Chem 442 Review for Exam 2. Exact separation of the Hamiltonian of a hydrogenic atom into center-of-mass (3D) and relative (3D) components.
|
|
- Brian Hardy
- 5 years ago
- Views:
Transcription
1 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 (3D) components The Schrödinger equation of a hydrogenic atom (3D) is exactly solvable Exact separation of the Schrödinger equation of a hydrogenic atom into radial (1D) and angular (D) components; the angular (D) component is further separable (two 1D) 0 The radial equation: mass Ze l( l 1) R R R ER ; is the reduced r r r 0r r The last term in the left-hand side arises from the kinetic energy of the angular part and increases with the angular momentum quantum number l The radial part of the hydrogenic wave function is proportional to (radius) l (associated / 0 Laguerre polynomial) (Slater-type orbital); a Slater-type orbital is e Zr na (spherical) The angular part of the hydrogenic wave function is the spherical harmonics: lm l, iml Y, is the product of an associated Legendre polynomial and e lm l Y ; The energy of the hydrogenic Schrödinger equation is Z E (in Ry) n The natural unit of length: a0 1 bohr 059 Angstrom (the most probable radius of the electron in hydrogen) The natural unit of energy: e0 1 Ry 136 ev (the negative of the energy of hydrogen in the ground state) The energy levels of the hydrogen atom are increasingly narrowly separated with n (the principal quantum number) The energy levels are continuous above the ionization limit 1
2 The ionization energy of the hydrogen atom is 136 ev (the negative of the ground-state energy) Quantum numbers of the hydrogenic wave function: n, l, and m l ; know the physical meaning and ranges of the quantum numbers Shells and subshells: K, L, M, etc shells; s, p, d, etc subshells; know the degrees of degeneracy The ns orbitals have n 1 nodal spheres; p orbitals have n nodal surfaces plus a nodal plane The wave function must change its sign across a nodal surface (not just vanish on the surface) The p 0 orbital is real and equal to p z orbital extending along the z axis The p +1 and p 1 orbitals are complex, but they can be linearly combined to form real p x and p y orbitals extending along x and y axes; the linear combination is justified because of degeneracy (they have the same energy) The p +1, p 0, and p 1 orbitals are eigenfunctions of the magnetic (z-component) orbital angular momentum operator with eigenvalues,0, They have the same energy and are triply degenerate The p x and p y orbitals are no longer eigenfunctions of the magnetic (z-component) orbital angular momentum operator but are still eigenfunctions of the Hamiltonian (have the same energy) The five complex d orbitals with well-defined z-component orbital angular momenta (d +, d +1, d 0, d 1, d ) can be linearly combined to form real d orbitals (d xy, d yz, d zx, d xx yy, d 3zz rr ) The average radius of the hydrogen: use the expectation value formula; use the normalized wave function; use ˆr for the radius operator; use spherical coordinates and corresponding volume element rdrsindd Radial distribution function: R r r () nl The most probable radius: find the maximum of the radial distribution function; find r where the first derivative of the radial distribution function vanishes; the zero derivatives may mean a maximum, minimum, or saddle point; also check the boundary points even if they do not have zero first derivatives The most probable point: find the maximum of the probability density
3 Atomic spectroscopic transitions: The transition probability is proportional to transition dipole moment (amplitude of the light s electric field) if and only if the energy conservation law is satisfied The transition dipole moment = * fzˆ id for z-polarized light The excitation and deexcitation are equally likely with all other conditions being the same Selection rules for atomic transition: l l l 1; m m m 0, 1; m 0 l l l s The unit cm 1 (reciprocal centimeter or inverse centimeter or wave number) is proportional to energy unit by the coefficient of proportionality hc 1 K = 07 cm 1 ; room temperature = 00 cm 1 ; 1 ev = 8000 cm 1 ; 1 kj/mol = 80 cm 1 Helium and heavier atoms: The orbital approximation: approximate separation of variables; fill electrons in hydrogenic atomic orbitals in polyelectron atoms; exact in the absence of the electron-electron repulsion term in the Hamiltonian A wave function of more than one electron must be antisymmetric because an electron is a fermion; antisymmetric means that the function changes sign upon interchange of two electron coordinates (x, y, z, as well as spin coordinate ); related to the Pauli exclusion principle He (1s) singlet: 1s 11s (1) () (1) () r r He (1s) 1 (s) 1 singlet: 1sr1s r sr1 1sr (1) () (1) () He (1s) 1 (s) 1 triplets: 1sr1s r sr1 1sr (1) () (1) (), 1sr1sr sr11sr (1) (), 1s 1s s 11s (1) () r r r r ; they are degenerate (the same energy) because they share the identical spatial part The energy ordering if the electron-electron repulsion term is neglected: (1s) singlet << (1s) 1 (s) 1 triplets = (1s) 1 (s) 1 singlet; the energy in each case is the sum of the energies of the hydrogenic atomic orbitals The energy ordering in reality (with the electron-electron repulsion reinstated): (1s) singlet << (1s) 1 (s) 1 triplets < (1s) 1 (s) 1 singlet; the second inequality because the triplets spatial part is antisymmetric, preventing two electrons from existing at the same spatial position (spin correlation) 3
4 The electron-electron repulsion (ignored in the orbital approximation) is responsible for shielding and spin correlation Shielding explains the energy ordering s < p < d < f in the same shell; outer orbitals have smaller attraction to nucleus due to shielding; s orbitals have less shielding (lower energy) because of greater probability density at nucleus; basis of aufbau (building up) principle Spin correlation (Pauli exclusion principle or antisymmetry) explains Hund s rule; spatial part antisymmetric in triplet states, leading to lower energy with unpaired electrons The effect of the electron-electron repulsion term in the orbital approximation can be quantified by approximating the energy by an expectation value The expectation value of energy in the He wave functions in the orbital approximation contains the so-called Coulomb and exchange terms in addition to the sums of the energies of hydrogenic orbitals; the Coulomb term explains shielding; the exchange term is tied to spin correlation The total z-component spin angular momentum operator of He: Sˆ sˆ 1 sˆ The singlet spin part is an eigenfunction of S ˆz : Sˆ (1) () (1) () 0 (1) () (1) () z The triplet spin parts are eigenfunctions of S ˆz : Sˆ (1) () (1) (); ˆ z Sz(1) () (1) (); Sˆ (1) () (1) () 0 (1) () (1) () z z z z The total spin angular momentum quantum numbers in He: singlet S = 0 (M S = 0); triplet: S = 1 (M S = +1, 0, 1) Spin multiplicities of atoms: singlet (S = 0, eg, ground state He), doublet (S = 1/, eg, the hydrogen atom or a single electron), triplet (S = 1, eg, an excited He), quartet (S = 3/, eg, an excited Li) Spectroscopic transitions between different spin multiplicities are forbidden because the operator in the transition dipole moment does not act on spin and the spin wave functions with different multiplicities are orthogonal to each other Spin-orbit effect: Spin and orbital angular momenta have a weak interaction: hca coupling constant given in cm 1 ŝl ˆ ; A is the spin-orbit 4
5 Level splitting by the spin-orbit interaction: jmax ls, jmin l s and all j s in between separated by 1; the degree of degeneracy is j The first-order perturbation theory gives E hca j j l l s s jls,, ( 1) ( 1) ( 1) Na D line is the paradigm of spin-orbit problem: (3s) 1 state (ground state) does not split; (3p) 1 state (excited state) splits into 3p 3/ and 3p 1/ spin-orbit coupled states; be able to compute energy shift by first-order perturbation theory and the degree of degeneracy for each state The spin-orbit interaction (and the concept of spin itself) comes from special relativity; the heavier the element the greater the value of A Phosphorescence versus fluorescence Intersystem crossing versus internal conversion Born-Oppenheimer principle: The full molecular Hamiltonian: n N n N N n ˆ e ZZe I J ZIe H e i N I m m r r r i1 e I1 i N ij I 0 ij IJ 0 IJ I i 0 Ii Electronic Schrödinger equation: n n N n e Ze I e ; ; i e Ee e i1 me ij 0r I i 4 i ij 0rIi rr R rr N I J Nuclear Schrödinger equation: E E Parameters versus variables I J Potential energy surface: E IJ 0 IJ NI e n n I1 mn IJ 0rIJ I N ZZe R R R N ZZe e R r Equilibrium molecular structures; binding and dissociation energies; vibrational and rotational energy levels The dynamical degrees of freedom of a molecule: 3n electronic; 3N nuclear (3 translational, 3 or rotational, 3N 6 or 3N 5 vibrational); separation between electronic and nuclear is the BO approximation; separation between translation and the rest is exact; separation between vibration and rotation is the rigid-rotor approximation 5
6 Valence bond theory: Singlet H in VB: A(1) B() B(1) A() (1) () (1) () Triplet H in VB: A(1) B() B(1) A() ; (1) () (1) () (1) () (1) () they are triply degenerate (the same energy) because they have the identical spatial part Energies: singlet < triplets because singlet (triplets) has enhanced (have depleted) electron density in between the nuclei The and bonds in VB Covalent bonds; Lewis structure VB descriptions of N, H O, NH 3 : 90 degree HOH and HNH angles using p orbitals Promotion and sp 1, sp, and sp 3 hybridization; VB description of CH 4, ethylene, and acetylene Lone pairs The sp 3 hybridized VB description of H O and NH 3 VB description of H O isovalence series and importance of lone-pair repulsion in determining molecular structure Molecular orbital theory: LCAO MO theory; the bonding and antibonding orbital of H : Singlet (X) H in MO: X(1) X() (1) () (1) () Triplet (X) 1 (Y) 1 H in MO: X(1) Y() Y(1) X() (1) () (1) () (1) () (1) () Singlet (X) 1 (Y) 1 H in MO: X(1) Y() Y(1) X() (1) () (1) () X AB and Y A B 6
7 The energy ordering: singlet (X) << triplet (X) 1 (Y) 1 < singlet (X) 1 (Y) 1 ; the first inequality because of different orbital configurations; the second inequality due to spin correlation or Hund s rule VB versus MO; VB: A(1) B() B(1) A() (1) () (1) () ; MO: X(1) X() (1) () (1) () = A(1) B() B(1) A() A(1) A() B(1) B() (1) () (1) () ; VB has 0% ionic (too small), while MO has 50% ionic (too large); these are the simplest descriptions in respective models ; the energy is approximated by an expectation value of energy in these normalized wave functions of H + : * ˆ e j k E Hd E1 s ; know the shapes (especially the asymptotic 0R 1 S behaviors) of S, j, and k; this explains the bound potential energy curve of the bonding orbital, the unbound (repulsive) potential energy curve of the antibonding orbital, both potential energy curves converging at E 1s at R = ; antibonding orbital more antibonding than bonding orbital bonding because of different denominators LCAO MO for H + ; MO s: N A B ; 1 N S Variational theorem: simple proof based on completeness and orthonormality Variational determination of LCAO MO expansion coefficients; Lagrange s undetermined multiplier method; matrix eigenvalue equation versus operator eigenvalue equation; undetermined multiplier becomes eigenvalue and thus energy Transformation of a matrix eigenvalue equation into quadratic equation; nonexistence of matrix inverse The and bonds in MO theory The MO description of H, He, He +, O, N, and HF Covalent bonds and ionic bonds Hückel theory: < 0; applications to conjugated electrons in ethylene, butadiene, cyclobutadiene; aromaticity 7
Potential energy, from Coulomb's law. Potential is spherically symmetric. Therefore, solutions must have form
Lecture 6 Page 1 Atoms L6.P1 Review of hydrogen atom Heavy proton (put at the origin), charge e and much lighter electron, charge -e. Potential energy, from Coulomb's law Potential is spherically symmetric.
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 informationChapter 9. Atomic structure and atomic spectra
Chapter 9. Atomic structure and atomic spectra -The structure and spectra of hydrogenic atom -The structures of many e - atom -The spectra of complex atoms The structure and spectra of hydrogenic atom
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 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 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 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 informationCHAPTER 8 Atomic Physics
CHAPTER 8 Atomic Physics 8.1 Atomic Structure and the Periodic Table 8.2 Total Angular Momentum 8.3 Anomalous Zeeman Effect What distinguished Mendeleev was not only genius, but a passion for the elements.
More informationAtomic Structure and Atomic Spectra
Atomic Structure and Atomic Spectra Atomic Structure: Hydrogenic Atom Reading: Atkins, Ch. 10 (7 판 Ch. 13) The principles of quantum mechanics internal structure of atoms 1. Hydrogenic atom: one electron
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 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 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 informationQUANTUM MECHANICS AND ATOMIC STRUCTURE
5 CHAPTER QUANTUM MECHANICS AND ATOMIC STRUCTURE 5.1 The Hydrogen Atom 5.2 Shell Model for Many-Electron Atoms 5.3 Aufbau Principle and Electron Configurations 5.4 Shells and the Periodic Table: Photoelectron
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 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 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 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 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 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 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 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 informationWe now turn to our first quantum mechanical problems that represent real, as
84 Lectures 16-17 We now turn to our first quantum mechanical problems that represent real, as opposed to idealized, systems. These problems are the structures of atoms. We will begin first with hydrogen-like
More informationAtomic Spectra in Astrophysics
Atomic Spectra in Astrophysics Potsdam University : Wi 2016-17 : Dr. Lidia Oskinova lida@astro.physik.uni-potsdam.de Complex Atoms Non-relativistic Schrödinger Equation 02 [ N i=1 ( ) 2 2m e 2 i Ze2 4πǫ
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 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 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 informationBasic Physical Chemistry Lecture 2. Keisuke Goda Summer Semester 2015
Basic Physical Chemistry Lecture 2 Keisuke Goda Summer Semester 2015 Lecture schedule Since we only have three lectures, let s focus on a few important topics of quantum chemistry and structural chemistry
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 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 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 informationI. CSFs Are Used to Express the Full N-Electron Wavefunction
Chapter 11 One Must be Able to Evaluate the Matrix Elements Among Properly Symmetry Adapted N- Electron Configuration Functions for Any Operator, the Electronic Hamiltonian in Particular. The Slater-Condon
More informationMultielectron Atoms.
Multielectron Atoms. Chem 639. Spectroscopy. Spring 00 S.Smirnov Atomic Units Mass m e 1 9.109 10 31 kg Charge e 1.60 10 19 C Angular momentum 1 1.055 10 34 J s Permittivity 4 0 1 1.113 10 10 C J 1 m 1
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 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 informationAtomic Structure. Chapter 8
Atomic Structure Chapter 8 Overview To understand atomic structure requires understanding a special aspect of the electron - spin and its related magnetism - and properties of a collection of identical
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 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 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 information14. Structure of Nuclei
14. Structure of Nuclei Particle and Nuclear Physics Dr. Tina Potter Dr. Tina Potter 14. Structure of Nuclei 1 In this section... Magic Numbers The Nuclear Shell Model Excited States Dr. Tina Potter 14.
More informationPreliminary Quantum Questions
Preliminary Quantum Questions Thomas Ouldridge October 01 1. Certain quantities that appear in the theory of hydrogen have wider application in atomic physics: the Bohr radius a 0, the Rydberg constant
More informationMultielectron Atoms and Periodic Table
GRE Question Multielectron Atoms and Periodic Table Helium Atom 2 2m e ( 2 1 + 2 2) + 2ke 2 2ke 2 + ke2 r 1 r 2 r 2 r 1 Electron-electron repulsion term destroys spherical symmetry. No analytic solution
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 informationLecture #13 1. Incorporating a vector potential into the Hamiltonian 2. Spin postulates 3. Description of spin states 4. Identical particles in
Lecture #3. Incorporating a vector potential into the Hamiltonian. Spin postulates 3. Description of spin states 4. Identical particles in classical and QM 5. Exchange degeneracy - the fundamental 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 informationOrganic Chemistry. Review Information for Unit 1. Atomic Structure MO Theory Chemical Bonds
Organic Chemistry Review Information for Unit 1 Atomic Structure MO Theory Chemical Bonds Atomic Structure Atoms are the smallest representative particle of an element. Three subatomic particles: protons
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 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 informationChapter 9. Molecular Geometry and Bonding Theories
Chapter 9. Molecular Geometry and Bonding Theories 9.1 Molecular Shapes Lewis structures give atomic connectivity: they tell us which atoms are physically connected to which atoms. The shape of a molecule
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 27st Page 1 Lecture 27 L27.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 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 informationChapter 1 Basic Concepts: Atoms
Chapter 1 Basic Concepts: Atoms CHEM 511 chapter 1 page 1 of 12 What is inorganic chemistry? The periodic table is made of elements, which are made of...? Particle Symbol Mass in amu Charge 1.0073 +1e
More informationMagnetism of Atoms and Ions. Wulf Wulfhekel Physikalisches Institut, Karlsruhe Institute of Technology (KIT) Wolfgang Gaede Str. 1, D Karlsruhe
Magnetism of Atoms and Ions Wulf Wulfhekel Physikalisches Institut, Karlsruhe Institute of Technology (KIT) Wolfgang Gaede Str. 1, D-76131 Karlsruhe 1 0. Overview Literature J.M.D. Coey, Magnetism and
More informationChemistry 218 Spring Molecular Structure
Chemistry 218 Spring 2015-2016 Molecular Structure R. Sultan COURSE SYLLABUS Email: rsultan@aub.edu.lb Homepage: http://staff.aub.edu.lb/~rsultan/ Lectures: 12:30-13:45 T, Th. 101 Chemistry Textbook: P.
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 informationApplied Statistical Mechanics Lecture Note - 3 Quantum Mechanics Applications and Atomic Structures
Applied Statistical Mechanics Lecture Note - 3 Quantum Mechanics Applications and Atomic Structures Jeong Won Kang Department of Chemical Engineering Korea University Subjects Three Basic Types of Motions
More informationCHAPTER STRUCTURE OF ATOM
12 CHAPTER STRUCTURE OF ATOM 1. The spectrum of He is expected to be similar to that [1988] H Li + Na He + 2. The number of spherical nodes in 3p orbitals are [1988] one three none two 3. If r is the radius
More information362 Lecture 6 and 7. Spring 2017 Monday, Jan 30
362 Lecture 6 and 7 Spring 2017 Monday, Jan 30 Quantum Numbers n is the principal quantum number, indicates the size of the orbital, has all positive integer values of 1 to (infinity) l is the angular
More informationElectronic Spectra of Complexes
Electronic Spectra of Complexes Interpret electronic spectra of coordination compounds Correlate with bonding Orbital filling and electronic transitions Electron-electron repulsion Application of MO theory
More informationChemistry 3502/4502. Exam III. All Hallows Eve/Samhain, ) This is a multiple choice exam. Circle the correct answer.
B Chemistry 3502/4502 Exam III All Hallows Eve/Samhain, 2003 1) This is a multiple choice exam. Circle the correct answer. 2) There is one correct answer to every problem. There is no partial credit. 3)
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 informationLecture January 18, Quantum Numbers Electronic Configurations Ionization Energies Effective Nuclear Charge Slater s Rules
Lecture 3 362 January 18, 2019 Quantum Numbers Electronic Configurations Ionization Energies Effective Nuclear Charge Slater s Rules Inorganic Chemistry Chapter 1: Figure 1.4 Time independent Schrödinger
More informationE = 2 (E 1)+ 2 (4E 1) +1 (9E 1) =19E 1
Quantum Mechanics and Atomic Physics Lecture 22: Multi-electron Atoms http://www.physics.rutgers.edu/ugrad/361 h / d/361 Prof. Sean Oh Last Time Multi-electron atoms and Pauli s exclusion principle Electrons
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 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 informationAtomic Structure Ch , 9.6, 9.7
Ch. 9.2-4, 9.6, 9.7 Magnetic moment of an orbiting electron: An electron orbiting a nucleus creates a current loop. A current loop behaves like a magnet with a magnetic moment µ:! µ =! µ B " L Bohr magneton:
More informationAtoms, Molecules and Solids (selected topics)
Atoms, Molecules and Solids (selected topics) Part I: Electronic configurations and transitions Transitions between atomic states (Hydrogen atom) Transition probabilities are different depending on the
More informationAtoms, Molecules and Solids (selected topics)
Atoms, Molecules and Solids (selected topics) Part I: Electronic configurations and transitions Transitions between atomic states (Hydrogen atom) Transition probabilities are different depending on the
More informationMulti-Electron Atoms II
Multi-Electron Atoms II LS Coupling The basic idea of LS coupling or Russell-Saunders coupling is to assume that spin-orbit effects are small, and can be neglected to a first approximation. If there is
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 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 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 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 informationPlease pass in only this completed answer sheet on the day of the test. LATE SUBMISSIONS WILL NOT BE ACCEPTED
CHM-201 General Chemistry and Laboratory I Unit #4 Take Home Test Due December 13, 2018 Please pass in only this completed answer sheet on the day of the test. LATE SUBMISSIONS WILL NOT BE ACCEPTED CHM-201
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 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 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 informationIntroduction to Heisenberg model. Javier Junquera
Introduction to Heisenberg model Javier Junquera Most important reference followed in this lecture Magnetism in Condensed Matter Physics Stephen Blundell Oxford Master Series in Condensed Matter Physics
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 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 informationDevelopment of atomic theory
Development of atomic theory The chapter presents the fundamentals needed to explain and atomic & molecular structures in qualitative or semiquantitative terms. Li B B C N O F Ne Sc Ti V Cr Mn Fe Co Ni
More informationFinal Exam Tuesday, May 8, 2012 Starting at 8:30 a.m., Hoyt Hall Duration: 2h 30m
Final Exam Tuesday, May 8, 2012 Starting at 8:30 a.m., Hoyt Hall. ------------------- Duration: 2h 30m Chapter 39 Quantum Mechanics of Atoms Units of Chapter 39 39-1 Quantum-Mechanical View of Atoms 39-2
More informationChapter Electron Spin. * Fine structure:many spectral lines consist of two separate. lines that are very close to each other.
Chapter 7 7. Electron Spin * Fine structure:many spectral lines consist of two separate lines that are very close to each other. ex. H atom, first line of Balmer series n = 3 n = => 656.3nm in reality,
More informationMANY ELECTRON ATOMS Chapter 15
MANY ELECTRON ATOMS Chapter 15 Electron-Electron Repulsions (15.5-15.9) The hydrogen atom Schrödinger equation is exactly solvable yielding the wavefunctions and orbitals of chemistry. Howev er, the Schrödinger
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 information221B Lecture Notes Many-Body Problems II Molecular Physics
1 Molecules 221B Lecture Notes Many-Body Problems II Molecular Physics In this lecture note, we discuss molecules. I cannot go into much details given I myself am not familiar enough with chemistry. But
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 informationElectronic Microstates & Term Symbols. Suggested reading: Shriver and Atkins, Chapter 20.3 or Douglas,
Lecture 4 Electronic Microstates & Term Symbols Suggested reading: Shriver and Atkins, Chapter 20.3 or Douglas, 1.4-1.5 Recap from last class: Quantum Numbers Four quantum numbers: n, l, m l, and m s Or,
More informationMolecular Geometry and Bonding Theories. Chapter 9
Molecular Geometry and Bonding Theories Chapter 9 Molecular Shapes CCl 4 Lewis structures give atomic connectivity; The shape of a molecule is determined by its bond angles VSEPR Model Valence Shell Electron
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 informationSchrodinger equation
CH1. Atomic Structure orbitals periodicity 1 Schrodinger equation - (h 2 /2p 2 m e2 ) [d 2 Y/dx 2 +d 2 Y/dy 2 +d 2 Y/dz 2 ] + V Y = E Y h = constant m e = electron mass V = potential E gives quantized
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 informationWhat Do Molecules Look Like?
What Do Molecules Look Like? The Lewis Dot Structure approach provides some insight into molecular structure in terms of bonding, but what about 3D geometry? Recall that we have two types of electron pairs:
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 information5.61 Physical Chemistry Final Exam 12/16/09. MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Chemistry Chemistry Physical Chemistry
5.6 Physical Chemistry Final Exam 2/6/09 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Chemistry Chemistry - 5.6 Physical Chemistry Final Examination () PRINT your name on the cover page. (2) It
More informationChapter 10: Modern Atomic Theory and the Periodic Table. How does atomic structure relate to the periodic table? 10.1 Electromagnetic Radiation
Chapter 10: Modern Atomic Theory and the Periodic Table How does atomic structure relate to the periodic table? 10.1 Electromagnetic Radiation Electromagnetic (EM) radiation is a form of energy that exhibits
More informationLast Name or Student ID
12/05/18, Chem433 Final Exam answers Last Name or Student ID 1. (2 pts) 12. (3 pts) 2. (6 pts) 13. (3 pts) 3. (3 pts) 14. (2 pts) 4. (3 pts) 15. (3 pts) 5. (4 pts) 16. (3 pts) 6. (2 pts) 17. (15 pts) 7.
More informationSelf-consistent Field
Chapter 6 Self-consistent Field A way to solve a system of many electrons is to consider each electron under the electrostatic field generated by all other electrons. The many-body problem is thus reduced
More informationParticle Behavior of Light 1. Calculate the energy of a photon, mole of photons 2. Find binding energy of an electron (know KE) 3. What is a quanta?
Properties of Electromagnetic Radiation 1. What is spectroscopy, a continuous spectrum, a line spectrum, differences and similarities 2. Relationship of wavelength to frequency, relationship of E to λ
More information(b) The wavelength of the radiation that corresponds to this energy is 6
Chapter 7 Problem Solutions 1. A beam of electrons enters a uniform 1.0-T magnetic field. (a) Find the energy difference between electrons whose spins are parallel and antiparallel to the field. (b) Find
More information