Ch120 - Study Guide 10
|
|
- Roland Fitzgerald
- 6 years ago
- Views:
Transcription
1 Ch120 - Study Guide 10 Adam Griffith October 17, 2005 In this guide: Symmetry; Diatomic Term Symbols; Molecular Term Symbols Last updated October 27, The Origin of m l States and Symmetry We are familiar that there are three different ways we can arrange four electrons into three different p orbitals, and this denotes the 3 P states of O. (See Figure 1.) From quantum mechanics we know there are associated values of m l that describe the relative angular momentum of the three degenerate states. Assigning these m l values to the familiar labels p x, p y, and p z is not trivial, however. Recall that the spatial part for the wavefunction for any orbital can be expressed in terms of its radial and angluar components. I.e. the expression, ψ n,l,ml (r, θ, φ) =R n,l (r)y m l l (θ, φ) (1) corresponds to the wavefunction for a p orbital once we insert the quantum numbers associated with the state we wish to describe. The R term is the radial function (a function of distance), and the Y term is a spherical harmonic (a function of θ and φ). The spherical harmonics for p orbitals are complex since the m l = ±1 values correspond to functions with the form e ±iφ. (See McQuarrie or the Goddard book in the suggested reading section). Since we cannot see the complex plane, we cannot easily visualize what these orbitals look like, however it is perfectly acceptable to create linear combinations of these complex spherical harmonics to form real functions that we can visualize. As a result, we can make new expressions that describe the 3 degenerate states with the following real wavefunctions: ψ px = A sin θ cos φ (2) ψ py = A sin θ sin φ (3) ψ pz = A cos θ (4) Where A is a coefficient that arises with particular values of Z and quantum number n. 2 Diatomic Term Symbols Now that we have three real functions to describe the p orbitals, we can think of how each of the three states of O behaves under rotation about the z-axis. We start with the xyz 2 state and assign a wavefunction to this state. Since there is one electron in each of the p x and p y orbitals, and assuming that Hund s rules apply, 1
2 we can write its wavefunction as (xy yx)(αα). However, we know that both p x and p y have angular dependencies, and in order to see how these orbitals behave under rotation we substitute: x =cosφ and y =sinφ since these are the only φ-dependent parts of the p x and p y wavefunctions. Therefore: (xy yx)(αα) =(cosφ 1 sin φ 2 sin φ 1 cos φ 2 )(αα) =sin(φ 2 φ 1 )(αα) (5) which is invariant under rotation of the z-axis by φ. Since it is invariant, we do not expect it to have any angular momentum. We assign this xyz 2 state as m l =0. We then arbitrarily assing the x 2 yz and xy 2 z states to have m l values of -1 and +1, respectively. We can see that the x 2 yz state transforms in the same manner as a pure p y orbital. Even though the p x orbital does change with rotation, the doubly-occupied p x orbital does not change since the two electrons contribute to two sign-changes overall. The same argument can be used to say that the xy z state behaves like a pure p x orbital under rotation about the z-axis. The same technique we just used to assign the spatial functions of a p-orbital to values of m l can be used for d orbitals as well. Now that we know how to identify states by their characteristic orbitals, we can assign terms (Σ, Π, and ) for diatomic molecules that resemble the different types of atomic orbitals we encounter (σ, π, and δ). We use these capital Greek characters as terms for only diatomic molecules because the nuclei of any two atoms side by side will always have the same basic symmetry element: a C axis of rotation (i.e. it is invariant of rotation about its z-axis). A few additional symmetries must be specified when dealing with diatomic molecules. First, the symmetry of the wavefunction will be either symmetric or anti-symmetric with respect to reflection through a mirror plane parallel to the axis of rotation. We denote symmetry of the wavefunction by a superscript + and antisymmetric with a superscript -. Since Π states are never symmetric with respect to reflection we do not bother writing + or - for the Π states. We also need to consider the effect of inversion symmetry in diatomics. Recall that the effect of inversion is to take (x, y, z) (-x, -y, -z). If the wavefunction remains the same sign after application of the inversion operator, then it is said to be symmetric under inversion, and is denoted with a g. Likewise, if the sign of the wavefunction changes after application of the inversion operator, then it is antisymmetric under inversion and is denoted with a u. 3 Molecular Term Symbols When we think about molecules that consist of more than two atoms we need to devise different terms to identify states since these molecules do not have the same C element that all diatomics have. This new paradigm for molecular terms needs to account for the structural geometry of any molecule, and so again we invoke symmetry operations on the Hamiltonian to do so. Our new system for determining molecular terms depend on the fundamental symmetry elements that exist in the symmetry point group for the molecule. To illustrate, we use the water molecule. (See Figure 2.) Water is of the C 2v point group because it has certain characteristic symmetry operators. These operators operate on the wavefunction of water and are the following: 1. A C 2 axis of rotation A C n axis shows a completely identical representation of the molecule occurs n times during a rotation of 360 about a particular axis. In the case of water, rotation about the z-axis results in an identical representation after rotations of 180 and 360. Since two 2
3 representations of the molecule occur under this rotation, water has a C 2 axis of symmetry. (See Figure 3.) A Quick Excercise for the Reader: What are all of the C n s for: (a) a book with no distinguishable markings on its cover? (b) an OREO TM cookie with no markings on the cookie? (c) a four-sided die in the shape of a pyramid without any numbers? In the case of water, a C 2 operation rotates the water molecule = 180. If we find that the wavefunction is symmetric upon application of this C 2 operator then the molecule is assigned the term A. If it is antisymmetric then it is assigned the term B. Application of the C 2 operator twice results in the exact same orientation we started with. This operation, the identity operator ( E for einheit) is also a symmetry element. 2. An identity operator, E All molecules have an identity operator. If the only symmetry operation that exists for a molecule is E, then it is said to be of the C 1 point group. Application of the E operator on a wavefunction will always result in the same wavefunction, so wavefunctions are always symmetric under the E operator. 3. One mirror plane that is parallel to the axis of rotation (σ v = σ xz ) The mirror plane along the xz-plane will take (x, y, z) (x, -y, z). If application of the σ xz operator results in the same wavefunction, then it is symmetric under σ xz.ifσ xz results in the negative of the wavefunction, then it is antisymmetric under σ xz. (See Figure 4.) A Quick Excercise for the Reader: (a) How many planes of symmetry are in an octagonal STOP sign, neglecting the word STOP? (b) How may are parallel to its highest order axis of rotation (C 8 in this case)? (c) How may are perpendicular to this axis of rotation? (Planes of symmetry parallel to the axis of rotation are σ v s. Planes perpendicular to the axis of rotation are σ h s.) 4. Another mirror plane that is parallel to the axis of rotation (σ v = σ yz ) The mirror plane along the yz-plane will take (x, y, z) (-x, y, z). If application of the σ yz operator results in the same wavefunction, then it is symmetric under σ yz.ifσ yz results in the negative of the wavefunction, then it is antisymmetric under σ yz. (See Figure 5.) These symmetry elements are not independent! One can test to see if any other symmetry elements exist in our point group by taking products of the operators to see if any new operations have not been accounted for. Taking the product of E with any other operator will always result in the operator itself. Additionally, in this point group, the product of any operator with the same operator also results in E. The product of σ xz C 2 = σ yz. Once we have all of the operators (also called generators), we can create a character table that contains all of the symmetry operations and their corresponding effect on the wavefunction (Figure 6). With this character table, by matching how the C 2 and σ xz operators affect H 2 O, we can identify its state as 1 A 1. 3
4 Depending on the term symbol of a molecule, we can identify its internal composition (i.e. whether the molecule has a pooched orbital or not, and thus how it prefers to bond to other atoms). NOTE: The C 2v point group has a unique property compared to other point groups. Generally, symmetry or antisymmetry of the wavefunction under reflection of a mirror plane is denoted by the subscripts of the A or B terms. The C 2v point group has two unique mirror planes, so therefore we need to decide which plane we choose to determine the unique terms. Standard convention is to define the primary mirror plane as the plane that intersects the most atoms, and so convention would be to place the H 2 O molecule in the xz plane. Professor Goddard prefers to define the primary mirror plane as the plane that reflects the most atoms onto other atoms, but in doing so he places the atoms in the yz plane. This is a subte detail that only arises in C 2v point groups, so be aware of the different conventions that are used. 4 Suggested Reading Atomic p Orbitals: McQuarrie: Hydrogen Atom Chapter 6.10 Term Symbols and Symmetry Operations for Diatomic Molecules: Goddard Book General Symmetry Operations and Point Groups: Miessler and Tarr: Chapter 4 5 Figures Figure 1: The x 2 yz, xyz 2, xy 2 z, states of 3 Π oxygen Figure 2: The water molecule and our selected coordinate axes 4
5 Figure 3: The C 2 symmetry operator applied to water Figure 4: The σ xz symmetry operator applied to water Figure 5: The σ yz symmetry operator applied to water Figure 6: The symmetry character table for the C 2v point group 5
Molecular Term Symbols
Molecular Term Symbols A molecular configuration is a specification of the occupied molecular orbitals in a molecule. For example, N : σ gσ uπ 4 uσ g A given configuration may have several different states
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 information26 Group Theory Basics
26 Group Theory Basics 1. Reference: Group Theory and Quantum Mechanics by Michael Tinkham. 2. We said earlier that we will go looking for the set of operators that commute with the molecular Hamiltonian.
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 informationQuantum Theory of Angular Momentum and Atomic Structure
Quantum Theory of Angular Momentum and Atomic Structure VBS/MRC Angular Momentum 0 Motivation...the questions Whence the periodic table? Concepts in Materials Science I VBS/MRC Angular Momentum 1 Motivation...the
More informationCHM Physical Chemistry II Chapter 9 - Supplementary Material. 1. Constuction of orbitals from the spherical harmonics
CHM 3411 - Physical Chemistry II Chapter 9 - Supplementary Material 1. Constuction of orbitals from the spherical harmonics The wavefunctions that are solutions to the time independent Schrodinger equation
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 informationOne-electron Atom. (in spherical coordinates), where Y lm. are spherical harmonics, we arrive at the following Schrödinger equation:
One-electron Atom The atomic orbitals of hydrogen-like atoms are solutions to the Schrödinger equation in a spherically symmetric potential. In this case, the potential term is the potential given by Coulomb's
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 informationConstruction of the C 2v character table
Construction of the C 2v character table The character table C 2v has the following form: C 2v E C 2 σ v (xz) σ v '(yz) Α 1 1 1 1 1 z x 2, y 2, z 2 Α 2 1 1-1 -1 R z xy Β 1 1-1 1-1 x, R y xz Β 2 1-1 -1
More informationONE AND MANY ELECTRON ATOMS Chapter 15
See Week 8 lecture notes. This is exactly the same as the Hamiltonian for nonrigid rotation. In Week 8 lecture notes it was shown that this is the operator for Lˆ 2, the square of the angular momentum.
More informationIV. Electronic Spectroscopy, Angular Momentum, and Magnetic Resonance
IV. Electronic Spectroscopy, Angular Momentum, and Magnetic Resonance The foundation of electronic spectroscopy is the exact solution of the time-independent Schrodinger equation for the hydrogen atom.
More informationCrystal field effect on atomic states
Crystal field effect on atomic states Mehdi Amara, Université Joseph-Fourier et Institut Néel, C.N.R.S. BP 66X, F-3842 Grenoble, France References : Articles - H. Bethe, Annalen der Physik, 929, 3, p.
More informationReading: Mathchapters F and G, MQ - Ch. 7-8, Lecture notes on hydrogen atom.
Chemistry 356 017: Problem set No. 6; Reading: Mathchapters F and G, MQ - Ch. 7-8, Lecture notes on hydrogen atom. The H atom involves spherical coordinates and angular momentum, which leads to the shapes
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 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 informationChem 673, Problem Set 5 Due Thursday, November 29, 2007
Chem 673, Problem Set 5 Due Thursday, November 29, 2007 (1) Trigonal prismatic coordination is fairly common in solid-state inorganic chemistry. In most cases the geometry of the trigonal prism is such
More informationLECTURE 2 DEGENERACY AND DESCENT IN SYMMETRY: LIGAND FIELD SPLITTINGS AND RELATED MATTERS
SYMMETRY II. J. M. GOICOECHEA. LECTURE 2. 1 LECTURE 2 DEGENERACY AND DESCENT IN SYMMETRY: LIGAND FIELD SPLITTINGS AND RELATED MATTERS 2.1 Degeneracy When dealing with non-degenerate symmetry adapted wavefunctions
More informationPage 712. Lecture 42: Rotations and Orbital Angular Momentum in Two Dimensions Date Revised: 2009/02/04 Date Given: 2009/02/04
Page 71 Lecture 4: Rotations and Orbital Angular Momentum in Two Dimensions Date Revised: 009/0/04 Date Given: 009/0/04 Plan of Attack Section 14.1 Rotations and Orbital Angular Momentum: Plan of Attack
More informationCHEM-UA 127: Advanced General Chemistry I
1 CHEM-UA 127: Advanced General Chemistry I Notes for Lecture 11 Nowthatwehaveintroducedthebasicconceptsofquantummechanics, wecanstarttoapplythese conceptsto build up matter, starting from its most elementary
More informationECEN 5005 Crystals, Nanocrystals and Device Applications Class 20 Group Theory For Crystals
ECEN 5005 Crystals, Nanocrystals and Device Applications Class 20 Group Theory For Crystals Laporte Selection Rule Polarization Dependence Spin Selection Rule 1 Laporte Selection Rule We first apply this
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 informationMolecular Physics. Attraction between the ions causes the chemical bond.
Molecular Physics A molecule is a stable configuration of electron(s) and more than one nucleus. Two types of bonds: covalent and ionic (two extremes of same process) Covalent Bond Electron is in a molecular
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 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 informationLecture 5: Harmonic oscillator, Morse Oscillator, 1D Rigid Rotor
Lecture 5: Harmonic oscillator, Morse Oscillator, 1D Rigid Rotor It turns out that the boundary condition of the wavefunction going to zero at infinity is sufficient to quantize the value of energy that
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 informationProblem 1: Spin 1 2. particles (10 points)
Problem 1: Spin 1 particles 1 points 1 Consider a system made up of spin 1/ particles. If one measures the spin of the particles, one can only measure spin up or spin down. The general spin state of a
More informationQuantum Numbers. principal quantum number: n. angular momentum quantum number: l (azimuthal) magnetic quantum number: m l
Quantum Numbers Quantum Numbers principal quantum number: n angular momentum quantum number: l (azimuthal) magnetic quantum number: m l Principal quantum number: n related to size and energy of orbital
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 informationLigand Field Theory Notes
Ligand Field Theory Notes Read: Hughbanks, Antisymmetry (Handout). Carter, Molecular Symmetry..., Sections 7.4-6. Cotton, Chemical Applications..., Chapter 9. Harris & Bertolucci, Symmetry and Spectroscopy...,
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 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 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 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 informationSymmetry: Translation and Rotation
Symmetry: Translation and Rotation The sixth column of the C 2v character table indicates the symmetry species for translation along (T) and rotation about (R) the Cartesian axes. y y y C 2 F v (x) T x
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 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 informationWhere have we been? Lectures 1 and 2 Bohr s Model/ Wave Mechanics/ Radial and Angular Wavefunctions/ Radial Distribution Functions/ s and p orbitals
Where have we been? Lectures 1 and 2 Bohr s Model/ Wave Mechanics/ Radial and Angular Wavefunctions/ Radial Distribution unctions/ s and p orbitals Where are we going? Lecture 3 Brief wavefunction considerations:
More informationQuantum Mechanics: The Hydrogen Atom
Quantum Mechanics: The Hydrogen Atom 4th April 9 I. The Hydrogen Atom In this next section, we will tie together the elements of the last several sections to arrive at a complete description of the hydrogen
More informationR BC. reaction coordinate or reaction progress R. 5) 8pts) (a) Which of the following molecules would give an infrared spectrum? HCl O 2 H 2 O CO 2
Physical Chemistry Spring 2006, Prof. Shattuck Final Name Part Ia. Answer 4 (four) of the first 5 (five) questions. If you answer more than 4, cross out the one you wish not to be graded. 1) 8pts) Of absorption
More informationChimica Inorganica 3
A symmetry operation carries the system into an equivalent configuration, which is, by definition physically indistinguishable from the original configuration. Clearly then, the energy of the system must
More informationB7 Symmetry : Questions
B7 Symmetry 009-10: Questions 1. Using the definition of a group, prove the Rearrangement Theorem, that the set of h products RS obtained for a fixed element S, when R ranges over the h elements of the
More informationΨ t = ih Ψ t t. Time Dependent Wave Equation Quantum Mechanical Description. Hamiltonian Static/Time-dependent. Time-dependent Energy operator
Time Dependent Wave Equation Quantum Mechanical Description Hamiltonian Static/Time-dependent Time-dependent Energy operator H 0 + H t Ψ t = ih Ψ t t The Hamiltonian and wavefunction are time-dependent
More information(3.1) Module 1 : Atomic Structure Lecture 3 : Angular Momentum. Objectives In this Lecture you will learn the following
Module 1 : Atomic Structure Lecture 3 : Angular Momentum Objectives In this Lecture you will learn the following Define angular momentum and obtain the operators for angular momentum. Solve the problem
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 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 informationExperiment 15: Atomic Orbitals, Bond Length, and Molecular Orbitals
Experiment 15: Atomic Orbitals, Bond Length, and Molecular Orbitals Introduction Molecular orbitals result from the mixing of atomic orbitals that overlap during the bonding process allowing the delocalization
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 informationNotation. Irrep labels follow Mulliken s convention: A and B label nondegenerate E irreps, doubly T degenerate, triply degenerate
Notation Irrep labels follow Mulliken s convention: A and B label nondegenerate E irreps, doubly T degenerate, triply degenerate Spectroscopists sometimes use F for triply degenerate; almost everyone G
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 informationChem Symmetry and Introduction to Group Theory. Symmetry is all around us and is a fundamental property of nature.
Chem 59-65 Symmetry and Introduction to Group Theory Symmetry is all around us and is a fundamental property of nature. Chem 59-65 Symmetry and Introduction to Group Theory The term symmetry is derived
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 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 informationPAPER No. 7: Inorganic Chemistry - II (Metal-Ligand Bonding, Electronic Spectra and Magnetic Properties of Transition Metal Complexes
Subject Chemistry Paper No and Title Module No and Title Module Tag 7, Inorganic chemistry II (Metal-Ligand Bonding, Electronic Spectra and Magnetic Properties of Transition Metal Complexes) 10, Electronic
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 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 informationd 3 r d 3 vf( r, v) = N (2) = CV C = n where n N/V is the total number of molecules per unit volume. Hence e βmv2 /2 d 3 rd 3 v (5)
LECTURE 12 Maxwell Velocity Distribution Suppose we have a dilute gas of molecules, each with mass m. If the gas is dilute enough, we can ignore the interactions between the molecules and the energy will
More informationQuantum Analogs Chapter 3 Student Manual
Quantum Analogs Chapter 3 Student Manual Broken Symmetry in the Spherical Resonator and Modeling a Molecule Professor Rene Matzdorf Universitaet Kassel 3. Broken symmetry in the spherical resonator and
More informationInorganic Chemistry with Doc M. Fall Semester, 2012 Day 9. Molecular Orbitals, Part 4. Beyond Diatomics, continued
Inorganic Chemistry with Doc M. Fall Semester, 2012 Day 9. Molecular Orbitals, Part 4. Beyond Diatomics, continued Topics: Name(s): Element: 1. Using p-orbitals for σ-bonding: molecular orbital diagram
More informationEXAM INFORMATION. Radial Distribution Function: B is the normalization constant. d dx. p 2 Operator: Heisenberg Uncertainty Principle:
EXAM INFORMATION Radial Distribution Function: P() r RDF() r Br R() r B is the normalization constant., p Operator: p ^ d dx Heisenberg Uncertainty Principle: n ax n! Integrals: xe dx n1 a x p Particle
More informationChem Symmetry and Introduction to Group Theory. Symmetry is all around us and is a fundamental property of nature.
Symmetry and Introduction to Group Theory Symmetry is all around us and is a fundamental property of nature. Symmetry and Introduction to Group Theory The term symmetry is derived from the Greek word symmetria
More informationActivity Molecular Orbital Theory
Activity 201 9 Molecular Orbital Theory Directions: This Guided Learning Activity (GLA) discusses the Molecular Orbital Theory and its application to homonuclear diatomic molecules. Part A describes the
More informationB F N O. Chemistry 6330 Problem Set 4 Answers. (1) (a) BF 4. tetrahedral (T d )
hemistry 6330 Problem Set 4 Answers (1) (a) B 4 - tetrahedral (T d ) B T d E 8 3 3 2 6S 4 6s d G xyz 3 0-1 -1 1 G unmoved atoms 5 2 1 1 3 G total 15 0-1 -1 3 If we reduce G total we find that: G total
More informationAtoms 2012 update -- start with single electron: H-atom
Atoms 2012 update -- start with single electron: H-atom x z φ θ e -1 y 3-D problem - free move in x, y, z - easier if change coord. systems: Cartesian Spherical Coordinate (x, y, z) (r, θ, φ) Reason: V(r)
More informationWhat are molecular orbitals? QUANTUM MODEL. notes 2 Mr.Yeung
What are molecular orbitals? QUANTUM MODEL notes 2 Mr.Yeung Recall, the quantum model is about electrons behaving both a wave and a particle. Electrons are in areas of calculated probability, these are
More information8.1 The hydrogen atom solutions
8.1 The hydrogen atom solutions Slides: Video 8.1.1 Separating for the radial equation Text reference: Quantum Mechanics for Scientists and Engineers Section 10.4 (up to Solution of the hydrogen radial
More informationPRACTICE PROBLEMS Give the electronic configurations and term symbols of the first excited electronic states of the atoms up to Z = 10.
PRACTICE PROBLEMS 2 1. Based on your knowledge of the first few hydrogenic eigenfunctions, deduce general formulas, in terms of n and l, for (i) the number of radial nodes in an atomic orbital (ii) the
More informationTheoretical Chemistry - Level II - Practical Class Molecular Orbitals in Diatomics
Theoretical Chemistry - Level II - Practical Class Molecular Orbitals in Diatomics Problem 1 Draw molecular orbital diagrams for O 2 and O 2 +. E / ev dioxygen molecule, O 2 dioxygenyl cation, O 2 + 25
More informationChem 3502/4502 Physical Chemistry II (Quantum Mechanics) 3 Credits Spring Semester 2006 Christopher J. Cramer. Lecture 20, March 8, 2006
Chem 3502/4502 Physical Chemistry II (Quantum Mechanics) 3 Credits Spring Semester 2006 Christopher J. Cramer Lecture 20, March 8, 2006 Solved Homework We determined that the two coefficients in our two-gaussian
More informationLecture 4 Quantum mechanics in more than one-dimension
Lecture 4 Quantum mechanics in more than one-dimension Background Previously, we have addressed quantum mechanics of 1d systems and explored bound and unbound (scattering) states. Although general concepts
More informationGenerators for Continuous Coordinate Transformations
Page 636 Lecture 37: Coordinate Transformations: Continuous Passive Coordinate Transformations Active Coordinate Transformations Date Revised: 2009/01/28 Date Given: 2009/01/26 Generators for Continuous
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 informationThe Hydrogen Atom. Dr. Sabry El-Taher 1. e 4. U U r
The Hydrogen Atom Atom is a 3D object, and the electron motion is three-dimensional. We ll start with the simplest case - The hydrogen atom. An electron and a proton (nucleus) are bound by the central-symmetric
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 information(2 pts) a. What is the time-dependent Schrödinger Equation for a one-dimensional particle in the potential, V (x)?
Part I: Quantum Mechanics: Principles & Models 1. General Concepts: (2 pts) a. What is the time-dependent Schrödinger Equation for a one-dimensional particle in the potential, V (x)? (4 pts) b. How does
More informationLecture #1. Review. Postulates of quantum mechanics (1-3) Postulate 1
L1.P1 Lecture #1 Review Postulates of quantum mechanics (1-3) Postulate 1 The state of a system at any instant of time may be represented by a wave function which is continuous and differentiable. Specifically,
More information5.4. Electronic structure of water
5.4. Electronic structure of water Water belongs to C 2v point group, we have discussed the corresponding character table. Here it is again: C 2v E C 2 σ v (yz) σ v (xz) A 1 1 1 1 1 A 2 1 1-1 -1 B 1 1-1
More informationSymmetry and Group Theory
Symmetry and Group Theory 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,
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 informationDegeneracy & in particular to Hydrogen atom
Degeneracy & in particular to Hydrogen atom In quantum mechanics, an energy level is said to be degenerate if it corresponds to two or more different measurable states of a quantum system. Conversely,
More informationWelcome back to PHY 3305
Welcome back to PHY 3305 Today s Lecture: Hydrogen Atom Part I John von Neumann 1903-1957 One-Dimensional Atom To analyze the hydrogen atom, we must solve the Schrodinger equation for the Coulomb potential
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 informationLittle Orthogonality Theorem (LOT)
Little Orthogonality Theorem (LOT) Take diagonal elements of D matrices in RG * D R D R i j G ij mi N * D R D R N i j G G ij ij RG mi mi ( ) By definition, D j j j R TrD R ( R). Sum GOT over β: * * ( )
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 informationPhysical Chemistry Quantum Mechanics, Spectroscopy, and Molecular Interactions. Solutions Manual. by Andrew Cooksy
Physical Chemistry Quantum Mechanics, Spectroscopy, and Molecular Interactions Solutions Manual by Andrew Cooksy February 4, 2014 Contents Contents i Objectives Review Questions 1 Chapter Problems 11 Notes
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 informationNPTEL/IITM. Molecular Spectroscopy Lecture 2. Prof.K. Mangala Sunder Page 1 of 14. Lecture 2 : Elementary Microwave Spectroscopy. Topics.
Lecture 2 : Elementary Microwave Spectroscopy Topics Introduction Rotational energy levels of a diatomic molecule Spectra of a diatomic molecule Moments of inertia for polyatomic molecules Polyatomic molecular
More information( ( ; R H = 109,677 cm -1
CHAPTER 9 Atomic Structure and Spectra I. The Hydrogenic Atoms (one electron species). H, He +1, Li 2+, A. Clues from Line Spectra. Reminder: fundamental equations of spectroscopy: ε Photon = hν relation
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 informationChem 467 Supplement to Lecture 19 Hydrogen Atom, Atomic Orbitals
Chem 467 Supplement to Lecture 19 Hydrogen Atom, Atomic Orbitals Pre-Quantum Atomic Structure The existence of atoms and molecules had long been theorized, but never rigorously proven until the late 19
More informationChapter 12. Linear Molecules
Chapter 1. Linear Molecules Notes: Most of the material presented in this chapter is taken from Bunker and Jensen (1998), Chap. 17. 1.1 Rotational Degrees of Freedom For a linear molecule, it is customary
More informationPAPER:2, PHYSICAL CHEMISTRY-I QUANTUM CHEMISTRY. Module No. 34. Hückel Molecular orbital Theory Application PART IV
Subject PHYSICAL Paper No and Title TOPIC Sub-Topic (if any), PHYSICAL -II QUANTUM Hückel Molecular orbital Theory Module No. 34 TABLE OF CONTENTS 1. Learning outcomes. Hückel Molecular Orbital (HMO) Theory
More informationIndicate if the statement is True (T) or False (F) by circling the letter (1 pt each):
Indicate if the statement is (T) or False (F) by circling the letter (1 pt each): False 1. In order to ensure that all observables are real valued, the eigenfunctions for an operator must also be real
More informationAlkali metals show splitting of spectral lines in absence of magnetic field. s lines not split p, d lines split
Electron Spin Electron spin hypothesis Solution to H atom problem gave three quantum numbers, n,, m. These apply to all atoms. Experiments show not complete description. Something missing. Alkali metals
More informationStatistical Mechanics
Statistical Mechanics Uncertainty Principle Demonstration Any experiment designed to observe the electron results in detection of a single electron particle and no interference pattern. Determinacy vs.
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 informationRotations in Quantum Mechanics
Rotations in Quantum Mechanics We have seen that physical transformations are represented in quantum mechanics by unitary operators acting on the Hilbert space. In this section, we ll think about the specific
More informationActivity Molecular Orbital Theory
Activity 201 9 Molecular Orbital Theory Directions: This Guided Learning Activity (GLA) discusses the Molecular Orbital Theory and its application to homonuclear diatomic molecules. Part A describes the
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 information