Molecules in strong magnetic fields

Size: px
Start display at page:

Download "Molecules in strong magnetic fields"

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 Trends in Quantum Chemistry Department of Chemistry, University of Aarhus, Århus, Denmark, December 12 14, 2008 Trygve Helgaker, Kai Lange, Alessandro Soncini, and Erik Tellgren Molecules (CTCC, in strong University magnetic of Oslo) fields Molecules in Strong Magnetic Fields 1 / 29

2 Outline 1 Overview 2 Molecular magnetism 3 London orbitals 4 The LONDON program 5 The field-dependence of the energy of closed-shell molecules 6 An analytical model for paramagnetic closed-shell molecules 7 Molecular properties in strong magnetic fields 8 Potential-energy surfaces in strong magnetic fields 9 Conclusions Helgaker et al. (CTCC, University of Oslo) Overview Molecules in Strong Magnetic Fields 2 / 29

3 Outline 1 Overview 2 Molecular magnetism 3 London orbitals 4 The LONDON program 5 The field-dependence of the energy of closed-shell molecules 6 An analytical model for paramagnetic closed-shell molecules 7 Molecular properties in strong magnetic fields 8 Potential-energy surfaces in strong magnetic fields 9 Conclusions Helgaker et al. (CTCC, University of Oslo) Overview Molecules in Strong Magnetic Fields 2 / 29

4 Outline 1 Overview 2 Molecular magnetism 3 London orbitals 4 The LONDON program 5 The field-dependence of the energy of closed-shell molecules 6 An analytical model for paramagnetic closed-shell molecules 7 Molecular properties in strong magnetic fields 8 Potential-energy surfaces in strong magnetic fields 9 Conclusions Helgaker et al. (CTCC, University of Oslo) Overview Molecules in Strong Magnetic Fields 2 / 29

5 Outline 1 Overview 2 Molecular magnetism 3 London orbitals 4 The LONDON program 5 The field-dependence of the energy of closed-shell molecules 6 An analytical model for paramagnetic closed-shell molecules 7 Molecular properties in strong magnetic fields 8 Potential-energy surfaces in strong magnetic fields 9 Conclusions Helgaker et al. (CTCC, University of Oslo) Overview Molecules in Strong Magnetic Fields 2 / 29

6 Outline 1 Overview 2 Molecular magnetism 3 London orbitals 4 The LONDON program 5 The field-dependence of the energy of closed-shell molecules 6 An analytical model for paramagnetic closed-shell molecules 7 Molecular properties in strong magnetic fields 8 Potential-energy surfaces in strong magnetic fields 9 Conclusions Helgaker et al. (CTCC, University of Oslo) Overview Molecules in Strong Magnetic Fields 2 / 29

7 Outline 1 Overview 2 Molecular magnetism 3 London orbitals 4 The LONDON program 5 The field-dependence of the energy of closed-shell molecules 6 An analytical model for paramagnetic closed-shell molecules 7 Molecular properties in strong magnetic fields 8 Potential-energy surfaces in strong magnetic fields 9 Conclusions Helgaker et al. (CTCC, University of Oslo) Overview Molecules in Strong Magnetic Fields 2 / 29

8 Outline 1 Overview 2 Molecular magnetism 3 London orbitals 4 The LONDON program 5 The field-dependence of the energy of closed-shell molecules 6 An analytical model for paramagnetic closed-shell molecules 7 Molecular properties in strong magnetic fields 8 Potential-energy surfaces in strong magnetic fields 9 Conclusions Helgaker et al. (CTCC, University of Oslo) Overview Molecules in Strong Magnetic Fields 2 / 29

9 Outline 1 Overview 2 Molecular magnetism 3 London orbitals 4 The LONDON program 5 The field-dependence of the energy of closed-shell molecules 6 An analytical model for paramagnetic closed-shell molecules 7 Molecular properties in strong magnetic fields 8 Potential-energy surfaces in strong magnetic fields 9 Conclusions Helgaker et al. (CTCC, University of Oslo) Overview Molecules in Strong Magnetic Fields 2 / 29

10 Outline 1 Overview 2 Molecular magnetism 3 London orbitals 4 The LONDON program 5 The field-dependence of the energy of closed-shell molecules 6 An analytical model for paramagnetic closed-shell molecules 7 Molecular properties in strong magnetic fields 8 Potential-energy surfaces in strong magnetic fields 9 Conclusions Helgaker et al. (CTCC, University of Oslo) Overview Molecules in Strong Magnetic Fields 2 / 29

11 Molecular para- and diamagnetism 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 permanent magnetic moments (unpaired spins) the molecule reorients itself and moves into the field temperature dependent Closed-shell molecules are nearly all diamagnetic no permanent magnetic moment induced magnetic dipole induced currents oppose the external field (Lenz law) temperature independent much weaker than open-shell paramagnetism Some closed-shell molecules are paramagnetic temperature independent much weaker than the temperature-dependent open-shell paramagnetism first discovered for MnO 4 (1914) much studied: BH and CH + Helgaker et al. (CTCC, University of Oslo) Molecular magnetism Molecules in Strong Magnetic Fields 3 / 29

12 Electronic Hamiltonian in an external magnetic field B The external magnetic field is represented by a vector potential B(r) = A(r), A(r) = 1 2 B r The non-relativistic electronic Hamiltonian (atomic units) H = H 0 + A (r) p + B (r) s A (r)2 = H B L + B s + 1 (B r) (B r) 8 H 0 is the field-free non-relativistic Hamiltonian p = i is the generalized momentum operator L = r p is the orbital angular momentum operator s = σ/2 is the spin angular momemtum operator The Hamiltonian depends both linearly and quadratically on the field the linear term may lower or raise the energy the quadratic term will always raise the energy We can expect a rather complicated dependence of the energy on the field for large B, we expect the second-order term to dominate but what happens for small and intermediate B? we consider only closed-shell states Helgaker et al. (CTCC, University of Oslo) Molecular magnetism Molecules in Strong Magnetic Fields 4 / 29

13 Perturbation theory Magnetic interactions are usually studied by perturbation theory E(B) = E(0) X µαb X 1 α 2 χ αβb αβ αb β + The first-order term represents interaction with the permanent magnetic moment µ = L + s temperature-dependent paramagnetism (vanishes for closed-shell systems) The second-order term represents interaction with the induced dipole moment D E χ = 1 X 0 4 rr T (r T r)i L n n LT 0 2 n E n E 0 the Langevin term arises from precessional motion of the electrons temperature-independent diamagnetism the sum-over-states term arises from orbital unquenching temperature-independent paramagnetism Helgaker et al. (CTCC, University of Oslo) Molecular magnetism Molecules in Strong Magnetic Fields 5 / 29

14 The need for a nonperturbative treatment It is possible to go to higher orders by including hypermagnetizabilities E(B) = E(0) 1 2 X αβb αβ αb β 1 24 X αβγδb αβγδ αb β B γb δ +... However, the field dependence on the energy can be very complicated the energy of C 20 (ring conformation) as a function of B (atomic units) Taylor expansions useless for strong fields How does this complicated field dependence arise? Helgaker et al. (CTCC, University of Oslo) Molecular magnetism Molecules in Strong Magnetic Fields 6 / 29

15 Nonperturbative treatment of molecules in strong fields Most studies of molecular magnetism are based on low-order expansions magnetizabilities and hypermagnetizabilities such studies provide much useful information However, expansions around zero field have many limitations the behaviour in strong fields cannot be studied many phenomena may remain unnoticed We have undertaken a nonperturbative study of molecules in strong fields this required the development of a new code complex orbitals and complex wave functions London atomic orbitals to remove gauge-origin dependence In the remainder of the talk, we will discuss the following points the need for London orbitals London-orbital integral evaluation in strong fields the energy in strong magnetic fields two-level analytic model for strong fields molecular properties in strong fields Helgaker et al. (CTCC, University of Oslo) Molecular magnetism Molecules in Strong Magnetic Fields 7 / 29

16 Gauge-origin dependence and London orbitals A uniform external field may be represented by any potential of the form A O (r) = 1 B (r O) 2 the vector potential vanishes as the gauge origin O the position of the origin is not unique In exact theory, this non-uniqueness does not matter a change in the origin represents a gauge transformation the exact wave function undergoes a corresponding gauge transformation Ψ O = exp[ia K (O) r]ψ K all choices of gauge origin O then lead to the same (observable) results In approximate calculations, our results in general do depend on the origin approx. wave functions are not sufficiently flexible to be properly gauge transformed This problem is solved by using gauge transforming the individual atomic orbitals (AOs) each AO has a unique best or favoured gauge origin: its atomic center ω lm = exp[ia K (O) r]χ lm (r K ) gauge transformation from AO to global origin each AO behaves as if the gauge origin were at its center the use of such London orbitals removes the gauge-origin dependence Helgaker et al. (CTCC, University of Oslo) London orbitals Molecules in Strong Magnetic Fields 8 / 29

17 The efficacy of London orbitals London orbitals are AOs with an attached complex phase factor ω lm = exp[ia K (O) r]χ lm (r K ) gauge factor removes gauge-origin dependence of magnetic properties London orbitals are correct to first-order in the external magnetic field for this reason, basis-set convergence is usually improved Calculations on the water molecule χ zz X zzzz Lon CM H Lon CM H STO-3G cc-pvdz cc-pvtz aug-cc-pvdz London orbitals greatly improve convergence of magnetizabilities they are less efficacious for hypermagnetizabilities Helgaker et al. (CTCC, University of Oslo) London orbitals Molecules in Strong Magnetic Fields 9 / 29

18 Hybrid plane-wave Gaussians (PWG) orbitals London AOs are in fact hybrid plane-wave Gaussian (PWG) orbitals ωκ,c(r) = exp(iκ r) S lm (r) exp( ara 2 {z } {z } ) plane wave solid-harmonic Gaussian the wave vector κ is the AO-centered vector potential at the gauge origin More generally, PWGs have several uses: mixed basis for periodic boundary conditions and scattering studies gauge-origin independent magnetic properties at zero field Requires a generalization of GTO integral-evaluation techniques at zero field, complex algebra may be avoided at finite field, complex algebra cannot be avoided We have developed and implemented a McMurchie Davidson PWG scheme Tellgren et al., JCP 129, (2008) previous work: M. Tachikawa and M. Shiga, Phys. Rev. E 64: (2001) Helgaker et al. (CTCC, University of Oslo) London orbitals Molecules in Strong Magnetic Fields 10 / 29

19 PWG product rule and overlap integrals In many respects, a straightforward generalization of GTO integral evaluation The Gaussian product rule still holds ab (r) = exp(iκ r) exp( ara) 2 exp(iλ r) exp( br 2 }{{}}{{ B) } PWG at A PWG at B Ω κλ = exp( ab a+b R2 AB) } {{ } prefactor exp[ i(κ λ) r] exp[ (a + b)rp] 2 }{{} PWG at P = (aa + bb)/(a + b) Integration over all space yields ab (r) dr = exp [ (κ λ)2 Ω κλ 4(a+b) ] + i(κ λ) P } {{ } phase-factor contribution ( ) π 3/2 exp ab a+b R2 AB (a + b) }{{ 3/2 } standard Gaussian overlap Helgaker et al. (CTCC, University of Oslo) London orbitals Molecules in Strong Magnetic Fields 11 / 29

20 PWG two-electron integrals As for standard Gaussians, Coulomb integrals reduce to the Boys function exp(iκ r1 ) exp( pr1p 2 J = ) exp(iλ r 2) exp( qr2q 2 ) dr 1 dr 2 r 12 [ ] = exp κ2 4p λ2 4q iκ P iλ Q 2π 5/2 [ pq p + q F pq 0 p+q (P Q ) 2] }{{} P = P iκ/2p, Q = Q iλ/2q The Boys function is given by F 0 (x) = 1 0 exp( xt 2 ) dt complex argument x evaluated in the usual manner by expansion and recursion For functions of higher angular momentum, recurrence relations are used some translational symmetry lost more complicated recurrence relations Helgaker et al. (CTCC, University of Oslo) London orbitals Molecules in Strong Magnetic Fields 12 / 29

21 The LONDON program an ab initio program for finite-field calculations with London orbitals some features of the present code Hartree Fock theory implemented recent Kohn Sham implementation many first-order properties (x Cx) m ( / x) n excitation energies using response theory some restrictions of the present code only uncontracted Cartesian basis functions closed-shell wave functions only code written by Erik Tellgren and Alessandro Soncini mostly C++, some Fortran 77 modular but not highly optimized yet C20 is a large system Helgaker et al. (CTCC, University of Oslo) The LONDON program Molecules in Strong Magnetic Fields 13 / 29

22 The dependence of total energy on the magnetic field RHF calculations of B dependence for different systems a) 0.1 b) x c) d) a benzene (aug-cc-pvdz): typical case of diamagnetic quadratic dependence b cyclobutadiene (aug-cc-pvdz): non-quadratic dependence on an out-of-plane field c BH (aug-cc-pvtz): paramagnetic dependence for a perpendicular field d BH (aug-cc-pvtz): larger range of the perpendicular field reveals nonperturbative behaviour Helgaker et al. (CTCC, University of Oslo) The field-dependence of closed-shell energies Molecules in Strong Magnetic Fields 14 / 29

23 The field dependence of paramagnetic molecules We have studied the field-dependence of a number of molecules O Mn B H C H O O O C 4H 4 C 8H 8 C 12H C 16H 10 C20H For all these systems, the field dependence takes on a sombrero shape we will give some examples we will explain this behaviour by means of simple analytical model Helgaker et al. (CTCC, University of Oslo) The field-dependence of closed-shell energies Molecules in Strong Magnetic Fields 15 / 29

24 W!W0!au" BH a) Small paramagnetic molecules Three closed-shell paramagnetic molecules: BH, CH +, and MnO!0.02! field applied perpendicularly!0.03 for BH and CH !0.02 B!au"!0.04!0.01 STO!3G, Bc " 0.24 DZ, Bc " 0.22!0.06 aug!dz, Bc " 0.23 aug-cc-pvdz on all atoms!0.04 except Wachters-f for Mn!0.1! STO3!G, Bc " 0.43 DZ, Bc " 0.44 aug!dz, Bc " 0.45 a) W!W0!au" BH b) B!au" !0.01 STO!3G, Bc " 0.24 DZ, Bc " 0.22 aug!dz, Bc " 0.23!0.02 W!W0!au" CH # !0.02 STO3!G, Bc " 0.43!0.04 DZ, Bc " 0.44!0.06 aug!dz, Bc " 0.45 B!au" c) W!W0!au" MnO4!!0.1! B!au" STO!3G, Bc " 0.45 Wachters, Bc " 0.50!0.03!0.08!0.3!0.04!0.1!0.4!0.12 W!W0!au" CH b) # W!W0!au" MnO4! B!au" c) Energy minimum occurs at a characteristic B critical!au" field B!0.02!0.1 STO!3G, Bc " 0.45 c Wachters, Bc " 0.50 STO3!G, Bc " 0.43!0.04!0.2 DZ, Bc " 0.44!0.06 Bc = aug!dz, a.u. Bc " 0.45 for!0.3 these small systems!0.08 strongest fields attainable:! T ( a.u.)!0.1 we may in principle separate molecules by applying a field gradient!0.12 c) W!W0!au" MnO4!!0.1! B!au" STO!3G, Bc " 0.45 Wachters, Bc " 0.50!0.3 Helgaker et al. (CTCC, University of Oslo) The field-dependence of closed-shell energies Molecules in Strong Magnetic Fields 16 / 29

25 !31G cc!pvdz! Antiaromatic closed-shell [4n]-carbocycles 0.015! !0.004 Their linear response is characterized by strong ring currents cyclobutadiene C4H 4, cyclo-octatetraene C 8H 8 B!au" and [12]-annulene C H 12 magnetic field along principal axis c) W!W0!au" C4H4: total energy W!W0!au" C8H8: total energy W!W0!au" C12H12: total energy STO!3G !31G STO!3G, STO!3G, Bc " Bc " cc!pvdz 6!31G, 6!31G, Bc " Bc " cc!pvdz, Bc " cc!pvdz, Bc " !0.005 d) B!au"! !0.002!0.001 B!au" The critical field is now one order of magnitude smaller Bc! " for C 8H 8; B c for C 12H 12 the critical field decreases with increasing size of the system Bc should vary as the inverse of the area of the molecule we estimate that Bc should be observable for C 72H 72 Helgaker et al. (CTCC, University of Oslo) The field-dependence of closed-shell energies Molecules in Strong Magnetic Fields 17 / 29

26 BH energy in a perpendicular magnetic field Polynomial fits to the BH energy in perpendicular magnetic field data points degree 6 degree 10 degree HF/aug-cc-pVDZ level of theory only even-order terms are included by symmetry expansions of order 14 or greater are needed Helgaker et al. (CTCC, University of Oslo) The field-dependence of closed-shell energies Molecules in Strong Magnetic Fields 18 / 29

27 Analytical model for the diamagnetic transition Molecular orbitals relevant for BH: 1s B, 2σ BH, 2p x, 2p y, 2p z (molecule along z axis) Ground and excited states: 0 = 1s 2 B2σ 2 BH2p 2 z, x = 1s 2 B2σ 2 BH2p z 2p x, y = 1s 2 B2σ 2 BH2p z 2p y Let us apply a perpendicular magnetic field in the y direction: H B = 1 2 BL y B2 (x 2 + z 2 ) This leads to the following Hamiltonian matrix: ( ) ( 0 H 0 0 H y E0 1 H(B) = = 2 χ ) 0B 2 iµb y H 0 y H y iµb E χ 1B 2 where we have introduced χ 0 = x 2 + z 2 0, χ 1 = 1 4 y x 2 + z 2 y, µ = 1 2 i 0 L y y Helgaker et al. (CTCC, University of Oslo) Two-level model for paramagnetic molecules Molecules in Strong Magnetic Fields 19 / 29

28 Energy levels of the two-level model Two-level Hamiltonian: 1 H(B) = 2 χ 0B 2 «iµb iµb 1 2 χ 1B 2 Eigenvalues: W 0/1 (B) = 1 2 (χ 0 + χ 1 )B Plots for different values of µ: « B 2 iµb = iµb B 2 q [2 + (χ 0 χ 1 )B 2 ] 2 + 4µ 2 B 2 uncoupled (0), diamagnetic (0.2), nonmagnetic (0.374), and paramagnetic (0.6) magnetizability: χ 0 + µ 2 /2 = 7, 5, 0, 11 Helgaker et al. (CTCC, University of Oslo) Two-level model for paramagnetic molecules Molecules in Strong Magnetic Fields 20 / 29

29 Two-level model fitted to experimental data The two-level model contains four parameters may be fitted to experimental data provides excellent fits, superior to polynomial fits BH singlet energies (aug!cc!pvdz) Energy E(B x ) g.s. exc!1 exc!2 fitted fitted! Field B x Helgaker et al. (CTCC, University of Oslo) Two-level model for paramagnetic molecules Molecules in Strong Magnetic Fields 21 / 29

30 Two-level model fitted to experimental data The two-level model contains four parameters may be fitted to experimental data %0.04 provides excellent fits, superior to polynomial fits comparisons with 6- and 8-order polynomial fits a) c) b) d) %0.01 %0.02 %0.03 %0.04 %0.02 BH: Polynomial and two%level model fits B CH $ : Polynomial and two%level BHmodel fits B DIAMAGNETIC % %0.04 Bc PARAMAGNETIC !! " 0!"# 2 CH c) CH $ : Polynomial and two%level model fits d) $ B e) f) C 16 H 2% 10 : Polynomial and two%level model fits 0.4 DIAMAGNETIC B %0.04 Helgaker et al. (CTCC, University of Oslo) Two-level model for paramagnetic %0.001molecules 0.3 Molecules in Strong Magnetic Fields 22 / 29 %0.02 %0.03 %0.06 %0.08 %0.1 %0.12

31 C 20 : more structure Helgaker et al. (CTCC, University of Oslo) Molecular properties in strong magnetic fields Molecules in Strong Magnetic Fields 23 / 29

32 Induced magnetic moment and angular momentum The induced magnetic moment M and angular momentum L are related as M = 1 2 L = 1 2 E (B), E = MB Diamagnetic molecules: M is always aligned against the field, increasing the energy Paramagnetic molecules: M first aligns with the field, decreasing the energy M reaches its maximum value at the inflection point E (B) = 0 M then decreases again until it vanishes at B c M then aligns against 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 induced angular momentum at the energy minimum Helgaker et al. (CTCC, University of Oslo) Molecular properties in strong magnetic fields Molecules in Strong Magnetic Fields 24 / 29

33 Oribtal and excitation energies in a magnetic field We have implemented linear response theory in finite magnetic fields BH orbital energies, HOMO LUMO gap and singlet excitations!(b x ) 0.1 0!0.1!0.2!0.3 Orbital energies LUMO HOMO ! gap (B x ) HOMO!LUMO gap "(B x ) Singlet excitation energies the HOMO and LUMO orbital energies decrease and increase, respectively, with B the HOMO LUMO gap opens up with increasing B most excitation energies increase with increasing B Helgaker et al. (CTCC, University of Oslo) Molecular properties in strong magnetic fields Molecules in Strong Magnetic Fields 25 / 29

34 H 2 potential-energy curve in a perpendicular magnetic field The magnetic field changes the shape of the potential-energy curve B 0.45 B 0.30 B B diamagnetic behaviour at all separations most pronounced for atoms (no paramagnetic term) The bond length of H 2 decreases with increasing magnetic field Helgaker et al. (CTCC, University of Oslo) Potential-energy surfaces in strong magnetic fields Molecules in Strong Magnetic Fields 26 / 29

35 F 2 potential-energy curve in a perpendicular magnetic field The magnetic field changes the shape of the potential-energy curve B 0.00 B B 0.05 B 0.10 B diamagnetic behaviour in the molecular limit paramagnetic behaviour in the atomic limit The bond length of F 2 increases with increasing magnetic field Helgaker et al. (CTCC, University of Oslo) Potential-energy surfaces in strong magnetic fields Molecules in Strong Magnetic Fields 27 / 29

36 BH potential-energy curve in a perpendicular magnetic field The magnetic field changes the shape of the potential-energy curve mixed para- and diamagnetic behaviour at all separations mostly diamagnetic in the atomic limit The bond length of BH decreases with increasing magnetic field Helgaker et al. (CTCC, University of Oslo) Potential-energy surfaces in strong magnetic fields Molecules in Strong Magnetic Fields 28 / 29

37 Conclusions We have developed the LONDON program complex orbitals and wave functions restricted Hartree Fock and Kohn Sham theories London atomic orbitals for gauge-origin independence expectation values of one-electron operators linear response theory The LONDON program may be used for finite-difference alternative to analytical derivatives studies of molecules in strong magnetic fields We have studied the behaviour of paramagnetic molecules in strong fields all paramagnetic molecules attain a global minimum at a characteristic field Bc Bc decreases with system size and should be observable for C 72H 72 This behaviour can be understood from a simple two-level model explains the existence of a global minimum at Bc at Bc, the induced angular momentum vanishes Helgaker et al. (CTCC, University of Oslo) Conclusions Molecules in Strong Magnetic Fields 29 / 29

Molecules in strong magnetic fields

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 information

Molecular electronic structure in strong magnetic fields

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 information

Molecular Magnetic Properties

Molecular 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 information

Chemical bonding in strong magnetic fields

Chemical 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 information

Molecular Magnetism. Molecules in an External Magnetic Field. Trygve Helgaker

Molecular 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 information

Diamagnetism and Paramagnetism in Atoms and Molecules

Diamagnetism 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 information

Molecular bonding in strong magnetic fields

Molecular 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 information

Molecular Magnetic Properties

Molecular 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 information

Molecules in strong magnetic fields

Molecules 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 information

Molecules in Magnetic Fields

Molecules 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 information

Molecular Magnetic Properties. The 11th Sostrup Summer School. Quantum Chemistry and Molecular Properties July 4 16, 2010

Molecular 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 information

Molecular Magnetism. Magnetic Resonance Parameters. Trygve Helgaker

Molecular 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 information

Non perturbative properties of

Non 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 information

Molecular Magnetic Properties

Molecular 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 information

Molecular Integral Evaluation

Molecular Integral Evaluation Molecular Integral Evaluation Trygve Helgaker Centre for Theoretical and Computational Chemistry Department of Chemistry, University of Oslo, Norway The 13th Sostrup Summer School Quantum Chemistry and

More information

DFT calculations of NMR indirect spin spin coupling constants

DFT 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 information

Oslo node. Highly accurate calculations benchmarking and extrapolations

Oslo 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 information

Basis sets for electron correlation

Basis 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 information

Molecular Magnetic Properties

Molecular 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 information

Highly accurate quantum-chemical calculations

Highly 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 information

Exercise 1: Structure and dipole moment of a small molecule

Exercise 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 information

The calculation of the universal density functional by Lieb maximization

The 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 information

Lecture 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- 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

Dalton Quantum Chemistry Program

Dalton 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 information

Molecular Magnetic Properties ESQC 07. Overview

Molecular 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 information

Computational Material Science Part II. Ito Chao ( ) Institute of Chemistry Academia Sinica

Computational 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 Hartree-Fock Molecular Orbital Theory Fundamental assumption of HF theory: each electron

More information

Chem 442 Review for Exam 2. Exact separation of the Hamiltonian of a hydrogenic atom into center-of-mass (3D) and relative (3D) components.

Chem 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 information

Molecular Magnetic Properties

Molecular 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 information

Molecular Integral Evaluation. The 9th Sostrup Summer School. Quantum Chemistry and Molecular Properties June 25 July 7, 2006.

Molecular Integral Evaluation. The 9th Sostrup Summer School. Quantum Chemistry and Molecular Properties June 25 July 7, 2006. 1 Molecular Integral Evaluation The 9th Sostrup Summer School Quantum Chemistry and Molecular Properties June 25 July 7, 2006 Trygve Helgaker Department of Chemistry, University of Oslo, Norway Overview

More information

PHY331 Magnetism. Lecture 3

PHY331 Magnetism. Lecture 3 PHY331 Magnetism Lecture 3 Last week Derived magnetic dipole moment of a circulating electron. Discussed motion of a magnetic dipole in a constant magnetic field. Showed that it precesses with a frequency

More information

MD simulation: output

MD 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 information

Electronic structure theory: Fundamentals to frontiers. 1. Hartree-Fock theory

Electronic 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 information

Quantum Chemistry in Magnetic Fields

Quantum 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 information

Lecture B6 Molecular Orbital Theory. Sometimes it's good to be alone.

Lecture 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 information

Time-independent molecular properties

Time-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 information

Introduction to computational chemistry Exercise I: Structure and electronic energy of a small molecule. Vesa Hänninen

Introduction to computational chemistry Exercise I: Structure and electronic energy of a small molecule. Vesa Hänninen Introduction to computational chemistry Exercise I: Structure and electronic energy of a small molecule Vesa Hänninen 1 Introduction In this exercise the equilibrium structure and the electronic energy

More information

PHY331 Magnetism. Lecture 4

PHY331 Magnetism. Lecture 4 PHY331 Magnetism Lecture 4 Last week Discussed Langevin s theory of diamagnetism. Use angular momentum of precessing electron in magnetic field to derive the magnetization of a sample and thus diamagnetic

More information

The Role of the Hohenberg Kohn Theorem in Density-Functional Theory

The Role of the Hohenberg Kohn Theorem in Density-Functional Theory The Role of the Hohenberg Kohn Theorem in Density-Functional Theory T. Helgaker, U. E. Ekström, S. Kvaal, E. Sagvolden, A. M. Teale,, E. Tellgren Centre for Theoretical and Computational Chemistry (CTCC),

More information

Molecular-Orbital Theory

Molecular-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 information

The Accurate Calculation of Molecular Energies and Properties: A Tour of High-Accuracy Quantum-Chemical Methods

The Accurate Calculation of Molecular Energies and Properties: A Tour of High-Accuracy Quantum-Chemical Methods 1 The Accurate Calculation of Molecular Energies and Properties: A Tour of High-Accuracy Quantum-Chemical Methods T. Helgaker Centre for Theoretical and Computational Chemistry Department of Chemistry,

More information

Quantum chemical modelling of molecular properties - parameters of EPR spectra

Quantum 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 information

Divergence in Møller Plesset theory: A simple explanation based on a two-state model

Divergence 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 information

Magnetism in low dimensions from first principles. Atomic magnetism. Gustav Bihlmayer. Gustav Bihlmayer

Magnetism 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 information

Importing 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 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 information

Chapter 2 Approximation Methods Can be Used When Exact Solutions to the Schrödinger Equation Can Not be Found.

Chapter 2 Approximation Methods Can be Used When Exact Solutions to the Schrödinger Equation Can Not be Found. Chapter 2 Approximation Methods Can be Used When Exact Solutions to the Schrödinger Equation Can Not be Found. In applying quantum mechanics to 'real' chemical problems, one is usually faced with a Schrödinger

More information

Physical Chemistry II Recommended Problems Chapter 12( 23)

Physical Chemistry II Recommended Problems Chapter 12( 23) Physical Chemistry II Recommended Problems Chapter 1( 3) Chapter 1(3) Problem. Overlap of two 1s orbitals recprobchap1.odt 1 Physical Chemistry II Recommended Problems, Chapter 1(3), continued Chapter

More information

The adiabatic connection

The 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 information

Chemistry 3502/4502. Final Exam Part I. May 14, 2005

Chemistry 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 information

Electromagnetism - Lecture 10. Magnetic Materials

Electromagnetism - Lecture 10. Magnetic Materials Electromagnetism - Lecture 10 Magnetic Materials Magnetization Vector M Magnetic Field Vectors B and H Magnetic Susceptibility & Relative Permeability Diamagnetism Paramagnetism Effects of Magnetic Materials

More information

Direct optimization of the atomic-orbital density matrix using the conjugate-gradient method with a multilevel preconditioner

Direct optimization of the atomic-orbital density matrix using the conjugate-gradient method with a multilevel preconditioner JOURNAL OF CHEMICAL PHYSICS VOLUME 115, NUMBER 21 1 DECEMBER 2001 Direct optimization of the atomic-orbital density matrix using the conjugate-gradient method with a multilevel preconditioner Helena Larsen,

More information

Introduction to Computational Chemistry

Introduction 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 information

Atoms, Molecules and Solids (selected topics)

Atoms, Molecules and Solids (selected topics) Atoms, Molecules and Solids (selected topics) Part I: Electronic configurations and transitions Transitions between atomic states (Hydrogen atom) Transition probabilities are different depending on the

More information

Density functional theory in magnetic fields

Density 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 information

The 3 dimensional Schrödinger Equation

The 3 dimensional Schrödinger Equation Chapter 6 The 3 dimensional Schrödinger Equation 6.1 Angular Momentum To study how angular momentum is represented in quantum mechanics we start by reviewing the classical vector of orbital angular momentum

More information

Jack Simons, Henry Eyring Scientist and Professor Chemistry Department University of Utah

Jack 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 information

Chemistry 543--Final Exam--Keiderling May 5, pm SES

Chemistry 543--Final Exam--Keiderling May 5, pm SES Chemistry 543--Final Exam--Keiderling May 5,1992 -- 1-5pm -- 174 SES Please answer all questions in the answer book provided. Make sure your name is clearly indicated and that the answers are clearly numbered,

More information

Ψ t = ih Ψ t t. Time Dependent Wave Equation Quantum Mechanical Description. Hamiltonian Static/Time-dependent. Time-dependent Energy operator

Ψ t = ih Ψ t t. Time Dependent Wave Equation Quantum Mechanical Description. Hamiltonian Static/Time-dependent. Time-dependent Energy operator Time Dependent Wave Equation Quantum Mechanical Description Hamiltonian Static/Time-dependent Time-dependent Energy operator H 0 + H t Ψ t = ih Ψ t t The Hamiltonian and wavefunction are time-dependent

More information

Lecture 9: Molecular integral. Integrals of the Hamiltonian matrix over Gaussian-type orbitals

Lecture 9: Molecular integral. Integrals of the Hamiltonian matrix over Gaussian-type orbitals Lecture 9: Molecular integral evaluation Integrals of the Hamiltonian matrix over Gaussian-type orbitals Gaussian-type orbitals The de-facto standard for electronic-structure calculations is to use Gaussian-type

More information

Intermission: Let s review the essentials of the Helium Atom

Intermission: 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 information

$ +! j. % i PERTURBATION THEORY AND SUBGROUPS (REVISED 11/15/08)

$ +! j. % i PERTURBATION THEORY AND SUBGROUPS (REVISED 11/15/08) PERTURBATION THEORY AND SUBGROUPS REVISED 11/15/08) The use of groups and their subgroups is of much importance when perturbation theory is employed in understanding molecular orbital theory and spectroscopy

More information

Same idea for polyatomics, keep track of identical atom e.g. NH 3 consider only valence electrons F(2s,2p) H(1s)

Same idea for polyatomics, keep track of identical atom e.g. NH 3 consider only valence electrons F(2s,2p) H(1s) XIII 63 Polyatomic bonding -09 -mod, Notes (13) Engel 16-17 Balance: nuclear repulsion, positive e-n attraction, neg. united atom AO ε i applies to all bonding, just more nuclei repulsion biggest at low

More information

Jack Simons, Henry Eyring Scientist and Professor Chemistry Department University of Utah

Jack 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 information

IV. Electronic Spectroscopy, Angular Momentum, and Magnetic Resonance

IV. Electronic Spectroscopy, Angular Momentum, and Magnetic Resonance IV. Electronic Spectroscopy, Angular Momentum, and Magnetic Resonance The foundation of electronic spectroscopy is the exact solution of the time-independent Schrodinger equation for the hydrogen atom.

More information

Problem 1: Spin 1 2. particles (10 points)

Problem 1: Spin 1 2. particles (10 points) Problem 1: Spin 1 particles 1 points 1 Consider a system made up of spin 1/ particles. If one measures the spin of the particles, one can only measure spin up or spin down. The general spin state of a

More information

The Gutzwiller Density Functional Theory

The 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 information

QUANTUM MECHANICS. Franz Schwabl. Translated by Ronald Kates. ff Springer

QUANTUM MECHANICS. Franz Schwabl. Translated by Ronald Kates. ff Springer Franz Schwabl QUANTUM MECHANICS Translated by Ronald Kates Second Revised Edition With 122Figures, 16Tables, Numerous Worked Examples, and 126 Problems ff Springer Contents 1. Historical and Experimental

More information

H 2 in the minimal basis

H 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 information

P. W. Atkins and R. S. Friedman. Molecular Quantum Mechanics THIRD EDITION

P. 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 information

Addition of Angular Momenta

Addition of Angular Momenta Addition of Angular Momenta What we have so far considered to be an exact solution for the many electron problem, should really be called exact non-relativistic solution. A relativistic treatment is needed

More information

Alkali metals show splitting of spectral lines in absence of magnetic field. s lines not split p, d lines split

Alkali 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

T. Helgaker, Department of Chemistry, University of Oslo, Norway. T. Ruden, University of Oslo, Norway. W. Klopper, University of Karlsruhe, Germany

T. Helgaker, Department of Chemistry, University of Oslo, Norway. T. Ruden, University of Oslo, Norway. W. Klopper, University of Karlsruhe, Germany 1 The a priori calculation of molecular properties to chemical accuarcy T. Helgaker, Department of Chemistry, University of Oslo, Norway T. Ruden, University of Oslo, Norway W. Klopper, University of Karlsruhe,

More information

Paramagnetism and Diamagnetism. Paramagnets (How do paramagnets differ fundamentally from ferromagnets?)

Paramagnetism and Diamagnetism. Paramagnets (How do paramagnets differ fundamentally from ferromagnets?) Paramagnetism and Diamagnetism Paramagnets (How do paramagnets differ fundamentally from ferromagnets?) The study of paramagnetism allows us to investigate the atomic magnetic moments of atoms almost in

More information

Problem 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, & & & & # % (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 information

Potential energy, from Coulomb's law. Potential is spherically symmetric. Therefore, solutions must have form

Potential energy, from Coulomb's law. Potential is spherically symmetric. Therefore, solutions must have form Lecture 6 Page 1 Atoms L6.P1 Review of hydrogen atom Heavy proton (put at the origin), charge e and much lighter electron, charge -e. Potential energy, from Coulomb's law Potential is spherically symmetric.

More information

Density-functional generalized-gradient and hybrid calculations of electromagnetic properties using Slater basis sets

Density-functional generalized-gradient and hybrid calculations of electromagnetic properties using Slater basis sets JOURNAL OF CHEMICAL PHYSICS VOLUME 120, NUMBER 16 22 APRIL 2004 ARTICLES Density-functional generalized-gradient and hybrid calculations of electromagnetic properties using Slater basis sets Mark A. Watson,

More information

Electric properties of molecules

Electric 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 information

Chapter 8. Molecular Shapes. Valence Shell Electron Pair Repulsion Theory (VSEPR) What Determines the Shape of a Molecule?

Chapter 8. Molecular Shapes. Valence Shell Electron Pair Repulsion Theory (VSEPR) What Determines the Shape of a Molecule? PowerPoint to accompany Molecular Shapes Chapter 8 Molecular Geometry and Bonding Theories Figure 8.2 The shape of a molecule plays an important role in its reactivity. By noting the number of bonding

More information

Origin of Chemical Shifts BCMB/CHEM 8190

Origin of Chemical Shifts BCMB/CHEM 8190 Origin of Chemical Shifts BCMB/CHEM 8190 Empirical Properties of Chemical Shift υ i (Hz) = γb 0 (1-σ i ) /2π σ i, shielding constant dependent on electronic structure, is ~ 10-6. Measurements are made

More information

Luigi Paolasini

Luigi Paolasini Luigi Paolasini paolasini@esrf.fr LECTURE 2: LONELY ATOMS - Systems of electrons - Spin-orbit interaction and LS coupling - Fine structure - Hund s rules - Magnetic susceptibilities Reference books: -

More information

2 Electronic structure theory

2 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 information

Electronic structure theory: Fundamentals to frontiers. 2. Density functional theory

Electronic structure theory: Fundamentals to frontiers. 2. Density functional theory Electronic structure theory: Fundamentals to frontiers. 2. Density functional theory MARTIN HEAD-GORDON, Department of Chemistry, University of California, and Chemical Sciences Division, Lawrence Berkeley

More information

PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 5 (TRANSITION PROBABILITIES AND TRANSITION DIPOLE MOMENT. OVERVIEW OF SELECTION RULES)

PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 5 (TRANSITION PROBABILITIES AND TRANSITION DIPOLE MOMENT. OVERVIEW OF SELECTION RULES) Subject Chemistry Paper No and Title Module No and Title Module Tag 8 and Physical Spectroscopy 5 and Transition probabilities and transition dipole moment, Overview of selection rules CHE_P8_M5 TABLE

More information

Density Functional Theory

Density 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 information

Preliminary Quantum Questions

Preliminary 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 information

3: Many electrons. Orbital symmetries. l =2 1. m l

3: Many electrons. Orbital symmetries. l =2 1. m l 3: Many electrons Orbital symmetries Atomic orbitals are labelled according to the principal quantum number, n, and the orbital angular momentum quantum number, l. Electrons in a diatomic molecule experience

More information

Convergence properties of the coupled-cluster method: the accurate calculation of molecular properties for light systems

Convergence properties of the coupled-cluster method: the accurate calculation of molecular properties for light systems 1 Convergence properties of the coupled-cluster method: the accurate calculation of molecular properties for light systems T. Helgaker Centre for Theoretical and Computational Chemistry, Department of

More information

COPYRIGHTED MATERIAL. Production of Net Magnetization. Chapter 1

COPYRIGHTED MATERIAL. Production of Net Magnetization. Chapter 1 Chapter 1 Production of Net Magnetization Magnetic resonance (MR) is a measurement technique used to examine atoms and molecules. It is based on the interaction between an applied magnetic field and a

More information

OVERVIEW OF QUANTUM CHEMISTRY METHODS

OVERVIEW 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 information

Quantum Chemical Simulations and Descriptors. Dr. Antonio Chana, Dr. Mosè Casalegno

Quantum Chemical Simulations and Descriptors. Dr. Antonio Chana, Dr. Mosè Casalegno Quantum Chemical Simulations and Descriptors Dr. Antonio Chana, Dr. Mosè Casalegno Classical Mechanics: basics It models real-world objects as point particles, objects with negligible size. The motion

More information

Electromagnetism II. Instructor: Andrei Sirenko Spring 2013 Thursdays 1 pm 4 pm. Spring 2013, NJIT 1

Electromagnetism 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 information

CHAPTER 6 CHEMICAL BONDING SHORT QUESTION WITH ANSWERS Q.1 Dipole moments of chlorobenzene is 1.70 D and of chlorobenzene is 2.5 D while that of paradichlorbenzene is zero; why? Benzene has zero dipole

More information

Chemistry 120A 2nd Midterm. 1. (36 pts) For this question, recall the energy levels of the Hydrogenic Hamiltonian (1-electron):

Chemistry 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 information

LUMO + 1 LUMO. Tómas Arnar Guðmundsson Report 2 Reikniefnafræði G

LUMO + 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 information

which implies that we can take solutions which are simultaneous eigen functions of

which implies that we can take solutions which are simultaneous eigen functions of Module 1 : Quantum Mechanics Chapter 6 : Quantum mechanics in 3-D Quantum mechanics in 3-D For most physical systems, the dynamics is in 3-D. The solutions to the general 3-d problem are quite complicated,

More information

Alcohols, protons α to ketones. Aromatics, Amides. Acids, Aldehydes. Aliphatic. Olefins. ppm TMS

Alcohols, protons α to ketones. Aromatics, Amides. Acids, Aldehydes. Aliphatic. Olefins. ppm TMS Interpretation of 1 spectra So far we have talked about different NMR techniques and pulse sequences, but we haven t focused seriously on how to analyze the data that we obtain from these experiments.

More information

General Physical Chemistry II

General 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 information

Ab-initio studies of the adiabatic connection in density-functional theory

Ab-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 information

Quantum Physics II (8.05) Fall 2002 Assignment 12 and Study Aid

Quantum 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 information

Lecture 33: Intermolecular Interactions

Lecture 33: Intermolecular Interactions MASSACHUSETTS INSTITUTE OF TECHNOLOGY 5.61 Physical Chemistry I Fall, 2017 Professors Robert W. Field Lecture 33: Intermolecular Interactions Recent Lectures Non-degenerate Perturbation Theory vs. Variational

More information

I. CSFs Are Used to Express the Full N-Electron Wavefunction

I. CSFs Are Used to Express the Full N-Electron Wavefunction Chapter 11 One Must be Able to Evaluate the Matrix Elements Among Properly Symmetry Adapted N- Electron Configuration Functions for Any Operator, the Electronic Hamiltonian in Particular. The Slater-Condon

More information