Molecules in strong magnetic fields
|
|
- Chester Knight
- 5 years ago
- Views:
Transcription
1 Molecules in strong magnetic fields Trygve Helgaker, Kai Lange, Alessandro Soncini, and Erik Tellgren Centre for Theoretical and Computational Chemistry Department of Chemistry, University of Oslo, Norway 14th European Symposium on Gas-Phase Electron Diffraction M. V. Lomonosov Moscow State University Moscow, June Trygve Helgaker, Kai Lange, Alessandro Soncini, and Erik Tellgren Molecules (CTCC, in strong University magnetic of Oslo) fields 14th European Symposium on GED 1 / 22
2 Outline 1 Molecular magnetism 2 The LONDON program 3 Molecular diamagnetism and paramagnetism 4 The helium atom 5 The H 2 and He 2 molecules 6 Excitations in strong fields 7 Conclusions Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields 14th European Symposium on GED 2 / 22
3 Molecules in weak magnetic fields Molecular magnetism is usually studied perturbatively E(B) = E(0) α mαbα 1 2 χ αβb αβ αb β + such an approach is highly successful and widely used in quantum chemistry molecular magnetic properties are accurately described by perturbation theory Example: 200 MHz NMR spectra of vinyllithium experiment RHF MCSCF B3LYP Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields 14th European Symposium on GED 3 / 22
4 Molecules in strong magnetic fields However, the field dependence of the energy can be complicated the energy of C 20 (ring conformation) plotted against B (atomic units) such behaviour cannot be described or understood perturbatively We have undertaken a nonperturbative study of molecular magnetism gives new insight into molecular electronic structure describes atoms and molecules observed in astrophysics (stellar atmospheres) provides a framework for studying the current dependence of the universal density functional enables evaluation of many properties by finite-difference techniques Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields 14th European Symposium on GED 4 / 22
5 The electronic Hamiltonian The non-relativistic electronic Hamiltonian (atomic units) in a magnetic field B is given by H = T + V Coulomb BLz + Bsz i (x2 i + yi 2 )B2 T is the kinetic-energy operator and V Coulomb = V en + V ee + V nn the Coulomb operator L and s are the orbital and spin angular momentum operators one atomic unit of B corresponds to T = G Coulomb regime: B 0 a.u. earth-like conditions: Coulomb interactions dominate magnetic interactions are treated perturbatively earth magnetism 10 10, loudspeaker 10 5, NMR 10 4 ; pulsed laboratory field 10 3 a.u. Landau regime: B 1 a.u. astrophysical conditions: magnetic interactions dominate Coulomb interactions are treated perturbatively harmonic-oscillator Hamiltonian with force constant B 2 /4: Landau levels the electrons are confined in the transverse directions by the magnetic field neutron stars a.u.; magnetars 10 5 a.u. Intermediate regime: B 1 a.u. the Coulomb and magnetic interactions are equally important complicated behaviour resulting from interplay of linear and quadratic terms white dwarves: up to about 1 a.u. We here consider the weak and intermediate regimes (B 15 a.u.) Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields 14th European Symposium on GED 5 / 22
6 The LONDON program The electronic Hamiltonian in a magnetic field H = T + V Coulomb B L + B s contains the orbital-angular momentum operator i (B r i ) (B r i ) L = i i (r i O) i (1) this operator is imaginary and depends on an arbitrary gauge origin O our results should be invariant to the choice of O Standard quantum-chemistry program cannot treat molecules in finite magnetic fields we must optimize a complex wave function we must use London atomic orbitals to ensure gauge-origin invariance ω lm (r K, B) = exp [ 1 2 ib (O K) r] χ lm (r K ) We have developed the LONDON code for this purpose complex wave functions with London atomic orbitals Hartree Fock (RHF, UHF, GHF) and Kohn Sham theories FCI and MCSCF theories first-order properties including molecular gradients for geometry optimizations linear response theory The code has been written by Erik Tellgren, Kai Lange, and Alessandro Soncini mostly C++, some Fortran 77 modular but not highly optimized yet C 20 is a large system Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields 14th European Symposium on GED 6 / 22
7 Diamagnetism and paramagnetism When a magnetic field is applied to a molecule, one of two things can happen: the energy is lowered: molecular paramagnetism the energy is raised: molecular diamagnetism Open-shell molecules are paramagnetic the molecule reorients the permanent magnetic moment and lowers the energy Closed-shell molecules may be diamagnetic or paramagnetic the induced magnetic moment may oppose or enhance the external field 0.02 most molecules are diamagnetic, opposing the external field by Lenz law in some systems, orbital unquenching results in paramagnetism RHF calculations of the field-dependence for three closed-shell systems: a) b) 14 x 10 3 a) c) b) d) x ac) benzene (aug-cc-pvdz): typical d) diamagnetic quadratic dependence on an out-of-plane field 0 b cyclobutadiene (aug-cc-pvdz): diamagnetic non-quadratic dependence on an out-of-plane field c BH (aug-cc-pvtz): paramagnetic dependence on a perpendicular field Helgaker et al (CTCC, University of Oslo) Molecules 0 in strong magnetic fields 14th European Symposium on GED 7 / 22
8 0.015! !0.004 Stabilizing field strength for paramagnetic molecules a) However, plots over a larger range reveal that all closed-shell systems become diamagnetic in sufficiently strong fields: c) d) W!W0!au" BH W!W0!au" C8H8: total energy W!W0!au" C12H12: total energy B!au" B!au"!0.005!0.01!0.02 STO!3G, Bc " 0.24 DZ, Bc " 0.22 aug!dz, Bc " STO!3G, Bc " !31G, Bc " cc!pvdz, Bc " STO!3G, Bc " !31G, Bc " cc!pvdz, Bc " 0.016! B!au"!0.0005!0.04!0.002!0.001 W!W0!au" CH b) # B!au" !0.02 STO3!G, Bc " 0.43!0.04 DZ, Bc " 0.44!0.06 aug!dz, Bc " 0.45!0.08!0.1!0.12 W!W0!au" MnO4! c) B!au" The transition occurs at a characteristic stabilizing critical field strength B c B c 0.22 perpendicular to principal axis for BH B c along the principal axis for antiaromatic octatetraene C 8H 8 B c along the principal axis for antiaromatic [12]-annulene C 12H 12 The stabilizing field strength decreases with increasing molecular size strongest laboratory fields attainable: 10 3 a.u. B c is inversely proportional to the area of the molecule normal to the field we estimate that B c should be observable for C 72H 72 We may in principle separate such molecules by applying a field gradient!0.1 STO!3G, Bc " 0.45 Helgaker et al. (CTCC, UniversityWachters, of Oslo) Bc " 0.50 Molecules in strong magnetic fields 14th European Symposium on GED 8 / 22
9 Analytical model for the diamagnetic transition The diamagnetic transition arises from an interplay between linear and quadratic terms it can be understood from a simple two-level model Molecular orbitals relevant for BH: Ground and excited states: 1s B, 2σ BH, 2p x, 2p y, 2p z (molecule along z axis) 0 = 1sB 2 2σ2 BH 2p2 z, 1 x = 1sB 2 2σ2 BH 2pz 2px, 1y = 1s2 B 2σ2 BH2pz 2py Let us apply a perpendicular magnetic field in the y direction: H B = 1 2 BLy B2 (x 2 + z 2 ) This leads to the following Hamiltonian matrix where 1 = 1 x : ( ) ( 0 H 0 0 H 1 E0 1 H(B) = = 2 χ 0B 2 ) iµb 1 H 0 1 H 1 iµb E χ 1B 2 where we have introduced Interpretation: χ 0 = x2 + z 2 0 < 0, χ 1 = x2 + z 2 1 < 0, µ = 1 i 0 Ly 1 2 the diabatic states 0 and 1 x are uncoupled at B = 0 and diamagnetic in a magnetic field, they become coupled by L y the resulting adiabatic ground state may be either diamagnitic or paramagnetic Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields 14th European Symposium on GED 9 / 22
10 Energy levels of the two-level model The excited state assumed E = 0.02 above the ground-state at B = 0 magnetizabilities of the diabatic states: χ 0 = 28 (red) and χ 1 = 16 (yellow) Plots for different values of the coupling-matrix element µ: uncoupled (0), diamagnetic (0.2), nonmagnetic (0.374), and paramagnetic (0.6) the magnetizability of the adiabatic ground state: χ GS = χ 0 + µ 2 / E = 7, 5, 0, 11 Paramagnetism results from avoided crossings at finite magnetic fields Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields 14th European Symposium on GED 10 / 22
11 Induced rotation of the electrons The magnetic field induces a rotation of the electrons about the field direction: the amount of rotation is the expectation value of the kinetic angular-momentum operator 0 Λ 0 = 2E (B) = 2M, Λ = r π, π = p + A positive rotation increases the energy; negative rotation reduces the energy Diamagnetic closed-shell molecules: 0 Λ 0 aligns with the field according to Lenz law, increasing the energy Λ 0 B Paramagnetic closed-shell molecules: 0 Λ 0 first aligns against the field, decreasing the energy 0 Λ 0 goes through a minimum (maximum rotation) at the inflection point E (B) = 0 0 Λ 0 then increases again until rotation vanishes at B c 0 Λ 0 finally aligns with the field, making the system diamagnetic E(B x ) Energy!25.11!25.12!25.13!25.14!25.15! L x (B x ) Angular momentum! L x / r! C nuc !0.2 Nuclear shielding integral Boron Hydrogen there is no net rotation at the stationary points B = 0 and B = B c Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields 14th European Symposium on GED 11 / 22
12 C 20 : more structure Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields 14th European Symposium on GED 12 / 22
13 Helium atom, RHF/augccpVTZ Field, B [au] The helium atom The helium energy behaves in simple manner in magnetic fields (left) an initial quadratic diamagnetic behaviour is followed by a linear increase in B this is expected from the Landau levels (harmonic potential increases as B 2 ) The orbital energies behave in a more complicated manner (middle) the initial behaviour is determined by the angular momentum beyond B 1, all energies increase with increasing field HOMO LUMO gap increases, suggesting a decreasing importance of electron correlation The atom becomes squeezed in the magnetic field (right) this is particularly true for the transverse directions the cyclotron radius is proportional to B 1/2 special basis sets are need for strong fields (Landau orbitals) 3 Helium atom, RHF/augccpVTZ 0.8 Helium atom, RHF/augccpVTZ Energy, E [Hartree] Orbital energy, [Hartree] Quadrupole moment [bohr 2 ] Qxx Qyy Qzz Field, B [au] Field, B [au] Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields 14th European Symposium on GED 13 / 22
14 The helium natural occupation numbers The natural occupation numbers are eigenvalues of the one-electron density matrix The FCI occupation numbers approach 2 and 0 strong fields decreasing importance of dynamical correlation in magnetic fields the two electrons rotate in the same direction about the field direction Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields 14th European Symposium on GED 14 / 22
15 The H 2 molecule Magnetic fields strongly affect chemical bonding and potetial energy surfaces Potential energy curves of H 2 in a parallel field B = 0, 5, 10 a.u. FCI/aug-cc-pVDZ calculations with energy set to zero at full dissociation H 2/aug-cc-pVDZ, lowest singlet, field parallel to bond 0.5 H 2/aug-cc-pVDZ, lowest triplet, field parallel to bond E (Ha) 1.5 E (Ha) B = 0.0 a.u. B = 5.0 a.u. B = 10.0 a.u R (bohr) B = 0.0 a.u. B = 5.0 a.u. B = 10.0 a.u R (bohr) H 2 becomes strongly bound in strong fields the basis set is too small to give quantitative results The low-spin triplet component (ββ) becomes the ground state at about 0.25 a.u. the Zeeman interaction contributes BM s to the energy Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields 14th European Symposium on GED 15 / 22
16 Electron rotation and correlation in H 2 The energy of H 2 in the singlet state increases in a parallel field (left) the field induces a rotation of the electrons 0 Λ z 0 about the molecular axis (right) The occupation numbers indicate reduced importance of dynamical correlation the two electrons rotate in the same direction, reducing the chance of near encounters 2.00 H 2/cc-pVTZ, optimized geometry, field parallel to bond Cummulative occupation number B (a.u.) Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields 14th European Symposium on GED 16 / 22
17 The covalently bound He 2 molecule We have also performed FCI/aug-cc-pVDZ calculations on the helium dimer energy set to zero for full dissociation in all cases We have considered the lowest singlet, triplet and quintet states in the absence of a field, only the triplet state is covalently bound in the presence of a field, all states become strongly bound He 2/aug-cc-pVDZ, lowest singlet, field parallel to bond He 2/aug-cc-pVDZ, lowest triplet, field parallel to bond He 2/aug-cc-pVDZ, lowest quintet, field parallel to bond E (Ha) 0.5 E (Ha) 0.4 E (Ha) B = 0.0 a.u. B = 5.0 a.u. B = 10.0 a.u. B = 15.0 a.u R (bohr) B = 0.0 a.u. B = 5.0 a.u. B = 10.0 a.u. B = 15.0 a.u R (bohr) B = 0.0 a.u. B = 5.0 a.u. B = 10.0 a.u. B = 15.0 a.u R (bohr) In magnetic fields, spin multiplets are split within each multiplet, the low-spin component has lowest energy Increasing magnetic fields favor low-spin components of high-spin states in weak fields, the ground state is a singlet in strong fields, the ground state is the low-spin quintet Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields 14th European Symposium on GED 17 / 22
18 Distortion of the H 2 O molecule H 2 O, HF/6 31G** H 2 O, HF/6 31G** oop in plane, C2 axis in plane, C2 axis oop in plane, C2 axis in plane, C2 axis d(o H) [pm] H O H angle [degrees] B [au] B [au] The molecule shrinks OH bonds contract by 0.25 pm in a field of 0.16 a.u. The molecule becomes more linear except for fields along the symmetry axis Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields 14th European Symposium on GED 18 / 22
19 Distortion of the NH 3 molecule Magnetic field applied along the symmetry axis NH 3, HF/6 31G** NH 3, HF/6 31G** d(n H) [pm] H N H angle [degrees] B [au] B [au] As expected the molecule shrinks bonds contract by 0.3 pm in a field of 0.15 a.u. Less intuitively, the molecule becomes more planar could this be caused by the shrinking lone pair? Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields 14th European Symposium on GED 19 / 22
20 Distortion of the C 6 H 6 molecule At B = 0.16, the molecule is 6.1 pm narrower and 3.5 pm longer in the field direction in agreement with perturbational estimates by Caputo and Lazzeretti, IJQC 111, 772 (2011) C 6 H 6, HF/6 31G* C H B C H other d(c H) [pm] B [au] C 6 H 6, HF/6 31G* C 6 H 6, HF/6 31G* C C C (a) C C H (b) d(c C) [pm] C C other (c) C C B (d) bond angle [degrees] B [au] B [au]
21 Linear response theory We have implemented linear response theory (RPA) in finite magnetic fields lowest CH 3 radical states in magnetic fields (middle) transitions to the green state are electric-dipole allowed oscillator strength for transition to green CH 3 state length (red) and velocity (blue) gauges En [Ha] B [a.u.] f E B [a.u.] Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields 14th European Symposium on GED 21 / 22
22 Conclusions We have developed the LONDON program complex wave functions London atomic orbitals HF and FCI theories molecular gradients and linear response theory The LONDON program may be used for finite-difference alternative to analytical derivatives studies of molecules in strong magnetic fields studies of exchange correlation functional in magnetic fields We have studied the behaviour of paramagnetic molecules in strong fields all paramagnetic molecules attain a global minimum at a characteristic field B c B c decreases with system size and should be observable for C 72H 72 We have studied H 2 and He 2 using FCI theory molecules become more strongly bound in magnetic fields the helium molecule becomes covalently bound We have studied molecular structure in magnetic fields bond distances are typically shortened in magnetic fields We have studied excitation energies in magnetic fields excitation energies typically increase in magnetic fields Helgaker et al. (CTCC, University of Oslo) Conclusions 14th European Symposium on GED 22 / 22
Molecular electronic structure in strong magnetic fields
Trygve Helgaker 1, Mark Hoffmann 1,2 Kai Molecules Lange in strong 1, Alessandro magnetic fields Soncini 1,3, andcctcc Erik Jackson Tellgren 211 1 (CTCC, 1 / 23 Uni Molecular electronic structure in strong
More informationMolecules in strong magnetic fields
Molecules in strong magnetic fields Trygve Helgaker, Kai Lange, Alessandro Soncini, and Erik Tellgren Centre for Theoretical and Computational Chemistry Department of Chemistry, University of Oslo, Norway
More informationChemical bonding in strong magnetic fields
Trygve Helgaker 1, Mark Hoffmann 1,2 Chemical Kai Lange bonding in 1, strong Alessandro magnetic fields Soncini 1,3, CMS212, and Erik JuneTellgren 24 27 212 1 (CTCC, 1 / 32 Uni Chemical bonding in strong
More informationMolecular Magnetism. Molecules in an External Magnetic Field. Trygve Helgaker
Molecular Magnetism Molecules in an External Magnetic Field Trygve Helgaker Centre for Theoretical and Computational Chemistry (CTCC), Department of Chemistry, University of Oslo, Norway Laboratoire de
More informationMolecular bonding in strong magnetic fields
Trygve Helgaker 1, Mark Hoffmann 1,2 Kai Molecules Lange in strong 1, Alessandro magnetic fields Soncini 1,3 AMAP,, andzurich, ErikJune Tellgren 1 4 212 1 (CTCC, 1 / 35 Uni Molecular bonding in strong
More informationMolecules in strong magnetic fields
Trygve Helgaker 1, Mark Hoffmann 1,2 Kai Molecules Lange in strong 1, Alessandro magnetic fields Soncini 1,3 Nottingham,, and Erik 3th October Tellgren 213 1 (CTCC, 1 / 27 Uni Molecules in strong magnetic
More informationMolecular Magnetic Properties
Molecular Magnetic Properties Trygve Helgaker Centre for Theoretical and Computational Chemistry (CTCC), Department of Chemistry, University of Oslo, Norway Raman Centre for Atomic, Molecular and Optical
More informationDiamagnetism and Paramagnetism in Atoms and Molecules
Diamagnetism and Paramagnetism in Atoms and Molecules Trygve Helgaker Alex Borgoo, Maria Dimitrova, Jürgen Gauss, Florian Hampe, Christof Holzer, Wim Klopper, Trond Saue, Peter Schwerdtfeger, Stella Stopkowicz,
More informationMolecular Magnetic Properties
Molecular Magnetic Properties Trygve Helgaker Centre for Theoretical and Computational Chemistry Department of Chemistry, University of Oslo, Norway The 12th Sostrup Summer School Quantum Chemistry and
More informationMolecules in Magnetic Fields
Molecules in Magnetic Fields Trygve Helgaker Hylleraas Centre, Department of Chemistry, University of Oslo, Norway and Centre for Advanced Study at the Norwegian Academy of Science and Letters, Oslo, Norway
More informationMolecular Magnetism. Magnetic Resonance Parameters. Trygve Helgaker
Molecular Magnetism Magnetic Resonance Parameters Trygve Helgaker Centre for Theoretical and Computational Chemistry (CTCC), Department of Chemistry, University of Oslo, Norway Laboratoire de Chimie Théorique,
More informationMolecular Magnetic Properties
Molecular Magnetic Properties Trygve Helgaker Hylleraas Centre, Department of Chemistry, University of Oslo, Norway and Centre for Advanced Study at the Norwegian Academy of Science and Letters, Oslo,
More informationDFT calculations of NMR indirect spin spin coupling constants
DFT calculations of NMR indirect spin spin coupling constants Dalton program system Program capabilities Density functional theory Kohn Sham theory LDA, GGA and hybrid theories Indirect NMR spin spin coupling
More informationMolecular Magnetic Properties. The 11th Sostrup Summer School. Quantum Chemistry and Molecular Properties July 4 16, 2010
1 Molecular Magnetic Properties The 11th Sostrup Summer School Quantum Chemistry and Molecular Properties July 4 16, 2010 Trygve Helgaker Centre for Theoretical and Computational Chemistry, Department
More informationMolecular Magnetic Properties
Molecular Magnetic Properties Trygve Helgaker Centre for Theoretical and Computational Chemistry (CTCC), Department of Chemistry, University of Oslo, Norway European Summer School in Quantum Chemistry
More informationThe calculation of the universal density functional by Lieb maximization
The calculation of the universal density functional by Lieb maximization Trygve Helgaker, Andy Teale, and Sonia Coriani Centre for Theoretical and Computational Chemistry (CTCC), Department of Chemistry,
More informationOslo node. Highly accurate calculations benchmarking and extrapolations
Oslo node Highly accurate calculations benchmarking and extrapolations Torgeir Ruden, with A. Halkier, P. Jørgensen, J. Olsen, W. Klopper, J. Gauss, P. Taylor Explicitly correlated methods Pål Dahle, collaboration
More informationBasis sets for electron correlation
Basis sets for electron correlation Trygve Helgaker Centre for Theoretical and Computational Chemistry Department of Chemistry, University of Oslo, Norway The 12th Sostrup Summer School Quantum Chemistry
More informationQuantum Chemistry in Magnetic Fields
Quantum Chemistry in Magnetic Fields Trygve Helgaker Hylleraas Centre of Quantum Molecular Sciences, Department of Chemistry, University of Oslo, Norway 11th Triennial Congress of the World Association
More informationImporting ab-initio theory into DFT: Some applications of the Lieb variation principle
Importing ab-initio theory into DFT: Some applications of the Lieb variation principle Trygve Helgaker, Andy Teale, and Sonia Coriani Centre for Theoretical and Computational Chemistry (CTCC), Department
More informationNon perturbative properties of
Non perturbative properties of molecules in strong magnetic fields Alessandro Soncini INPAC Institute for NanoscalePhysics and Chemistry, University of Leuven, elgium Lb ti Nti ld Ch M éti I t Laboratoire
More informationQuantum chemical modelling of molecular properties - parameters of EPR spectra
Quantum chemical modelling of molecular properties - parameters of EPR spectra EPR ( electric paramagnetic resonance) spectra can be obtained only for open-shell systems, since they rely on transitions
More informationMolecular Magnetic Properties
Molecular Magnetic Properties Trygve Helgaker Centre for Theoretical and Computational Chemistry (CTCC), Department of Chemistry, University of Oslo, Norway European Summer School in Quantum Chemistry
More informationElectromagnetism II. Instructor: Andrei Sirenko Spring 2013 Thursdays 1 pm 4 pm. Spring 2013, NJIT 1
Electromagnetism II Instructor: Andrei Sirenko sirenko@njit.edu Spring 013 Thursdays 1 pm 4 pm Spring 013, NJIT 1 PROBLEMS for CH. 6 http://web.njit.edu/~sirenko/phys433/phys433eandm013.htm Can obtain
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 informationElectric properties of molecules
Electric properties of molecules For a molecule in a uniform electric fielde the Hamiltonian has the form: Ĥ(E) = Ĥ + E ˆµ x where we assume that the field is directed along the x axis and ˆµ x is the
More informationChem 442 Review for Exam 2. Exact separation of the Hamiltonian of a hydrogenic atom into center-of-mass (3D) and relative (3D) components.
Chem 44 Review for Exam Hydrogenic atoms: The Coulomb energy between two point charges Ze and e: V r Ze r Exact separation of the Hamiltonian of a hydrogenic atom into center-of-mass (3D) and relative
More informationMolecular Magnetic Properties ESQC 07. Overview
1 Molecular Magnetic Properties ESQC 07 Trygve Helgaker Department of Chemistry, University of Oslo, Norway Overview the electronic Hamiltonian in an electromagnetic field external and nuclear magnetic
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 informationψ s a ˆn a s b ˆn b ψ Hint: Because the state is spherically symmetric the answer can depend only on the angle between the two directions.
1. Quantum Mechanics (Fall 2004) Two spin-half particles are in a state with total spin zero. Let ˆn a and ˆn b be unit vectors in two arbitrary directions. Calculate the expectation value of the product
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 informationNuclear vibrations and rotations
Nuclear vibrations and rotations Introduction to Nuclear Science Simon Fraser University Spring 2011 NUCS 342 February 2, 2011 NUCS 342 (Lecture 9) February 2, 2011 1 / 29 Outline 1 Significance of collective
More informationLecture B6 Molecular Orbital Theory. Sometimes it's good to be alone.
Lecture B6 Molecular Orbital Theory Sometimes it's good to be alone. Covalent Bond Theories 1. VSEPR (valence shell electron pair repulsion model). A set of empirical rules for predicting a molecular geometry
More informationElectronic structure theory: Fundamentals to frontiers. 1. Hartree-Fock theory
Electronic structure theory: Fundamentals to frontiers. 1. Hartree-Fock theory MARTIN HEAD-GORDON, Department of Chemistry, University of California, and Chemical Sciences Division, Lawrence Berkeley National
More informationLUMO + 1 LUMO. Tómas Arnar Guðmundsson Report 2 Reikniefnafræði G
Q1: Display all the MOs for N2 in your report and classify each one of them as bonding, antibonding or non-bonding, and say whether the symmetry of the orbital is σ or π. Sketch a molecular orbital diagram
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 informationAn Introduction to Quantum Chemistry and Potential Energy Surfaces. Benjamin G. Levine
An Introduction to Quantum Chemistry and Potential Energy Surfaces Benjamin G. Levine This Week s Lecture Potential energy surfaces What are they? What are they good for? How do we use them to solve chemical
More informationOVERVIEW OF QUANTUM CHEMISTRY METHODS
OVERVIEW OF QUANTUM CHEMISTRY METHODS Outline I Generalities Correlation, basis sets Spin II Wavefunction methods Hartree-Fock Configuration interaction Coupled cluster Perturbative methods III Density
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 informationThe adiabatic connection
The adiabatic connection Trygve Helgaker, Andy Teale, and Sonia Coriani Centre for Theoretical and Computational Chemistry (CTCC), Department of Chemistry, University of Oslo, Norway Dipartimento di Scienze
More informationMagentic Energy Diagram for A Single Electron Spin and Two Coupled Electron Spins. Zero Field.
7. Examples of Magnetic Energy Diagrams. There are several very important cases of electron spin magnetic energy diagrams to examine in detail, because they appear repeatedly in many photochemical systems.
More informationDensity functional theory in magnetic fields
Density functional theory in magnetic fields U. E. Ekström, T. Helgaker, S. Kvaal, E. Sagvolden, E. Tellgren, A. M. Teale Centre for Theoretical and Computational Chemistry (CTCC), Department of Chemistry,
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 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 informationExchange Mechanisms. Erik Koch Institute for Advanced Simulation, Forschungszentrum Jülich. lecture notes:
Exchange Mechanisms Erik Koch Institute for Advanced Simulation, Forschungszentrum Jülich lecture notes: www.cond-mat.de/events/correl Magnetism is Quantum Mechanical QUANTUM MECHANICS THE KEY TO UNDERSTANDING
More informationDalton Quantum Chemistry Program
1 Quotation from home page: Dalton Quantum Chemistry Program Dalton QCP represents a powerful quantum chemistry program for the calculation of molecular properties with SCF, MP2, MCSCF or CC wave functions.
More informationNuclear models: Collective Nuclear Models (part 2)
Lecture 4 Nuclear models: Collective Nuclear Models (part 2) WS2012/13: Introduction to Nuclear and Particle Physics,, Part I 1 Reminder : cf. Lecture 3 Collective excitations of nuclei The single-particle
More informationCondensed Matter Physics Prof. G. Rangarajan Department of Physics Indian Institute of Technology, Madras
(Refer Slide Time: 00:22) Condensed Matter Physics Prof. G. Rangarajan Department of Physics Indian Institute of Technology, Madras Lecture 25 Pauli paramagnetism and Landau diamagnetism So far, in our
More informationHighly accurate quantum-chemical calculations
1 Highly accurate quantum-chemical calculations T. Helgaker Centre for Theoretical and Computational Chemistry, Department of Chemistry, University of Oslo, Norway A. C. Hennum and T. Ruden, University
More informationAn Introduction to Hyperfine Structure and Its G-factor
An Introduction to Hyperfine Structure and Its G-factor Xiqiao Wang East Tennessee State University April 25, 2012 1 1. Introduction In a book chapter entitled Model Calculations of Radiation Induced Damage
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 informationH 2 in the minimal basis
H 2 in the minimal basis Alston J. Misquitta Centre for Condensed Matter and Materials Physics Queen Mary, University of London January 27, 2016 Overview H 2 : The 1-electron basis. The two-electron basis
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 informationCOUPLED-CLUSTER CALCULATIONS OF GROUND AND EXCITED STATES OF NUCLEI
COUPLED-CLUSTER CALCULATIONS OF GROUND AND EXCITED STATES OF NUCLEI Marta Włoch, a Jeffrey R. Gour, a and Piotr Piecuch a,b a Department of Chemistry,Michigan State University, East Lansing, MI 48824 b
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 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 informationPreface Introduction to the electron liquid
Table of Preface page xvii 1 Introduction to the electron liquid 1 1.1 A tale of many electrons 1 1.2 Where the electrons roam: physical realizations of the electron liquid 5 1.2.1 Three dimensions 5 1.2.2
More informationQuantum Theory of Many-Particle Systems, Phys. 540
Quantum Theory of Many-Particle Systems, Phys. 540 Questions about organization Second quantization Questions about last class? Comments? Similar strategy N-particles Consider Two-body operators in Fock
More informationAb-initio studies of the adiabatic connection in density-functional theory
Ab-initio studies of the adiabatic connection in density-functional theory Trygve Helgaker, Andy Teale, and Sonia Coriani Centre for Theoretical and Computational Chemistry (CTCC), Department of Chemistry,
More informationQUANTUM CHEMISTRY FOR TRANSITION METALS
QUANTUM CHEMISTRY FOR TRANSITION METALS Outline I Introduction II Correlation Static correlation effects MC methods DFT III Relativity Generalities From 4 to 1 components Effective core potential Outline
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 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 informationTime-independent molecular properties
Time-independent molecular properties Trygve Helgaker Hylleraas Centre, Department of Chemistry, University of Oslo, Norway and Centre for Advanced Study at the Norwegian Academy of Science and Letters,
More informationONE-ELECTRON AND TWO-ELECTRON SPECTRA
ONE-ELECTRON AND TWO-ELECTRON SPECTRA (A) FINE STRUCTURE AND ONE-ELECTRON SPECTRUM PRINCIPLE AND TASK The well-known spectral lines of He are used for calibrating the diffraction spectrometer. The wavelengths
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 informationExercise 1: Structure and dipole moment of a small molecule
Introduction to computational chemistry Exercise 1: Structure and dipole moment of a small molecule Vesa Hänninen 1 Introduction In this exercise the equilibrium structure and the dipole moment of a small
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 informationChapter 10: Multi- Electron Atoms Optical Excitations
Chapter 10: Multi- Electron Atoms Optical Excitations To describe the energy levels in multi-electron atoms, we need to include all forces. The strongest forces are the forces we already discussed in Chapter
More informationwith the larger dimerization energy also exhibits the larger structural changes.
A7. Looking at the image and table provided below, it is apparent that the monomer and dimer are structurally almost identical. Although angular and dihedral data were not included, these data are also
More information4) protons experience a net magnetic field strength that is smaller than the applied magnetic field.
1) Which of the following CANNOT be probed by an spectrometer? See sect 15.1 Chapter 15: 1 A) nucleus with odd number of protons & odd number of neutrons B) nucleus with odd number of protons &even number
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 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 informationCHEMISTRY. Chapter 8 ADVANCED THEORIES OF COVALENT BONDING Kevin Kolack, Ph.D. The Cooper Union HW problems: 6, 7, 12, 21, 27, 29, 41, 47, 49
CHEMISTRY Chapter 8 ADVANCED THEORIES OF COVALENT BONDING Kevin Kolack, Ph.D. The Cooper Union HW problems: 6, 7, 12, 21, 27, 29, 41, 47, 49 2 CH. 8 OUTLINE 8.1 Valence Bond Theory 8.2 Hybrid Atomic Orbitals
More information4) protons experience a net magnetic field strength that is smaller than the applied magnetic field.
1) Which of the following CANNOT be probed by an NMR spectrometer? See sect 15.1 Chapter 15: 1 A) nucleus with odd number of protons & odd number of neutrons B) nucleus with odd number of protons &even
More informationDivergence in Møller Plesset theory: A simple explanation based on a two-state model
JOURNAL OF CHEMICAL PHYSICS VOLUME 112, NUMBER 22 8 JUNE 2000 Divergence in Møller Plesset theory: A simple explanation based on a two-state model Jeppe Olsen and Poul Jørgensen a) Department of Chemistry,
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 informationHONOUR SCHOOL OF NATURAL SCIENCE. Final Examination GENERAL PHYSICAL CHEMISTRY I. Answer FIVE out of nine questions
HONOUR SCHOOL OF NATURAL SCIENCE Final Examination GENERAL PHYSICAL CHEMISTRY I Monday, 12 th June 2000, 9.30 a.m. - 12.30 p.m. Answer FIVE out of nine questions The numbers in square brackets indicate
More informationNPTEL/IITM. Molecular Spectroscopy Lectures 1 & 2. Prof.K. Mangala Sunder Page 1 of 15. Topics. Part I : Introductory concepts Topics
Molecular Spectroscopy Lectures 1 & 2 Part I : Introductory concepts Topics Why spectroscopy? Introduction to electromagnetic radiation Interaction of radiation with matter What are spectra? Beer-Lambert
More informationThe Zeeman Effect in Atomic Mercury (Taryl Kirk )
The Zeeman Effect in Atomic Mercury (Taryl Kirk - 2001) Introduction A state with a well defined quantum number breaks up into several sub-states when the atom is in a magnetic field. The final energies
More informationQuantum Mechanics: Fundamentals
Kurt Gottfried Tung-Mow Yan Quantum Mechanics: Fundamentals Second Edition With 75 Figures Springer Preface vii Fundamental Concepts 1 1.1 Complementarity and Uncertainty 1 (a) Complementarity 2 (b) The
More informationcharges q r p = q 2mc 2mc L (1.4) ptles µ e = g e
APAS 5110. Atomic and Molecular Processes. Fall 2013. 1. Magnetic Moment Classically, the magnetic moment µ of a system of charges q at positions r moving with velocities v is µ = 1 qr v. (1.1) 2c charges
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 informationCrystal Field Theory
Crystal Field Theory It is not a bonding theory Method of explaining some physical properties that occur in transition metal complexes. Involves a simple electrostatic argument which can yield reasonable
More informationIntermission: Let s review the essentials of the Helium Atom
PHYS3022 Applied Quantum Mechanics Problem Set 4 Due Date: 6 March 2018 (Tuesday) T+2 = 8 March 2018 All problem sets should be handed in not later than 5pm on the due date. Drop your assignments in the
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 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 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 informationMD simulation: output
Properties MD simulation: output Trajectory of atoms positions: e. g. diffusion, mass transport velocities: e. g. v-v autocorrelation spectrum Energies temperature displacement fluctuations Mean square
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 informationDensity Functional Theory
Density Functional Theory March 26, 2009 ? DENSITY FUNCTIONAL THEORY is a method to successfully describe the behavior of atomic and molecular systems and is used for instance for: structural prediction
More informationIntroduction to Computational Chemistry
Introduction to Computational Chemistry Vesa Hänninen Laboratory of Physical Chemistry Chemicum 4th floor vesa.hanninen@helsinki.fi September 10, 2013 Lecture 3. Electron correlation methods September
More informationThe Superfluid Phase s of Helium 3
The Superfluid Phase s of Helium 3 DIETER VOLLHARD T Rheinisch-Westfälische Technische Hochschule Aachen, Federal Republic of German y PETER WÖLFL E Universität Karlsruhe Federal Republic of Germany PREFACE
More information2m 2 Ze2. , where δ. ) 2 l,n is the quantum defect (of order one but larger
PHYS 402, Atomic and Molecular Physics Spring 2017, final exam, solutions 1. Hydrogenic atom energies: Consider a hydrogenic atom or ion with nuclear charge Z and the usual quantum states φ nlm. (a) (2
More informationQuantum Physics II (8.05) Fall 2002 Assignment 12 and Study Aid
Quantum Physics II (8.05) Fall 2002 Assignment 12 and Study Aid Announcement This handout includes 9 problems. The first 5 are the problem set due. The last 4 cover material from the final few lectures
More informationBonding forces and energies Primary interatomic bonds Secondary bonding Molecules
Chapter 2. Atomic structure and interatomic bonding 2.1. Atomic structure 2.1.1.Fundamental concepts 2.1.2. Electrons in atoms 2.1.3. The periodic table 2.2. Atomic bonding in solids 2.2.1. Bonding forces
More informationCHEM J-5 June 2014
CHEM1101 2014-J-5 June 2014 The molecular orbital energy level diagrams for H 2, H 2 +, H 2 and O 2 are shown below. Fill in the valence electrons for each species in its ground state and label the types
More informationJanuary 30, 2018 Chemistry 328N
Lecture 4 Some More nmr January 30, 2018 Tricks for solving unknowns Review. Empirical formula is lowest common denominator ratio of atomic composition From Homework: unknown has an empirical formula of
More informationThe Gutzwiller Density Functional Theory
The Gutzwiller Density Functional Theory Jörg Bünemann, BTU Cottbus I) Introduction 1. Model for an H 2 -molecule 2. Transition metals and their compounds II) Gutzwiller variational theory 1. Gutzwiller
More informationc E If photon Mass particle 8-1
Nuclear Force, Structure and Models Readings: Nuclear and Radiochemistry: Chapter 10 (Nuclear Models) Modern Nuclear Chemistry: Chapter 5 (Nuclear Forces) and Chapter 6 (Nuclear Structure) Characterization
More informationLecture 5: More about one- Final words about the Hartree-Fock theory. First step above it by the Møller-Plesset perturbation theory.
Lecture 5: More about one- determinant wave functions Final words about the Hartree-Fock theory. First step above it by the Møller-Plesset perturbation theory. Items from Lecture 4 Could the Koopmans theorem
More information