Two-parameter Study of the 1s2s Excited State of He and Li + - Hund's Rule
|
|
- Jonah Cox
- 5 years ago
- Views:
Transcription
1 Two-parameter Study of the 1ss xcited State of He and Li + - Hund's Rule The trial variational wave functions for the 1s 1 s 1 excited state of helium atom and lithium ion are scaled hydrogen 1s and s orbitals: Ψ 1s ( r) 3 π exp( r) Ψ s( r) 3 3π ( r) exp r Nuclear charge: Z 3 Seed values for and : Z 1 Z 1 The purpose of this analysis is to illustrate Hund's rule by calculating the energy of the singlet and triplet states for the 1s 1 s 1 electronic configuration. The singlet state has a symmetric orbital wave function and the triplet state has an antisymmetric orbital wave function: 1s( 1) s Ψ 1s( ) s ( 1 S = 1s( 1) s( ) 1s( ) s( 1) Ψ T = S 1ss S 1ss The integrals required for variational calculations on these states are given below: T 1s ( ) T s ( ) 8 V N1s ( ) Z V Ns ( ) Z 4 T 1ss ( ) ( ) 4 V 11 ( ) ( ) 7 V N1ss ( ) 4Z 3 3 ( ) S 1ss ( ) ( ) V 11 ( ) ( ) 5 The next step in this calculation is to collect these terms in an expression for the energy of the singlet and triplet states and then minimize the energy with respect to the variational parameters, and. The results of this minimization procedure are shown below. state calculation: S ( ) T 1s ( ) T s ( ) T 1ss ( ) S 1ss ( ) V N1s ( ) V Ns ( ) V N1ss ( ) S 1ss ( ) V 11 ( ) V 11 ( ) 1 S 1ss ( ) Minimize S 3.19
2 Break down the total energy into kinetic, electron-nuclear potential, and electron-electron potential energy. T T 1s ( ) T s ( ) T 1ss ( ) S 1ss ( ) 1 S 1ss ( ) V N1s ( ) V Ns ( ) V ne V N1ss ( ) S 1ss ( ) V ee V 11 ( ) V 11 ( ) 1 S 1ss ( ) 1 S 1ss ( ) T 5.91 V ne 1.69 V ee.448 S ( ) 5.91 Calculate < r 1s >, < r s > and the absolute magnitude of the 1s-s overlap: rψ 1s ( r) 4πr dr.497 rψ s ( r) 4πr dr Ψ 1s ( r) Ψ s ( r) 4πr dr.51 Display the radial distribution function for the 1s and s orbitals: r Ψ 1s ( r) r Ψ s ( r) r Display contour plot of singlet wave function: i 8 r1.85i j 8 r.85j i j Ψ( r1r) Ψ 1s ( r1) Ψ s ( r) 1 S 1ss ( ) Ψ 1s ( r) Ψ s ( r1) r1r M ij Ψr1 r i j r M r 1
3 state calculation: T ( ) T 1s ( ) T s ( ) T 1ss ( ) S 1ss ( ) V N1s ( ) V Ns ( ) V N1ss ( ) S 1ss ( ) V 11 ( ) V 11 ( ) 1 S 1ss ( ) Minimization of (, ) simultaneously with respect to and. Z Z 1 Minimize T Break down the total energy into kinetic, electron-nuclear potential, and electron-electron potential energy. T T 1s ( ) T s ( ) T 1ss ( ) S 1ss ( ) 1 S 1ss ( ) V N1s ( ) V Ns ( ) V ne V N1ss ( ) S 1ss ( ) V ee V 11 ( ) V 11 ( ) 1 S 1ss ( ) 1 S 1ss ( ) T 5.13 V ne V ee.479 T ( ) 5.13 Calculate < r 1s >, < r s > and the absolute magnitude of the 1s-s overlap: rψ 1s ( r) 4πr dr.5 rψ s ( r) 4πr dr.334 Ψ 1s ( r) Ψ s ( r) 4πr dr.38 Display the radial distribution function for the 1s and s orbitals: r Ψ 1s ( r) r Ψ s ( r) 4 6 r Display contour plot of triplet wave function: Ψ( r1r) Ψ 1s ( r1) Ψ s ( r) 1 S 1ss ( ) Ψ 1s ( r) Ψ s ( r1) r1r M ij Ψr1 r i j
4 r M r 1 Summary of the calculations for the helium atom and lithium ion: HundsRule r 1s r s 1ss dτ Helium o Atom Lithium o Ion The triplet state has a lower energy than the singlet state because electron-nuclear attraction increases more than electron-electron repulsion. Atomic size decreases in the triplet state due to the large decrease in the size of the s orbital. The 1s orbital's size is basically the same in the singlet and triplet states. The sharp decrease in the size of the s orbital is responsible for the increase in the electron-nuclear attraction, electron-electron repulsion and the absolute magnitude of the 1s-s orbital overlap. Note that this two-parameter calculation does not show the He triplet state lower in energy than the singlet state. However, it does show that the absolute magnitudes of V ne and V ee are greater in the triplet state. This is important because this is the trend that the exact wave function (see reference) reveals (V ee :.5 vs.68; V ne : vs ). The calculation for the lithium ion does show the triplet state lower in energy and the same trend in V ne and V ee as for the helium atom and the exact calculation. In other words, both electron-electron repulsion and electron-nuclear attraction are stronger in the triplet state, and the real reason the triplet state lies below the singlet in energy is because the decrease in V ne overwhelms the increase in V ee. Thus, the common explanation that the triplet state is favored because of reduced electron-electron repulsion is without merit.
5 The key phenomenon that accounts for these effects is the dramatic decrease in the size of the s orbital in going from the singlet to the triplet state. This accounts for the increases electron-electron repulsion and electron-nuclear attraction. The size of the 1s orbital is about the same in the singlet and triplet states. The significant increase in the absolute magnitude of the overlap integral in the triplet state is consistent with its higher electron-electron repulsion. Reference: Snow, R. L.; Bills, J. L. "The Pauli Principle and lectron Repulsion in Helium," Journal of Chemical ducation 1974, T Initial Intermediate Final to singlet with s orbital contraction: T, V ee and increase while V ne decreases. to triplet with frozen orbitals: T, V ee and decrease while V ne increases. T Initial Intermediate Final to triplet with frozen orbitals: T and V ee decrease while V ne and increase. to triplet with s orbital contraction: T and V ee increase while V ne and decrease.
Atomic 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 informationMolecular-Orbital Theory
Prof. Dr. I. Nasser atomic and molecular physics -551 (T-11) April 18, 01 Molecular-Orbital Theory You have to explain the following statements: 1- Helium is monatomic gas. - Oxygen molecule has a permanent
More informationChemistry 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 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 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 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 informationATOMIC STRUCTURE. Atomic Structure. Atomic orbitals and their energies (a) Hydrogenic radial wavefunctions
ATOMIC STRUCTURE Atomic orbitals and their energies (a) Hydrogenic radial wavefunctions Bundet Boekfa Chem Div, Fac Lib Arts & Sci Kasetsart University Kamphaeng Saen Campus 1 2 Atomic orbitals and their
More informationChem 261 Sept 8, 2014
Chem 261 Sept 8, 2014 Atomic theory: - Neils Bohr (1913) won his Nobel prize for his atomic theory NOT fully correct - the neutrons (no charge) and protons (positively charged) occupy a dense central region
More informationonly two orbitals, and therefore only two combinations to worry about, but things get
131 Lecture 1 It is fairly easy to write down an antisymmetric wavefunction for helium since there are only two orbitals, and therefore only two combinations to worry about, but things get complicated
More 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 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 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 informationMany-Electron Atoms. Thornton and Rex, Ch. 8
Many-Electron Atoms Thornton and Rex, Ch. 8 In principle, can now solve Sch. Eqn for any atom. In practice, -> Complicated! Goal-- To explain properties of elements from principles of quantum theory (without
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 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 informationLec. 8: Hydrogen Atom and Band Theory
Solid State Electronics EC210 AAST Cairo Fall 2014 Lec. 8: Hydrogen Atom and Band Theory Fig 3.20 1 1 These PowerPoint color diagrams can only be used by instructors if the 3 rd Edition has been adopted
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 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 informationHelium and two electron atoms
1 Helium and two electron atoms e 2 r 12 e 1 r 2 r 1 +Ze Autumn 2013 Version: 04.12.2013 2 (1) Coordinate system, Schrödinger Equation 3 slides Evaluation of repulsion term 2 slides Radial Integral - details
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 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 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 informationAb initio asymptotic-expansion coefficients for pair energies in Møller-Plesset perturbation theory for atoms
Ab initio asymptotic-expansion coefficients for pair energies in Møller-Plesset perturbation theory for atoms K. JANKOWSKI a, R. SŁUPSKI a, and J. R. FLORES b a Nicholas Copernicus University 87-100 Toruń,
More informationMany-Electron Atoms. Thornton and Rex, Ch. 8
Many-Electron Atoms Thornton and Rex, Ch. 8 In principle, can now solve Sch. Eqn for any atom. In practice, -> Complicated! Goal-- To explain properties of elements from principles of quantum theory (without
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 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 information( ) dσ 1 dσ 2 + α * 2
Chemistry 36 Dr. Jean M. Standard Problem Set Solutions. The spin up and spin down eigenfunctions for each electron in a many-electron system are normalized and orthogonal as given by the relations, α
More informationr 2 dr h2 α = 8m2 q 4 Substituting we find that variational estimate for the energy is m e q 4 E G = 4
Variational calculations for Hydrogen and Helium Recall the variational principle See Chapter 16 of the textbook The variational theorem states that for a Hermitian operator H with the smallest eigenvalue
More informationFor the case of S = 1/2 the eigenfunctions of the z component of S are φ 1/2 and φ 1/2
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department 8.044 Statistical Physics I Spring Term 03 Excited State Helium, He An Example of Quantum Statistics in a Two Particle System By definition He has
More informationDalton s Postulates about atoms are inconsistent with later observations :
Name : Netya Shoma Siwi Pertiwi NIM : 21030110141022 1. Which of Dalton s postulates about atoms are inconsistent with later observations? Do these inconsistencies mean that Dalton was wrong? Is Dalton
More informationThe periodic system of the elements. Predict. (rather than passively learn) Chemical Properties!
The periodic system of the elements Study of over 100 elements daunting task! Nature has provided the periodic table Enables us to correlate an enormous amount of information Predict (rather than passively
More informationPotential 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 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 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 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 informationPrinciples of Quantum Mechanics
Principles of Quantum Mechanics - indistinguishability of particles: bosons & fermions bosons: total wavefunction is symmetric upon interchange of particle coordinates (space,spin) fermions: total wavefuncftion
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 informationOrigin of the first Hund rule in He-like atoms and 2-electron quantum dots
in He-like atoms and 2-electron quantum dots T Sako 1, A Ichimura 2, J Paldus 3 and GHF Diercksen 4 1 Nihon University, College of Science and Technology, Funabashi, JAPAN 2 Institute of Space and Astronautical
More informationPHY331 Magnetism. Lecture 8
PHY331 Magnetism Lecture 8 Last week. We discussed domain theory of Ferromagnetism. We saw there is a motion of domain walls with applied magnetic field. Stabilization of domain walls due to competition
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 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 informationMagnetism in low dimensions from first principles. Atomic magnetism. Gustav Bihlmayer. Gustav Bihlmayer
IFF 10 p. 1 Magnetism in low dimensions from first principles Atomic magnetism Gustav Bihlmayer Institut für Festkörperforschung, Quantum Theory of Materials Gustav Bihlmayer Institut für Festkörperforschung
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 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 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 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 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 informationAtomic Spectroscopy II
Applied Spectroscopy Atomic Spectroscopy II Multielectron Atoms Recommended Reading: Banwell And McCash Chapter 5 The Building-Up (aufbau) Principle How do the electrons in multi-electron atoms get distributed
More informationThe Quantum Mechanical Atom
The Quantum Mechanical Atom CHAPTER 8 Chemistry: The Molecular Nature of Matter, 6 th edition By Jesperson, Brady, & Hyslop CHAPTER 8: Quantum Mechanical Atom Learning Objectives q Light as Waves, Wavelength
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 informationElectronic Configuration of the Elements
Electronic Configuration of the Elements As the number of electrons increases with the number of protons of a neutral atom, they occupy orbitals of increasing energy: The possibilities are: n l m l m s
More informationQuestion: How are electrons arranged in an atom?
Honors Chemistry: Coulomb s Law and periodic trends Question: How are electrons arranged in an atom? Coulomb s Law equation: 1. A) Define what each of the following variables in the equation represents.
More informationELECTRON CONFIGURATION OF ATOMS
ELECTRON CONFIGURATION OF ATOMS 1 Electron Configuration? is the distribution of electrons within the orbitals of its atoms, in relation with chemical and physical properties Objectives: to show how the
More information7.1 Variational Principle
7.1 Variational Principle Suppose that you want to determine the ground-state energy E g for a system described by H, but you are unable to solve the time-independent Schrödinger equation. It is possible
More 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 1 - Basic Concepts: atoms
Chapter 1 - Basic Concepts: atoms Discovery of atomic structure Rutherford (1910) JJ Thomson (1897) Milliken (1909) Rutherford (1911) 1 Symbol p + e - n 0 Mass (amu) 1.0073 0.000549 1.00870 Discovery 1919,
More informationElectron Configurations
Ch08 Electron Configurations We now understand the orbital structure of atoms. Next we explore how electrons filling that structure change it. version 1.5 Nick DeMello, PhD. 2007-2016 2 Ch08 Putting Electrons
More informations or Hz J atom J mol or -274 kj mol CHAPTER 4. Practice Exercises ΔE atom = ΔE mol =
CHAPTER 4 Practice Exercises 4.1 10 1 2.1410 s or Hz 4.3 ΔE atom = ΔE mol = 4.5610 J atom 19 1 2.7410 J mol or -274 kj mol 5 1-1 4.5 excitation energy = 471 kj mol 1 + 275 kj mol 1 = 746 kj mol 1 Hg 4.7
More informationComputational Methods. Chem 561
Computational Methods Chem 561 Lecture Outline 1. Ab initio methods a) HF SCF b) Post-HF methods 2. Density Functional Theory 3. Semiempirical methods 4. Molecular Mechanics Computational Chemistry " Computational
More informatione L 2m e the Bohr magneton
e L μl = L = μb 2m with : μ B e e 2m e the Bohr magneton Classical interation of magnetic moment and B field: (Young and Freedman, Ch. 27) E = potential energy = μ i B = μbcosθ τ = torque = μ B, perpendicular
More informationPauli Deformation APPENDIX Y
APPENDIX Y Two molecules, when isolated say at infinite distance, are independent and the wave function of the total system might be taken as a product of the wave functions for the individual molecules.
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 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 informationJack Simons Henry Eyring Scientist and Professor Chemistry Department University of Utah
1. Born-Oppenheimer approx.- energy surfaces 2. Mean-field (Hartree-Fock) theory- orbitals 3. Pros and cons of HF- RHF, UHF 4. Beyond HF- why? 5. First, one usually does HF-how? 6. Basis sets and notations
More informationPHL424: Nuclear Shell Model. Indian Institute of Technology Ropar
PHL424: Nuclear Shell Model Themes and challenges in modern science Complexity out of simplicity Microscopic How the world, with all its apparent complexity and diversity can be constructed out of a few
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 informationThe Electronic Theory of Chemistry
JF Chemistry CH1101 The Electronic Theory of Chemistry Dr. Baker bakerrj@tcd.ie Module Aims: To provide an introduction to the fundamental concepts of theoretical and practical chemistry, including concepts
More informationwould represent a 1s orbital centered on the H atom and φ 2px )+ 1 r 2 sinθ
Physical Chemistry for Engineers CHEM 4521 Homework: Molecular Structure (1) Consider the cation, HeH +. (a) Write the Hamiltonian for this system (there should be 10 terms). Indicate the physical meaning
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 informationOrigin of Hund s multiplicity rule in singly-excited helium: Existence of a conjugate Fermi hole in the lower spin state
Origin of Hund s rule in He Origin of Hund s multiplicity rule in singly-excited helium: Existence of a conjugate Fermi hole in the lower spin state Tokuei Sako Laboratory of Physics, College of Science
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 informationA One-Slide Summary of Quantum Mechanics
A One-Slide Summary of Quantum Mechanics Fundamental Postulate: O! = a! What is!?! is an oracle! operator wave function (scalar) observable Where does! come from?! is refined Variational Process H! = E!
More informationChemistry 1A. Chapter 7
Chemistry 1A Chapter 7 Atomic Theory To see a World in a Grain of Sand And a Heaven in a Wild Flower Hold Infinity in the palm of your hand And Eternity in an hour William Blake Auguries of Innocence Thus,
More information( )! rv,nj ( R N )! ns,t
Chapter 8. Nuclear Spin Statistics Notes: Most of the material presented in this chapter is taken from Bunker and Jensen (2005) Chap. 9 and Bunker and Jensen (1998) Chap. 8. 8.1 The Complete Internal Wave
More informationX-Ray transitions to low lying empty states
X-Ray Spectra: - continuous part of the spectrum is due to decelerated electrons - the maximum frequency (minimum wavelength) of the photons generated is determined by the maximum kinetic energy of the
More informationMulti-Particle Wave functions
Multi-Particle Wave functions Multiparticle Schroedinger equation (N particles, all of mass m): 2 2m ( 2 1 + 2 2 +... 2 N ) Multiparticle wave function, + U( r 1, r 2,..., r N )=i (~r 1,~r 2,...,~r N,t)
More informationQuantum theory predicts that an atom s electrons are found in: To account that orbitals hold two electrons, we need:
Quantum theory predicts that an atom s electrons are found in: To account that orbitals hold two electrons, we need: Shells (principle quantum number, n) Subshells (angular momentum, l) Orbitals (magnetic
More informationr R A 1 r R B + 1 ψ(r) = αψ A (r)+βψ B (r) (5) where we assume that ψ A and ψ B are ground states: ψ A (r) = π 1/2 e r R A ψ B (r) = π 1/2 e r R B.
Molecules Initial questions: What are the new aspects of molecules compared to atoms? What part of the electromagnetic spectrum can we probe? What can we learn from molecular spectra? How large a molecule
More informationRemember Bohr s Explanation: Energy Levels of Hydrogen: The Electronic Structure of the Atom 11/28/2011
The Electronic Structure of the Atom Bohr based his theory on his experiments with hydrogen he found that when energy is added to a sample of hydrogen, energy is absorbed and reemitted as light When passed
More informationATOMIC STRUCRURE
ATOMIC STRUCRURE Long Answer Questions: 1. What are quantum numbers? Give their significance? Ans. The various orbitals in an atom qualitatively distinguished by their size, shape and orientation. The
More informationChemistry 1B Fall 2013
Chemistry 1B Fall 2013 lectures 5-6 Chapter 12 pp. 557-569 th 7 *(569-571) 7th 4 2012 Nobel in QUANTUM physics 5 many-electron atoms and Schrödinger Equation (ppn557-558) Although the Schrödinger equation
More informationIntroduction to Quantum Mechanics (Prelude to Nuclear Shell Model) Heisenberg Uncertainty Principle In the microscopic world,
Introduction to Quantum Mechanics (Prelude to Nuclear Shell Model) Heisenberg Uncertainty Principle In the microscopic world, x p h π If you try to specify/measure the exact position of a particle you
More informationCHEM3023: Spins, Atoms and Molecules
CHEM3023: Spins, Atoms and Molecules Lecture 4 Molecular orbitals C.-K. Skylaris Learning outcomes Be able to manipulate expressions involving spin orbitals and molecular orbitals Be able to write down
More information6.4 Electronic Structure of Atoms (Electron Configurations)
Chapter 6 Electronic Structure and Periodic Properties of Elements 317 Orbital n l m l degeneracy Radial nodes (no.) 4f 4 3 7 0 4p 4 1 3 2 7f 7 3 7 3 5d 5 2 5 2 Check Your Learning How many orbitals have
More informationLecture 16 February 20 Transition metals, Pd and Pt
Lecture 16 February 20 Transition metals, Pd and Pt Nature of the Chemical Bond with applications to catalysis, materials science, nanotechnology, surface science, bioinorganic chemistry, and energy Course
More informationSupplemental Activities. Module: Atomic Theory. Section: Periodic Properties and Trends - Key
Supplemental Activities Module: Atomic Theory Section: Periodic Properties and Trends - Key Periodic Table and Reactivity Activity 1 1. Consider lithium metal. a. Why don t we find lithium metal in its
More informationLecture 3: Helium Readings: Foot Chapter 3
Lecture 3: Helium Readings: Foot Chapter 3 Last Week: the hydrogen atom, eigenstate wave functions, and the gross and fine energy structure for hydrogen-like single-electron atoms E n Z n = hcr Zα / µ
More informationElectronic Structure of Atoms and the Periodic table. Electron Spin Quantum # m s
Electronic Structure of Atoms and the Periodic table Chapter 6 & 7, Part 3 October 26 th, 2004 Homework session Wednesday 3:00 5:00 Electron Spin Quantum # m s Each electron is assigned a spinning motion
More informationAtomic Structure Atoms are very small ~ metres All atoms are made up of three sub-atomic particles: protons, neutrons and electrons
IB Chemistry (unit ) ATOMIC THEORY Atomic Structure - Recap Questions Define the following words: Atom Element Molecule Compound Atomic Structure Atoms are very small ~ - metres All atoms are made up of
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 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 informationMendeleev s Periodic Law
Mendeleev s Periodic Law Periodic Law When the elements are arranged in order of increasing atomic mass, certain sets of properties recur periodically. Mendeleev s Periodic Law allows us to predict what
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 informationGeneral Chemistry. Contents. Chapter 9: Electrons in Atoms. Contents. 9-1 Electromagnetic Radiation. EM Radiation. Frequency, Wavelength and Velocity
General Chemistry Principles and Modern Applications Petrucci Harwood Herring 8 th Edition Chapter 9: Electrons in Atoms Philip Dutton University of Windsor, Canada N9B 3P4 Contents 9-1 Electromagnetic
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 informationChapter 8. Periodic Properties of the Element
Chapter 8 Periodic Properties of the Element Mendeleev (1834 1907) Ordered elements by atomic mass Saw a repeating pattern of properties Periodic law when the elements are arranged in order of increasing
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 informationChapter 8. Periodic Properties of the Elements
Chapter 8 Periodic Properties of the Elements Mendeleev (1834 1907) Ordered elements by atomic mass. Saw a repeating pattern of properties. Periodic Law When the elements are arranged in order of increasing
More information2.4. Quantum Mechanical description of hydrogen atom
2.4. Quantum Mechanical description of hydrogen atom Atomic units Quantity Atomic unit SI Conversion Ang. mom. h [J s] h = 1, 05459 10 34 Js Mass m e [kg] m e = 9, 1094 10 31 kg Charge e [C] e = 1, 6022
More informationContour Plots Electron assignments and Configurations Screening by inner and common electrons Effective Nuclear Charge Slater s Rules
Lecture 4 362 January 23, 2019 Contour Plots Electron assignments and Configurations Screening by inner and common electrons Effective Nuclear Charge Slater s Rules How to handle atoms larger than H? Effective
More information