WE WILL BE COVERING CHAPTER 14 IN A DIFFERENT ORDER THAN THE TEXT: first we
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 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 informationPAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE NO. : 23 (NORMAL MODES AND IRREDUCIBLE REPRESENTATIONS FOR POLYATOMIC MOLECULES)
Subject Chemistry Paper No and Title Module No and Title Module Tag 8/ Physical Spectroscopy 23/ Normal modes and irreducible representations for polyatomic molecules CHE_P8_M23 TABLE OF CONTENTS 1. Learning
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 informationPhysical Chemistry I Fall 2016 Second Hour Exam (100 points) Name:
Physical Chemistry I Fall 2016 Second Hour Exam (100 points) Name: (20 points) 1. Quantum calculations suggest that the molecule U 2 H 2 is planar and has symmetry D 2h. D 2h E C 2 (z) C 2 (y) C 2 (x)
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 informationUsing Symmetry to Generate Molecular Orbital Diagrams
Using Symmetry to Generate Molecular Orbital Diagrams review a few MO concepts generate MO for XH 2, H 2 O, SF 6 Formation of a bond occurs when electron density collects between the two bonded nuclei
More informationChemistry 431. Lecture 14. Wave functions as a basis Diatomic molecules Polyatomic molecules Huckel theory. NC State University
Chemistry 431 Lecture 14 Wave functions as a basis Diatomic molecules Polyatomic molecules Huckel theory NC State University Wave functions as the basis for irreducible representations The energy of the
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 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 informationSpectra of Atoms and Molecules. Peter F. Bernath
Spectra of Atoms and Molecules Peter F. Bernath New York Oxford OXFORD UNIVERSITY PRESS 1995 Contents 1 Introduction 3 Waves, Particles, and Units 3 The Electromagnetic Spectrum 6 Interaction of Radiation
More informationP. W. Atkins and R. S. Friedman. Molecular Quantum Mechanics THIRD EDITION
P. W. Atkins and R. S. Friedman Molecular Quantum Mechanics THIRD EDITION Oxford New York Tokyo OXFORD UNIVERSITY PRESS 1997 Introduction and orientation 1 Black-body radiation 1 Heat capacities 2 The
More information5.111 Lecture Summary #13 Monday, October 6, 2014
5.111 Lecture Summary #13 Monday, October 6, 2014 Readings for today: Section 3.8 3.11 Molecular Orbital Theory (Same in 5 th and 4 th ed.) Read for Lecture #14: Sections 3.4, 3.5, 3.6 and 3.7 Valence
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 informationLecture 12. Symmetry Operations. NC State University
Chemistry 431 Lecture 12 Group Theory Symmetry Operations NC State University Wave functions as the basis for irreducible representations The energy of the system will not change when symmetry Operations
More informationMolecular Orbital Approach to Bonding
Molecular Orbital Approach to Bonding Chemistry 362; spring 2019 Marcetta Y. Darensbourg, Professor Xuemei Yang, Graduate Assistant Kyle Burns, Graduate Assistant The following slides were modified from
More informationChapter 18 Molecular orbitals and spectroscopy Conjugation of bonds and resonance structures
Chapter 18 Molecular orbitals and spectroscopy 18.1 Diatomic molecules 18.2 Polyatomic molecules 18.3 Conjugation of bonds and resonance structures 18.4 The interaction of light and matter (spectroscopy)
More informationElectronic transitions: Vibrational and rotational structure
Electronic transitions: Vibrational and rotational structure An electronic transition is made up of vibrational bands, each of which is in turn made up of rotational lines Vibrational structure Vibrational
More informationChemistry 483 Lecture Topics Fall 2009
Chemistry 483 Lecture Topics Fall 2009 Text PHYSICAL CHEMISTRY A Molecular Approach McQuarrie and Simon A. Background (M&S,Chapter 1) Blackbody Radiation Photoelectric effect DeBroglie Wavelength Atomic
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 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 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 informationContent. 1. Overview of molecular spectra 2. Rotational spectra 3. Vibrational spectra 4. Electronic spectra
Content 1. Overview of molecular spectra 2. Rotational spectra 3. Vibrational spectra 4. Electronic spectra Molecular orbital theory Electronic quantum numbers Vibrational structure of electronic transitions
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 information1s atomic orbital 2s atomic orbital 2s atomic orbital (with node) 2px orbital 2py orbital 2pz orbital
Atomic Orbitals 1s atomic orbital 2s atomic orbital 2s atomic orbital (with node) 2px orbital 2py orbital 2pz orbital Valence Bond Theory and ybridized Atomic Orbitals Bonding in 2 1s 1s Atomic Orbital
More informationChemistry 1B, Fall 2012 Lectures 15-16
Chemistry 1B Fall 2012 Quantum Mechanics of the Covalent Bond for chapter 14 animations and links see: http://switkes.chemistry.ucsc.edu/teaching/chem1b/www_other_links/ch14_links.htm 1 LISTEN UP!!! WE
More informationMolecular Orbital Theory
Molecular Orbital Theory Paramagnetic properties of O 2 pranjoto utomo Covalent Bonding Theory Valence Bond Theory useful for deriving shapes/polarity simple but inaccurate/deficient Molecular Orbital
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 informationCHM Physical Chemistry II Chapter 12 - Supplementary Material. 1. Einstein A and B coefficients
CHM 3411 - Physical Chemistry II Chapter 12 - Supplementary Material 1. Einstein A and B coefficients Consider two singly degenerate states in an atom, molecule, or ion, with wavefunctions 1 (for the lower
More informationRethinking Hybridization
Rethinking Hybridization For more than 60 years, one of the most used concepts to come out of the valence bond model developed by Pauling was that of hybrid orbitals. The ideas of hybridization seemed
More information5.111 Principles of Chemical Science
MIT OpenCourseWare http://ocw.mit.edu 5.111 Principles of Chemical Science Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 5.111 Lecture Summary
More informationI 2 Vapor Absorption Experiment and Determination of Bond Dissociation Energy.
I 2 Vapor Absorption Experiment and Determination of Bond Dissociation Energy. What determines the UV-Vis (i.e., electronic transitions) band appearance? Usually described by HOMO LUMO electron jump LUMO
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 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 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 informationChemistry 881 Lecture Topics Fall 2001
Chemistry 881 Lecture Topics Fall 2001 Texts PHYSICAL CHEMISTRY A Molecular Approach McQuarrie and Simon MATHEMATICS for PHYSICAL CHEMISTRY, Mortimer i. Mathematics Review (M, Chapters 1,2,3 & 4; M&S,
More informationOrbital approximation
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
More information7. Arrange the molecular orbitals in order of increasing energy and add the electrons.
Molecular Orbital Theory I. Introduction. A. Ideas. 1. Start with nuclei at their equilibrium positions. 2. onstruct a set of orbitals that cover the complete nuclear framework, called molecular orbitals
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 informationBonding in Molecules Prof John McGrady Michaelmas Term 2009
Bonding in Molecules Prof John McGrady Michaelmas Term 2009 6 lectures building on material presented in Introduction to Molecular Orbitals (HT Year 1). Provides a basis for analysing the shapes, properties,
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 informationExp. 4. Quantum Chemical calculation: The potential energy curves and the orbitals of H2 +
Exp. 4. Quantum Chemical calculation: The potential energy curves and the orbitals of H2 + 1. Objectives Quantum chemical solvers are used to obtain the energy and the orbitals of the simplest molecules
More informationChemistry 795T. NC State University. Lecture 4. Vibrational and Rotational Spectroscopy
Chemistry 795T Lecture 4 Vibrational and Rotational Spectroscopy NC State University The Dipole Moment Expansion The permanent dipole moment of a molecule oscillates about an equilibrium value as the molecule
More informationChapter 5. Molecular Orbitals
Chapter 5. Molecular Orbitals MO from s, p, d, orbitals: - Fig.5.1, 5.2, 5.3 Homonuclear diatomic molecules: - Fig. 5.7 - Para- vs. Diamagnetic Heteronuclear diatomic molecules: - Fig. 5.14 - ex. CO Hybrid
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 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 informationCHAPTER 13 Molecular Spectroscopy 2: Electronic Transitions
CHAPTER 13 Molecular Spectroscopy 2: Electronic Transitions I. General Features of Electronic spectroscopy. A. Visible and ultraviolet photons excite electronic state transitions. ε photon = 120 to 1200
More informationA Quantum Mechanical Model for the Vibration and Rotation of Molecules. Rigid Rotor
A Quantum Mechanical Model for the Vibration and Rotation of Molecules Harmonic Oscillator Rigid Rotor Degrees of Freedom Translation: quantum mechanical model is particle in box or free particle. A molecule
More informationLecture 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 informationChem 442 Review of Spectroscopy
Chem 44 Review of Spectroscopy General spectroscopy Wavelength (nm), frequency (s -1 ), wavenumber (cm -1 ) Frequency (s -1 ): n= c l Wavenumbers (cm -1 ): n =1 l Chart of photon energies and spectroscopies
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 informationeigenvalues eigenfunctions
Born-Oppenheimer Approximation Atoms and molecules consist of heavy nuclei and light electrons. Consider (for simplicity) a diatomic molecule (e.g. HCl). Clamp/freeze the nuclei in space, a distance r
More informationChem120a : Exam 3 (Chem Bio) Solutions
Chem10a : Exam 3 (Chem Bio) Solutions November 7, 006 Problem 1 This problem will basically involve us doing two Hückel calculations: one for the linear geometry, and one for the triangular geometry. We
More informationMOLECULAR SPECTROSCOPY
MOLECULAR SPECTROSCOPY First Edition Jeanne L. McHale University of Idaho PRENTICE HALL, Upper Saddle River, New Jersey 07458 CONTENTS PREFACE xiii 1 INTRODUCTION AND REVIEW 1 1.1 Historical Perspective
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 informationChemistry 543--Final Exam--Keiderling May 5, pm SES
Chemistry 543--Final Exam--Keiderling May 5,1992 -- 1-5pm -- 174 SES Please answer all questions in the answer book provided. Make sure your name is clearly indicated and that the answers are clearly numbered,
More informationTentative content material to be covered for Exam 2 (Wednesday, November 2, 2005)
Tentative content material to be covered for Exam 2 (Wednesday, November 2, 2005) Chapter 16 Quantum Mechanics and the Hydrogen Atom 16.1 Waves and Light 16.2 Paradoxes in Classical Physics 16.3 Planck,
More informationChemistry 3502/4502. Final Exam Part I. May 14, 2005
Chemistry 3502/4502 Final Exam Part I May 14, 2005 1. For which of the below systems is = where H is the Hamiltonian operator and T is the kinetic-energy operator? (a) The free particle (e) The
More informationLECTURE 3 DIRECT PRODUCTS AND SPECTROSCOPIC SELECTION RULES
SYMMETRY II. J. M. GOICOECHEA. LECTURE 3 1 LECTURE 3 DIRECT PRODUCTS AND SPECTROSCOPIC SELECTION RULES 3.1 Direct products and many electron states Consider the problem of deciding upon the symmetry of
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 informationProblem Set 5 Solutions
Chemistry 362 Dr Jean M Standard Problem Set 5 Solutions ow many vibrational modes do the following molecules or ions possess? [int: Drawing Lewis structures may be useful in some cases] In all of the
More informationSection 2 Simple Molecular Orbital Theory
Section 2 Simple Molecular Orbital Theory In this section, the conceptual framework of molecular orbital theory is developed. Applications are presented and problems are given and solved within qualitative
More informationCB VII. Molecular Orbital (MO) Theory. General. Basic Principles. Basic Ideas. further improvement on Lewis, VSEPR & VB theory;
chem101/3, D1 fa010 po 14 1 CB VII Molecular Orbital (MO) Theory chem101/3, D1 fa010 po 14 General further improvement on Lewis, VSEPR & VB theory; resulting in better info on: bond energy bond order magnetic
More informationVibrational and Rotational Analysis of Hydrogen Halides
Vibrational and Rotational Analysis of Hydrogen Halides Goals Quantitative assessments of HBr molecular characteristics such as bond length, bond energy, etc CHEM 164A Huma n eyes Near-Infrared Infrared
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 informationV( x) = V( 0) + dv. V( x) = 1 2
Spectroscopy 1: rotational and vibrational spectra The vibrations of diatomic molecules Molecular vibrations Consider a typical potential energy curve for a diatomic molecule. In regions close to R e (at
More informationPrinciples of Molecular Spectroscopy
Principles of Molecular Spectroscopy What variables do we need to characterize a molecule? Nuclear and electronic configurations: What is the structure of the molecule? What are the bond lengths? How strong
More informationChemistry 1B, Fall 2013 Lectures 15-16
Chemistry 1, Fall 2013 Lectures 1516 Chemistry 1 Fall 2013 Lectures 1516 Quantum Mechanics of the Covalent ond LISTEN UP!!! WE WILL E COVERING SECOND PRT OF CHPTER 14 (pp 676688) FIRST You will go CRZY
More informationChemistry 2000 Lecture 2: LCAO-MO theory for homonuclear diatomic molecules
Chemistry 2000 Lecture 2: LCAO-MO theory for homonuclear diatomic molecules Marc R. Roussel January 5, 2018 Marc R. Roussel Homonuclear diatomics January 5, 2018 1 / 17 MO theory for homonuclear diatomic
More informationA Rigorous Introduction to Molecular Orbital Theory and its Applications in Chemistry. Zachary Chin, Alex Li, Alex Liu
A Rigorous Introduction to Molecular Orbital Theory and its Applications in Chemistry Zachary Chin, Alex Li, Alex Liu Quantum Mechanics Atomic Orbitals and Early Bonding Theory Quantum Numbers n: principal
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 informationValence bond theory accounts, at least qualitatively, for the stability of the covalent bond in terms of overlapping atomic orbitals.
Molecular Orbital Theory Valence bond theory accounts, at least qualitatively, for the stability of the covalent bond in terms of overlapping atomic orbitals. Using the concept of hybridization, valence
More informationATOMS. Central field model (4 quantum numbers + Pauli exclusion) n = 1, 2, 3,... 0 l n 1 (0, 1, 2, 3 s, p, d, f) m l l, m s = ±1/2
ATOMS Central field model (4 quantum numbers + Pauli exclusion) n = 1, 2, 3,... 0 l n 1 (0, 1, 2, 3 s, p, d, f) m l l, m s = ±1/2 Spectroscopic notation: 2S+1 L J (Z 40) L is total orbital angular momentum
More informationChemistry 21b Spectroscopy
Chemistry 21b Spectroscopy 05Feb2018 Lecture # 14 Electronic Structure & Nomenclature of Diatomic Molecules; the Franck-Condon Approximation Before looking at the electronic structure of simple molecules,
More informationChemical Bonding & Structure
Chemical Bonding & Structure Further aspects of covalent bonding and structure Hybridization Ms. Thompson - HL Chemistry Wooster High School Topic 14.2 Hybridization A hybrid orbital results from the mixing
More information