Molecular electronic structure in strong magnetic fields


 Darcy Craig
 5 months ago
 Views:
Transcription
1 Trygve Helgaker 1, Mark Hoffmann 1,2 Kai Molecules Lange in strong 1, Alessandro magnetic fields Soncini 1,3, andcctcc Erik Jackson Tellgren (CTCC, 1 / 23 Uni Molecular electronic structure in strong magnetic fields Trygve Helgaker 1, Mark Hoffmann 1,2 Kai Lange 1, Alessandro Soncini 1,3, and Erik Tellgren 1 1 CTCC, Department of Chemistry, University of Oslo, Norway 2 Department of Chemistry, University of North Dakota, Grand Forks, USA 3 School of Chemistry, University of Melbourne, Australia 2th Conference on Current Trends in Computational Chemistry (CCTCC), Hilton Jackson Hotel and Conference Center, Jackson, Mississippi, USA, October 27 29, 211
2 Introduction perturbative vs. nonperturbative studies Molecular magnetism is usually studied perturbatively such an approach is highly successful and widely used in quantum chemistry molecular magnetic properties are accurately described by perturbation theory example: 2 MHz NMR spectra of vinyllithium experiment RHF MCSCF B3LYP 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 finitedifference techniques Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields CCTCC Jackson / 23
3 Introduction the electronic Hamiltonian The nonrelativistic electronic Hamiltonian (a.u.) in a magnetic field B along the z axis: H = H BLz + Bsz B2 (x 2 + y 2 ) linear and quadratic B terms H is the fieldfree operator and L and s the orbital and spin angular momentum operators one atomic unit of B corresponds to T = G Coulomb regime: B a.u. earthlike conditions: Coulomb interactions dominate magnetic interactions are treated perturbatively earth magnetism 1 1, NMR 1 4 ; pulsed laboratory field 1 3 a.u. Intermediate regime: B 1 a.u. the Coulomb and magnetic interactions are equally important complicated behaviour resulting from an interplay of linear and quadratic terms white dwarves: up to about 1 a.u. Landau regime: B 1 a.u. astrophysical conditions: magnetic interactions dominate Landau levels: harmonicoscillator Hamiltonian with force constant B 2 /4 Coulomb interactions are treated perturbatively relativity becomes important for B α a.u. neutron stars a.u. We here consider the weak and intermediate regimes (B < 1 a.u.) For a review, see D. Lai, Rev. Mod. Phys. 73, 629 (21) Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields CCTCC Jackson / 23
4 Introduction London orbitals and size extensivity The nonrelativistic electronic Hamiltonian in a magnetic field: H = H BLz + Bsz B2 (x 2 + y 2 ) linear and quadratic B terms The orbitalangular momentum operator is imaginary and gaugeorigin dependent L = i (r O), A O = 1 B (r O) 2 we must optimize a complex wave function and also ensure gaugeorigin invariance Gaugeorigin invariance is ensured by using London atomic orbitals (LAOs) ω lm (r K, B) = exp [ 1 2 ib (O K) r] χ lm (r K ) special integral code needed London orbitals are necessary to ensure size extensivity and correct dissociation H 2 dissociation FCI/unaugccpVTZ B 2.5 diamagnetic system full lines: with LAOs dashed lines: AOs with midbond gauge origin Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields CCTCC Jackson / 23
5 Introduction the London program and overview We have developed the London code for calculations in finite magnetic fields complex wave functions and London atomic orbitals Hartree Fock theory (RHF, UHF, GHF), FCI theory, and Kohn Sham theory energy, gradients, excitation energies London atomic orbitals require a generalized integral code 1 F n(z) = exp( zt 2 )t 2n dt complex argument z C++ code written by Erik Tellgren, Kai Lange, and Alessandro Soncini C 2 is a large system Overview: closedshell paramagnetic molecules: transition to diamagnetism helium atom in strong fields: atomic distortion and electron correlation H 2 and He 2 in strong magnetic fields: bonding, structure and orientation molecular structure in strong magnetic fields conclusion Previous work in this area: much work has been done on small atoms FCI on twoelectron molecules H 2 and H but without London orbitals Nakatsuji s free complement method no general molecular code with London orbitals Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields CCTCC Jackson / 23
6 Closedshell paramagnetic molecules molecular diamagnetism and paramagnetism.1 The Hamiltonian has paramagnetic and diamagnetic parts: a) b) x H = H BLz + Bsz B2 (x y 2 ) linear and quadratic B terms.6 Most closedshell molecules are diamagnetic their energy increases in an applied magnetic field.4 induced currents oppose the field according to Lenz s law Some closedshell systems are paramagnetic their energy decreases in a magnetic field relaxation of the wave function lowers the energy RHF calculations of the fielddependence for two closedshell systems: a) c) b) x d) left: benzene: diamagnetic dependence on an outofplane field, χ < right: BH: c) paramagnetic dependence on a perpendicular d) field, χ > Helgaker et al. (CTCC, University.2 of Oslo) Molecules in strong magnetic.2 fields CCTCC Jackson / 23
7 Closedshell paramagnetic molecules diamagnetic transition at stabilizing field strength B c However, all systems become diamagnetic in sufficiently strong fields: The transition occurs at a characteristic stabilizing critical field strength B c B c.1 for C 2 (ring conformation) above 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 Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields CCTCC Jackson / 23
8 Closedshell paramagnetic molecules paramagnetism and double minimum explained Ground and (singlet) excited states of BH along the z axis zz = 1sB 2 2σ2 BH 2p2 z, zx = 1sB 2 2σ2 BH 2pz 2px, zy = 1s2 B 2σ2 BH2pz 2py All expectation values increase quadratically in a perpendicular field in the y direction: H BLy B2 (x 2 + z 2 ) = E x 2 + z 2 B 2 = E 1 2 χ B 2 The zz ground state is coupled to the lowlying zx excited state by this field: zz H BLy B2 (x 2 + z 2 ) xz = 1 zz Ly xz B A paramagnetic groundstate with a double minimum is generated by strong coupling Tellgren et al., PCCP 11, 5489 (29) Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields CCTCC Jackson / 23
9 Closedshell paramagnetic molecules induced electron rotation The magnetic field induces a rotation of the electrons about the field direction: the amount of rotation is the expectation value of the kinetic angularmomentum operator Λ = 2E (B), Λ = r π, π = p + A Paramagnetic closedshell molecules (here BH): E(B x ) E(B x ) Energy!25.11!25.12!25.13!25.14!25.15 L (B ) L x (B x ) x x Angular momentum !25.16!.5! no rotation at B = paramagnetic rotation against Orbital the energies field reduces the energy HOMO!LUMO for < B < gapb c maximum paramagnetic rotation at the inflection point.5 E (B) = no rotation and lowest.1 energy at B = B c.48 diamagnetic rotation with the field increases the energy for B > B c.46 Diamagnetic closedshell molecules:!.1 LUMO.44 diamagnetic rotation always increases HOMO the energy according to Lenz s law!(b x ) L x / r! C nuc L x / r! C nuc Nuclear shielding inte Boron Hydrogen!.2! Singlet excitation en !.2.42 Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields CCTCC.2 Jackson / 23! gap (B x ) "(B x )
10 The helium atom total energy and orbital energies 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 (right) 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 3 Helium atom, RHF/augccpVTZ 1 Helium atom, RHF/augccpVTZ 2 8 Energy, E [Hartree] 1 1 Orbital energy, [Hartree] Field, B [au] Field, B [au] Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields CCTCC Jackson / 23
11 The helium atom natural occupation numbers and electron correlation The FCI occupation numbers approach 2 and strong fields diminishing 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 CCTCC Jackson / 23
12 The helium atom atomic size and atomic distortion Atoms become squeezed and distorted in magnetic fields Helium 1s 2 1 S (left) becomes squeezed and prolate Helium 1s2p 3 P (right) is oblate in weak fields and prolate in strong fields transversal size proportional to 1/ B, longitudinal size proportional to 1/ log B Atomic distortion affects chemical bonding which orientation will be favored? Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields CCTCC Jackson / 23
13 The H 2 molecule potentialenergy curves of the 1 Σ + g (1σ2 g ) and 3 Σ + u (1σg1σ u ) states (M S = ) FCI/unaugccpVTZ curves in parallel (full) and perpendicular (dashed) orientations 1..5 B B B B The energy increases diamagnetically for both states in all orientations The singlet triplet separation is greatest in the parallel orientation (stronger overlap) the singlet state favors a parallel orientation (red full line) the triplet state favors a perpendicular orientation (blue dashed line) and becomes bound parallel orientation studied by Schmelcher et al., PRA 61, (2); 64, 2341 (21) Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields CCTCC Jackson / 23
14 The H2 molecule lowest singlet and triplet potentialenergy surfaces E (R, Θ) I Polar plots of the singlet (left) and triplet (right) energy E (R, Θ) at B = 1 a.u I Bond distance Re (pm), orientation Θe ( ), diss. energy D, and rot. barrier E (kj/mol) B. 1. Re Helgaker et al. (CTCC, University of Oslo) θe singlet D E 83 Re 136 Molecules in strong magnetic fields triplet θe D 9 12 E 12 CCTCC Jackson / 23
15 The H 2 molecule Zeeman splitting of the lowest triplet state The spin Zeeman interaction contributes BM s to the energy, splitting the triplet lowest singlet (red) and triplet (blue) energy of H 2: B..5 B B B The ββ triplet component becomes the ground state at B.25 a.u. eventually, all triplet components will be pushed up in energy diamagnetically Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields CCTCC Jackson / 23
16 The H 2 molecule evolution of lowest three triplet states We often observe a complicated evolution of electronic states a weakly bound 3 Σ + u (1σg1σ u ) ground state in intermediate fields a covalently bound 3 Π u(1σ g2π u) ground state in strong fields E (Ha) R (bohr) Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields CCTCC Jackson / 23
17 The H 2 molecule electron rotation and correlation The field induces a rotation of the electrons Λ z about the molecular axis increased rotation increases kinetic energy, raising the energy concerted rotation reduces the chance of near encounters natural occupation numbers indicate reduced importance of dynamical correlation Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields CCTCC Jackson / 23
18 The helium dimer the 1 Σ + g (1σg1σ 2 u 2 ) singlet state The fieldfree He 2 is bound by dispersion in the ground state our FCI/unaugccpVTZ calculations give D =.8 kj/mol at R e = 33 pm (too short) In a magnetic field, He 2 becomes smaller and more strongly bound this effect is particularly strong in the perpendicular orientation (dashed lines) for B = 2.5, D = 31 kj/mol at R e = 94 pm and Θ e = B B B 1. B.5 B. Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields CCTCC Jackson / 23
19 The helium dimer the 3 Σ + u (1σ2 g 1σ u 2σg) triplet state He 2 in the covalently bound triplet state becomes further stabilized in a magnetic field D = 178 kj/mol at R e = 14 pm at B = D = 655 kj/mol at R e = 8 pm at B = 2.25 (parallel orientation) D = 379 kj/mol at R e = 72 pm at B = 2.25 (perpendicular orientation) The molecule begins a transition to diamagnetism at B B B B B 2.5 In strong magnetic fields, He 2 has a quintet ground state Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields CCTCC Jackson / 23
20 The helium dimer the lowest singlet, triplet and quintet states We have considered the lowest singlet, triplet and quintet states in the absence of a field, only the triplet state is covalently bound however, in sufficiently strong fields, the ground state is a bound quintet state at B = 2.5, it has a perpendicular minimum of D = 1 kj/mol at R e = 118 pm Below, we have plotted singlet (left), triplet (middle) and quintet (right) states bound states contract with shrinking size of the atoms and become more strongly bound noncovalently bound molecules energy minimum in perpendicular orientation In strong fields, anisotropic Gaussians are needed for a compact description for B 1 without such basis sets, calculations become speculative Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields CCTCC Jackson / 23
21 Molecular structure FCI calculations on the H + 3 ion We are investigating H + 3 in a magnetic field Warke and Dutta, PRA 16, 1747 (1977) equilateral triangle for B < 1; linear chain for B > 1 ground state singlet for B <.5; triplet for B >.5 the triplet state does not become bound until fields B > 1 Potential energy curves for lowest singlet (left) and triplet (right) states electronic energy as function of bond distances for equilateral triangles E (Ha) E (Ha) R (bohr) R (bohr). Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields CCTCC Jackson / 23
22 Molecular structure Hartree Fock calculations on larger molecules Ammonia in a field of.15 along the symmetry axis 1.15 NH 3, HF/6 31G** 18.3 NH 3, HF/6 31G** d(n H) [pm] H N H angle [degrees] B [au] it shrinks: bonds contract by.3 pm it becomes more planar from shrinking lone pair? B [au] Benzene in a field of.16 along two CC bonds, it becomes 6.1 pm narrower and 3.5 pm longer in the field direction agrees with perturbational estimates by Caputo and Lazzeretti, IJQC 111, 772 (211) C 6 H 6, HF/6 31G* d(c C) [pm] C C other (c) C C B (d) B [au] Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields CCTCC Jackson / 23
23 Conclusions Summary and outlook We have developed the LONDON program for molecules in magnetic field We have studied closedshell 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 He, H 2 and He 2 in magnetic fields atoms and molecules shrink molecules are stabilized by magnetic fields preferred orientation in the field varies from system to system We have studied molecular structure in magnetic fields bond distances are typically shortened in magnetic fields We have not studied molecules in the Landau regime B 1 a.u. all molecules become linear with singly occupied orbitals of β spin An important goal is to study the universal density functional in magnetic fields ( ) F [ρ, j] = sup u,a E[u, A] ρ(r)u(r) dr j(r) A(r) dr Support: The Norwegian Research Council and the European Research Council Postdoc position is available in my group Helgaker et al. (CTCC, University of Oslo) Conclusions CCTCC Jackson / 23
Molecules 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationImporting abinitio theory into DFT: Some applications of the Lieb variation principle
Importing abinitio 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 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 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 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 kineticenergy operator? (a) The free particle
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 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 nonbonding, and say whether the symmetry of the orbital is σ or π. Sketch a molecular orbital diagram
More informationOrigin of the first Hund rule in Helike atoms and 2electron quantum dots
in Helike atoms and 2electron 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 informationAbinitio studies of the adiabatic connection in densityfunctional theory
Abinitio studies of the adiabatic connection in densityfunctional theory Trygve Helgaker, Andy Teale, and Sonia Coriani Centre for Theoretical and Computational Chemistry (CTCC), Department of Chemistry,
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 openshell systems, since they rely on transitions
More informationThe Role of the Hohenberg Kohn Theorem in DensityFunctional Theory
The Role of the Hohenberg Kohn Theorem in DensityFunctional Theory T. Helgaker, U. E. Ekström, S. Kvaal, E. Sagvolden, A. M. Teale,, E. Tellgren Centre for Theoretical and Computational Chemistry (CTCC),
More informationElectronic structure theory: Fundamentals to frontiers. 1. HartreeFock theory
Electronic structure theory: Fundamentals to frontiers. 1. HartreeFock theory MARTIN HEADGORDON, Department of Chemistry, University of California, and Chemical Sciences Division, Lawrence Berkeley National
More informationBonding and Physical Properties The Molecular Orbital Theory
Bonding and Physical Properties The Molecular Orbital Theory Ø Developed by F. Hund and R. S. Mulliken in 1932 Ø Diagram of molecular energy levels Ø Magnetic and spectral properties Paramagnetic vs. Diamagnetic
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 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 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 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 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 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 informationComputational Material Science Part II. Ito Chao ( ) Institute of Chemistry Academia Sinica
Computational Material Science Part II Ito Chao ( ) Institute of Chemistry Academia Sinica Ab Initio Implementations of HartreeFock Molecular Orbital Theory Fundamental assumption of HF theory: each electron
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 informationHighly accurate quantumchemical calculations
1 Highly accurate quantumchemical calculations T. Helgaker Centre for Theoretical and Computational Chemistry, Department of Chemistry, University of Oslo, Norway A. C. Hennum and T. Ruden, University
More informationBe H. Delocalized Bonding. Localized Bonding. σ 2. σ 1. Two (sp1s) BeH 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 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 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 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 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 informationChapter 9. Chemical Bonding II: Molecular Geometry and Bonding Theories
Chapter 9 Chemical Bonding II: Molecular Geometry and Bonding Theories Topics Molecular Geometry Molecular Geometry and Polarity Valence Bond Theory Hybridization of Atomic Orbitals Hybridization in Molecules
More informationRDCH 702 Lecture 4: Orbitals and energetics
RDCH 702 Lecture 4: Orbitals and energetics Molecular symmetry Bonding and structure Molecular orbital theory Crystal field theory Ligand field theory Provide fundamental understanding of chemistry dictating
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 information2 Electronic structure theory
Electronic structure theory. Generalities.. BornOppenheimer approximation revisited In Sec..3 (lecture 3) the BornOppenheimer approximation was introduced (see also, for instance, [Tannor.]). We are
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 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 kineticenergy operator? (a) The free particle (e) 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 informationChemistry 120A 2nd Midterm. 1. (36 pts) For this question, recall the energy levels of the Hydrogenic Hamiltonian (1electron):
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 informationJack Simons, Henry Eyring Scientist and Professor Chemistry Department University of Utah
1. BornOppenheimer approx. energy surfaces 2. Meanfield (HartreeFock) theory orbitals 3. Pros and cons of HF RHF, UHF 4. Beyond HF why? 5. First, one usually does HFhow? 6. Basis sets and notations
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 informationChem 442 Review for Exam 2. Exact separation of the Hamiltonian of a hydrogenic atom into centerofmass (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 centerofmass (3D) and relative
More informationThe energy level structure of lowdimensional multielectron quantum dots
The energy level structure of lowdimensional multielectron quantum dots Tokuei Sako, a Josef Paldus b and Geerd H. F. Diercksen c a Laboratory of Physics, College of Science and Technology, Nihon University,
More informationValence Bond Theory Considers the interaction of separate atoms brought together as they form a molecule. Lewis structures Resonance considerations
CHEM 511 chapter 2 page 1 of 11 Chapter 2 Molecular Structure and Bonding Read the section on Lewis dot structures, we will not cover this in class. If you have problems, seek out a general chemistry text.
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 informationChapter 21 dblock metal chemistry: coordination complexes
Chapter 21 dblock metal chemistry: coordination complexes Bonding: valence bond, crystal field theory, MO Spectrochemical series Crystal field stabilization energy (CFSE) Electronic Spectra Magnetic Properties
More informationone ν im: transition state saddle point
Hypothetical Potential Energy Surface Ethane conformations HartreeFock theory, basis set stationary points all ν s >0: minimum eclipsed one ν im: transition state saddle point multiple ν im: hilltop 1
More informationQuantum Theory of ManyParticle Systems, Phys. 540
Quantum Theory of ManyParticle Systems, Phys. 540 Questions about organization Second quantization Questions about last class? Comments? Similar strategy Nparticles Consider Twobody operators in Fock
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 informationMolecularOrbital Theory
Prof. Dr. I. Nasser atomic and molecular physics 551 (T11) April 18, 01 MolecularOrbital Theory You have to explain the following statements: 1 Helium is monatomic gas.  Oxygen molecule has a permanent
More informationChapter 10 Chemical Bonding II
Chapter 10 Chemical Bonding II Valence Bond Theory Valence Bond Theory: A quantum mechanical model which shows how electron pairs are shared in a covalent bond. Bond forms between two atoms when the following
More informationCHEM1901/ J5 June 2013
CHEM1901/3 2013J5 June 2013 Oxygen exists in the troposphere as a diatomic molecule. 4 (a) Using arrows to indicate relative electron spin, fill the leftmost valence orbital energy diagram for O 2,
More informationLecture 4: Band theory
Lecture 4: Band theory Very short introduction to modern computational solid state chemistry Band theory of solids Molecules vs. solids Band structures Analysis of chemical bonding in Reciprocal space
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 informationOrbitals and energetics
Orbitals and energetics Bonding and structure Molecular orbital theory Crystal field theory Ligand field theory Provide fundamental understanding of chemistry dictating radionuclide complexes Structure
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 informationElectron Correlation
Series in Modern Condensed Matter Physics Vol. 5 Lecture Notes an Electron Correlation and Magnetism Patrik Fazekas Research Institute for Solid State Physics & Optics, Budapest lb World Scientific h Singapore
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 Blackbody radiation 1 Heat capacities 2 The
More informationCHAPTER 8. EXPLORING THE EFFECTS OF STRUCTURAL INSTABILITIES AND OF TRIPLET INSTABILITY ON THE MI PHASE TRANSITION IN (EDOTTF)2PF6.
CHAPTER 8. EXPLORING THE EFFECTS OF STRUCTURAL INSTABILITIES AND OF TRIPLET INSTABILITY ON THE MI PHASE TRANSITION IN (EDOTTF)2PF6. The thermal metalinsulator phase transition in (EDOTTF)2PF6 is attributed
More informationH B. θ = 90 o. Lecture notes Part 4: SpinSpin Coupling. θ θ
Lecture notes Part 4: SpinSpin Coupling F. olger Försterling October 4, 2011 So far, spins were regarded spins isolated from each other. owever, the magnetic moment of nuclear spins also have effect on
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 informationSpin Interactions. Giuseppe Pileio 24/10/2006
Spin Interactions Giuseppe Pileio 24/10/2006 Magnetic moment µ = " I ˆ µ = " h I(I +1) " = g# h Spin interactions overview Zeeman Interaction Zeeman interaction Interaction with the static magnetic field
More informationExcited States Calculations for Protonated PAHs
52 Chapter 3 Excited States Calculations for Protonated PAHs 3.1 Introduction Protonated PAHs are closed shell ions. Their electronic structure should therefore be similar to that of neutral PAHs, but
More informationOVERVIEW OF QUANTUM CHEMISTRY METHODS
OVERVIEW OF QUANTUM CHEMISTRY METHODS Outline I Generalities Correlation, basis sets Spin II Wavefunction methods HartreeFock Configuration interaction Coupled cluster Perturbative methods III Density
More information3.15 Nuclear Magnetic Resonance Spectroscopy, NMR
3.15 Nuclear Magnetic Resonance Spectroscopy, NMR What is Nuclear Magnetic Resonance  NMR Developed by chemists and physicists together it works by the interaction of magnetic properties of certain nuclei
More informationWe can keep track of the mixing of the 2s and 2p orbitals in beryllium as follows:
We can keep track of the mixing of the 2s and 2p orbitals in beryllium as follows: The beryllium sp orbitals overlap with hydrogen Is orbitals (the hydrogen's electrons are shown in the above orbital diagram
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? BeerLambert
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.condmat.de/events/correl Magnetism is Quantum Mechanical QUANTUM MECHANICS THE KEY TO UNDERSTANDING
More informationModels for TimeDependent Phenomena
Models for TimeDependent Phenomena I. Phenomena in lasermatter interaction: atoms II. Phenomena in lasermatter interaction: molecules III. Model systems and TDDFT Manfred Lein p.1 Outline Phenomena
More informationGeneral Physical Chemistry II
General Physical Chemistry II Lecture 13 Aleksey Kocherzhenko October 16, 2014" Last time " The Hückel method" Ø Used to study π systems of conjugated molecules" Ø π orbitals are treated separately from
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 informationMD simulation: output
Properties MD simulation: output Trajectory of atoms positions: e. g. diffusion, mass transport velocities: e. g. vv autocorrelation spectrum Energies temperature displacement fluctuations Mean square
More informationCHEMUA 127: Advanced General Chemistry I
CHEMUA 7: Advanced General Chemistry I I. LINEAR COMBINATION OF ATOMIC ORBITALS Linear combination of atomic orbitals (LCAO) is a simple method of quantum chemistry that yields a qualitative picture of
More information5.80 SmallMolecule Spectroscopy and Dynamics
MIT OpenCourseWare http://ocw.mit.edu 5.80 SmallMolecule Spectroscopy and Dynamics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 5.80 Lecture
More informationDivergence in Møller Plesset theory: A simple explanation based on a twostate model
JOURNAL OF CHEMICAL PHYSICS VOLUME 112, NUMBER 22 8 JUNE 2000 Divergence in Møller Plesset theory: A simple explanation based on a twostate model Jeppe Olsen and Poul Jørgensen a) Department of Chemistry,
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 informationStudy of Ozone in Tribhuvan University, Kathmandu, Nepal. Prof. S. Gurung Central Department of Physics, Tribhuvan University, Kathmandu, Nepal
Study of Ozone in Tribhuvan University, Kathmandu, Nepal Prof. S. Gurung Central Department of Physics, Tribhuvan University, Kathmandu, Nepal 1 Country of the Mt Everest 2 View of the Mt Everest 3 4 5
More informationA Rigorous Introduction to Molecular Orbital Theory and its Applications in Chemistry. Zachary Chin, Alex Li, Alex Liu
A Rigorous Introduction to Molecular Orbital Theory and its Applications in Chemistry Zachary Chin, Alex Li, Alex Liu Quantum Mechanics Atomic Orbitals and Early Bonding Theory Quantum Numbers n: principal
More 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 spinhalf 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 informationAnswers Quantum Chemistry NWIMOL406 G. C. Groenenboom and G. A. de Wijs, HG00.307, 8:3011:30, 21 jan 2014
Answers Quantum Chemistry NWIMOL406 G. C. Groenenboom and G. A. de Wijs, HG00.307, 8:3011:30, 21 jan 2014 Question 1: Basis sets Consider the split valence SV321G one electron basis set for formaldehyde
More informationChapter 9  Covalent Bonding: Orbitals
Chapter 9  Covalent Bonding: Orbitals 9.1 Hybridization and the Localized Electron Model A. Hybridization 1. The mixing of two or more atomic orbitals of similar energies on the same atom to produce new
More informationChapter 9. Molecular Geometry and Bonding Theories
Chapter 9. Molecular Geometry and Bonding Theories PART I Molecular Shapes Lewis structures give atomic connectivity: they tell us which atoms are physically connected to which atoms. The shape of a molecule
More informationProblem Set 2 Due Thursday, October 1, & & & & # % (b) Construct a representation using five d orbitals that sit on the origin as a basis:
Problem Set 2 Due Thursday, October 1, 29 Problems from Cotton: Chapter 4: 4.6, 4.7; Chapter 6: 6.2, 6.4, 6.5 Additional problems: (1) Consider the D 3h point group and use a coordinate system wherein
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 1electron basis. The twoelectron basis
More informationLecture 5: More about one Final words about the HartreeFock theory. First step above it by the MøllerPlesset perturbation theory.
Lecture 5: More about one determinant wave functions Final words about the HartreeFock theory. First step above it by the MøllerPlesset perturbation theory. Items from Lecture 4 Could the Koopmans theorem
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 information