Quantum Chemistry in Magnetic Fields

Transcription

1 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 of Theoretical and Computational Chemists (WATOC 2017) Gasteig Cultural Centre, Munich, Germany, 27 August 1 September 2017 T. Helgaker (Hylleraas Centre, University of Oslo) Quantum Chemistry in Magnetic Fields WATOC 27 August / 35

2 Introduction Introduction Molecular magnetism is usually studied perturbatively such an approach is highly successful and widely used in quantum chemistry for example, for the calculation of NMR parameters We have undertaken a nonperturbative study of molecular magnetism gives new insight into molecular electronic structure provides direct route to many magnetic phenomena describes atoms and molecules observed in astrophysics it is a stress test for quantum chemistry in a different environment Status today: chemistry in magnetic fields can now be studied reliably by quantum chemistry the greatest challenge is DFT, on which we will focus interesting new phenomena have been uncovered Important previous work: Literature fundamental work by Lorenz Cederbaum and Peter Schmelcher Matter in Strong Magnetic Fields, D. Lai, Rev. Mod. Phys. 73, 629 (2001) Atoms and Molecules in Strong External Fields, P. Schmelcher and W. Schweizer, eds., (Kluwer, 2002) Perspective: Coupled cluster theory for atoms and molecules in strong magnetic fields, S. Stopkowicz, Int. J. Quantum Chem. (2017) Helgaker (Hylleraas Centre, University of Oslo) Quantum Chemistry in Magnetic Fields WATOC 27 August / 35

3 Background Electronic Hamiltonian The one-electron Hamiltonian in the absence of a magnetic field (a.u.): H 0 = 1 2 p2 + V, p = i In a magnetic field B = A, the kinetic-energy operator is modified and spin added: H = 1 (p + A) (p + A) + B S + V 2 For a uniform magnetic field in the z direction, expansion of the kinetic energy gives H = H BLz + Bsz B2 (x 2 + y 2 ) The linear paramagnetic Zeeman terms are easily understood: the magnetic moments associated with L z and s z interact with B The quadratic diamagnetic term may be understood in the following manner: 1 the field B induces a precession of the electrons with Larmor frequency B/4π 2 this precession generates an induced magnetic moment proportional to the field charge frequency area = B 4π π(x2 + y 2 ) 3 this induced magnetic moment interacts with B, raising the energy quadratically Helgaker (Hylleraas Centre, University of Oslo) Quantum Chemistry in Magnetic Fields WATOC 27 August / 35

4 Background Three field regimes One atomic unit of magnetic field strength B corresponds to B 0 = T From a consideration of the electronic Hamiltonian H = H BLz + Bsz B2 (x 2 + y 2 ), we may then distinguish three field regimes: 1 Coulomb regime: B B 0 earth magnetism 10 10, NMR 10 4 ; pulsed fields 10 2 Coulomb interactions dominate familiar chemistry of spherical atoms 2 Intermediate regime: B B 0 magnetic white dwarfs of field strength B 0 Coulomb and magnetic interactions compete exotic chemistry of small, ellipsoidal atoms 3 Landau regime: B B 0 neutron stars: B 0 magnetic interactions dominate alien chemistry of tiny, needle-like atoms Helgaker (Hylleraas Centre, University of Oslo) Quantum Chemistry in Magnetic Fields WATOC 27 August / 35

5 Background London code Standard quantum-chemistry programs cannot be used for molecules in magnetic fields the wave function is complex complex algebra cannot be avoided London orbitals (GIAOs) must be used for gauge-origin invariance LONDON is the first general molecular code for molecules in magnetic fields Erik Tellgren, Kai Lange, Stella Stopkowicz, Andy Teale HF, FCI, MCSCF, MP2, CCSD(T), DFT BAGEL Toru Shiozaki s group, Northwestern QUEST Andy Teale s group, Nottingham Helgaker (Hylleraas Centre, University of Oslo) Quantum Chemistry in Magnetic Fields WATOC 27 August / 35

6 Para- and diamagnetism Para- and diamagnetism The Hamiltonian contains paramagnetic terms and diamagnetic terms: H = H 0 + Bs z BLz + 1 ( 8 B2 x 2 + y 2) Important consequences of the paramagnetic Zeeman terms: they reduce symmetry and split energy levels energy is raised or lowered, depending on orientation 1s 1s α 2p +1 1s β 2p 2p 0 2p -1 Important consequences of the diamagnetic term: 1 it raises the energy of all systems 2 it squeezes all systems ground-state helium atom transversal size 1/ B longitudinal size 1/ log B Helgaker (Hylleraas Centre, University of Oslo) Quantum Chemistry in Magnetic Fields WATOC 27 August / 35

7 Para- and diamagnetism Quadratic Zeeman effect For open-shell atoms, we observe the quadratic Zeeman effect initial energy lowering by Zeeman terms counteracted by the diamagnetic term H = H 0 + Bs z BLz + 1 ( 8 B2 x 2 + y 2) Lowest states of the fluorine atom (left) and sodium atom (right) in a magnetic field CCSD(T) calculations in uncontracted aug-cc-pcvqz basis (atomic units) Stopkowicz, Gauss, Lange, Tellgren, and Helgaker, JCP 143, (2015) Helgaker (Hylleraas Centre, University of Oslo) Quantum Chemistry in Magnetic Fields WATOC 27 August / 35

8 Para- and diamagnetism Closed-shell diamagnetism In a closed-shell system, ground-state energy should increase diamagnetically: 0 H 0 = 0 H B2 0 x 2 + y 2 0, 0 L z 0 = 0 S z 0 = 0 Energy of benzene in a perpendicular magnetic field (atomic units): a) 0.1 b) x Helgaker (Hylleraas Centre, University of Oslo) Quantum Chemistry in Magnetic Fields WATOC 27 August / 35

9 Para- and diamagnetism Closed-shell paramagnetism Nevertheless, closed-shell paramagnetic molecules such as C 20 do exist Zeeman Zeeman Paramagnetism results from Zeeman coupling of ground and excited states in the field in the absence of coupling, the diamagnetic diabatic ground and excited states cross the Zeeman interaction generates adiabatic states with an avoided crossing a sufficiently strong coupling creates a double minimum (cmp. Renner Teller) Tellgren, Helgaker and Soncini, PCCP 11, 5489 (2009) Helgaker (Hylleraas Centre, University of Oslo) Quantum Chemistry in Magnetic Fields WATOC 27 August / 35

10 Para- and diamagnetism Induced molecular currents Magnetic fields induce currents in molecules, often with interesting behaviour Consider the currents in the antiaromatic tetraoxaisophlorin molecule (µa) anti-clockwise inner path below 4 kt clockwise outer path above 4 kt Tellgren, Fliegl, and Helgaker (in preparation) Helgaker (Hylleraas Centre, University of Oslo) Quantum Chemistry in Magnetic Fields WATOC 27 August / 35

11 Perpendicular paramagnetic bonding Bonding in magnetic fields We have seen an interesting interplay of para- and diamagnetism in atoms and molecules we shall now see how para- and diamagnetism may affect bonding How will the squeezed and distorted atoms bind in a magnetic field? will diatomic molecules bond parallel or perpendicular to the field? We shall illustrate binding by considering the hydrogen molecule in its two lowest states the bound singlet ground state 1σ 2 g the unbound lowest triplet state 1σ g1σ u Lange, Tellgren, Hoffmann and Helgaker, Science (2012) Helgaker (Hylleraas Centre, University of Oslo) Quantum Chemistry in Magnetic Fields WATOC 27 August / 35

12 Perpendicular paramagnetic bonding H 2 in the lowest singlet state The total energy of singlet H 2 1σ 2 g increases diamagnetically The molecule shrinks, becomes more strongly bound, and aligns with the field B R e D e E rot pm 455 kj/mol pm 834 kj/mol 239 kj/mol Helgaker (Hylleraas Centre, University of Oslo) Quantum Chemistry in Magnetic Fields WATOC 27 August / 35

13 Perpendicular paramagnetic bonding H 2 in the lowest triplet state The ββ triplet 1σ g1σ u decreases paramagnetically, becoming the ground state at 0.4B 0 This molecule also shrinks, but orients itself perpendicular to the field and becomes bound B R e D e E rot kj/mol pm 38 kj/mol 34 kj/mol Helgaker (Hylleraas Centre, University of Oslo) Quantum Chemistry in Magnetic Fields WATOC 27 August / 35

14 Perpendicular paramagnetic bonding Paramagnetic stabilization of antibonding orbitals Consider the evolution of orbitals from separate atoms to the united atom limit H + H H 2 He 1σ g transforms into helium 1s, acquiring no angular momentum 1σu transforms into helium 2p, acquiring angular momentum 1 With the bond parallel to the field, the 1σ u orbital transforms into 2p 0 - = = 2p 0 the 2p 0 electron rotates perpendicular to the field axis and is not stabilized 2 With the bond normal to the field, the 1σ u transforms into 2p 1 - = = 2p -1 the 2p 1 electron rotates about the field axis and is stabilized Bonding occurs as the kinetic energy is lowered by favourable rotation in the field Helgaker (Hylleraas Centre, University of Oslo) Quantum Chemistry in Magnetic Fields WATOC 27 August / 35

15 Perpendicular paramagnetic bonding Induced rotation in the antibonding H 2 orbital Emergence of rotation in antibonding H 2 triplet orbital Plots of absolute values of complex orbitals Helgaker (Hylleraas Centre, University of Oslo) Quantum Chemistry in Magnetic Fields WATOC 27 August / 35

16 Perpendicular paramagnetic bonding Diamagnetic and paramagnetic effects on MOs H + 2 orbital energies plotted against atomic separation at field strength 1.0B 0 field-free orbital energies plotted in grey for comparison -0.4 antibonding orbital destabilized -0.4 antibonding orbital stabilized at short distances bonding orbital destabilized, more so at large distances bonding orbital destabilized, more so at large distances bond axis parallel with field bond axis perpendicular to field The orbitals become destabilized in all orientations except the antibonding orbital at short bond distances in a perpendicular orientation Helgaker (Hylleraas Centre, University of Oslo) Quantum Chemistry in Magnetic Fields WATOC 27 August / 35

17 Perpendicular paramagnetic bonding Perpendicular paramagnetic bonding Stabilization of antibonding orbitals yield zero-bond-border bonds in magnetic fields Bonding occurs by a modification of the MO level diagram: correlation not needed EêkJmol EêkJmol -1 No Field H He 2 Field strength 1.0 a.u HF FCI Energy (H) He He Field strength 1.0 a.u Rêpm No Field Previous studies of H 2 in magnetic fields: parallel orientation studied by Schmelcher et al., PRA 61, (2000); 64, (2001) Hartree Fock studies by Žaucer and and Ažman (1977) and by Kubo (2007) Helgaker (Hylleraas Centre, University of Oslo) Quantum Chemistry in Magnetic Fields WATOC 27 August / 35

18 Perpendicular paramagnetic bonding Paramagnetic bonding in He 2 The field-free 1 Σ + g (1σ 2 g 1σ u 2 ) helium dimer is bound by dispersion Energies aligned at dissociation illustrate emergence of paramagnetic bonding in the field Helgaker (Hylleraas Centre, University of Oslo) Quantum Chemistry in Magnetic Fields WATOC 27 August / 35

19 Perpendicular paramagnetic bonding Importance of correlation for paramagnetic bonding Paramagnetic bonding of triangular He 3 is significantly enhanced by electron correlation Helgaker (Hylleraas Centre, University of Oslo) Quantum Chemistry in Magnetic Fields WATOC 27 August / 35

20 Perpendicular paramagnetic bonding Paramagnetic bonding in LiH Lowest triplet state of LiH becomes bound in a perpendicular magnetic field at 0.2B 0 : the antibonding HOMO (red) is a deformed sp hybrid (unbound) at 0.6B 0 : the HOMO (green) has a large contribution from the Li 2p orbital (bound) Stopkowicz, Gauss, Lange, Tellgren, and Helgaker, JCP 143, (2015) Helgaker (Hylleraas Centre, University of Oslo) Quantum Chemistry in Magnetic Fields WATOC 27 August / 35

21 DFT in magnetic fields Current-Density-Functional Theory (CDFT) To study large systems and magnetic size effects, we must adapt DFT to magnetic fields In standard DFT, everything depends on the density ρ and universal density functional F : E(v) = inf ρ ( F (ρ) + (v ρ) ) the density ρ interacts with the external scalar potential v as (v ρ) = v(r)ρ(r)dr In an external magnetic field B = A, currents are induced in the electronic system density and current density in benzene left: in the molecular plane right: above the molecular plane In CDFT, the paramagnetic current density κ is incorporated in the description ( E(v, A) = inf ρ,κ F (ρ, κ) + (v A2 ρ) + (A κ) ) while the density ρ interacts with v A2, the current density κ interacts with A Vignale and Rasolt (1987) ours is the first molecular magnetic code The CDFT density functional F is unknown we need high-level theory for benchmarking Helgaker (Hylleraas Centre, University of Oslo) Quantum Chemistry in Magnetic Fields WATOC 27 August / 35

22 DFT in magnetic fields CDFT with the VRG functional The Vignale Rasolt Geldart (VRG) local vorticity functional (1988) E VRG (ν) = g(ρ(r)) ν(r) 2 dr, ν(r) = κ(r) ρ(r) based on uniform electron gas Aligned dissociation curves of He 2 in a perpendicular field of strength B 0 (u-aug-cc-pvtz) Tellgren, Teale, Furness, Lange, Ekström, and Helgaker, JCP (2014) our experience is in line with those of Lee, Cowell, and Handy (1994) for magnetizabilities and Trickey for atoms (2014) Helgaker (Hylleraas Centre, University of Oslo) Quantum Chemistry in Magnetic Fields WATOC 27 August / 35

23 DFT in magnetic fields Current meta-gga functionals Higher up on Jacob s ladder, the meta-gga functionals contain the kinetic energy density: τ σ(r) = Nocc i=1 ϕiσ(r) 2 Fermi hole curvature, Dobson (1993) generalized this to current-carrying states: τ σ(r) = τ σ(r) κ σ(r) 2 /2ρ σ(r) Becke (2002), Tao (2005), Maximoff and Scuseria (2006) The resulting current meta-ggas in particular, ctpss perform very well aligned aug-cc-pcvtz curves of Ne 2 in a perpendicular field of strength B 0 Interaction Energy ê meh Á Ú Ï Ì Á Ú Ï Ê Ù Ì Á Ú Ï Ê Ù Á Ì ÏÚ Ê Ù Á Ì Ï Ú Ê Ù Á Ì Ï Ú Ê Á Ù Ì Ï Ú Ê Á ÙÌ Ï Ú Ê Ê Ê Ê ÏÊ Á Ù ÏÊ Á Ù ÏÊ Á ÚÙ Ì Ê Á ÏÁ ÙÌ Ï Ú ÙÌ ÚÌ Ú Á Ì ÏÙ Á Ì Ú Á ÏÙ Ì Ú ÙÌ Ï Ú Ú Ê HF LDA Ï PBE Ú KT3 Ù cb98-8 Á ctpss ctpsshhl Ì CCSDHTL r ê a0 Furness, Verbeke, Tellgren, Stopkowicz, Ekström, Helgaker, Teale, JCTC 11, 4169 (2015) excitation energies: Bates and Furche JCP (2012); atoms: Zhu, Zhang and Trickey (2014) Helgaker (Hylleraas Centre, University of Oslo) Quantum Chemistry in Magnetic Fields WATOC 27 August / 35

24 DFT in magnetic fields Iso-orbital indicator The fermionic kinetic-energy density is upper bound to the corresponding bosonic density: Nocc τ σ(r) = ϕ iσ(r) 2 (fermionic) τ W σ (r) = 1 4 i=1 ρ σ(r) 2 (bosonic) ρ σ(r) equality in iso-orbital regions one-electron regions (tails, single bonds, lone pairs) The bosonic/fermionic ratio is a useful iso-orbital indicator to avoid self interaction z σ(r) = τ W σ (r) τ σ(r), 0 zσ(r) 1 Becke (1998), Tao Perdew Staroverov Scuseria (2003) Self-interaction problems exacerbated in magnetic fields by confinement (here He 2 ): The electron localization function (ELF) in a magnetic field volumes of He 2 with f ELF 0.8 at zero field (lighter) and at B 0 (darker) Furness, Ekström, Helgaker and Teale, Mol. Phys. 114, 1415 (2016) Helgaker (Hylleraas Centre, University of Oslo) Quantum Chemistry in Magnetic Fields WATOC 27 August / 35

25 DFT in magnetic fields Helium flakes Helium forms flakes consisting of equilateral triangles in a perpendicular field moreover, HF binding energy per atom increases with cluster size suggestive of hexagonal 2D crystal MOLDEN MOLDEN MOLDEN defaults used first point Helgaker (Hylleraas Centre, University of Oslo) Quantum Chemistry in Magnetic Fields WATOC 27 August / 35

26 DFT in magnetic fields Symmetric in-plane dissociation of He 7 flake Planar He 7 flake in perpendicular fields from 0 to 1B 0 interatomic distance plotted against the field strength calculations with QUEST in u-aug-cc-pvtz basis We have paramagnetic bonding dispersion unimportant good agreement between MP2 and ctpss HF MP ctpss ctpss-d3 Helgaker (Hylleraas Centre, University of Oslo) Quantum Chemistry in Magnetic Fields WATOC 27 August / 35

27 DFT in magnetic fields Interlayer dissociation of He 17 cluster hcp-like He 17 cluster in perpendicular fields from 0 to 1B 0 interlayer distance plotted against the field strength calculations with QUEST in u-aug-cc-pvtz basis We have weak dispersion only no paramagnetic bonding good agreement between MP2 and ctpss-d HF MP ctpss ctpss-d Helgaker (Hylleraas Centre, University of Oslo) Quantum Chemistry in Magnetic Fields WATOC 27 August / 35

28 DFT in magnetic fields Magnetic-field DFT (BDFT) An alternative to CDFT in a magnetic field is BDFT (Grayce and Harrison 1994) the universal density functional depends on the magnetic field B: E(v, B) = inf ρ (F (ρ, B) + (v ρ)) We have also implemented the Lieb variation principle for BDFT F (ρ, B) = sup v (E(v, B) (v ρ)) calculation at CC level accurate benchmarking of Kohn Sham components with high-level theory F (ρ, B) = T s(ρ, B) + J(ρ) + E xc(ρ, B) We have recently studied BDFT and its relationship to CDFT (ρ κ) ρ κ F (ρ ) E( κ) ( ) Reimann, Borgoo, Tellgren, Teale, and Helgaker, JCTC (ASAP 2017) Helgaker (Hylleraas Centre, University of Oslo) Quantum Chemistry in Magnetic Fields WATOC 27 August / 35

29 DFT in magnetic fields Kohn Sham decomposition of H 2 O in a magnetic field Changes in the CCSD and ctpss Kohn Sham energy components with field strength H2O CCSD H2O ctpss E total T s v J E xc E total T s v J E xc Water is diamagnetic, with an overall quadratic energy increase in the field the changes are driven by the increasing kinetic energy T s(ρ, B) a resulting compression of the density increases J(ρ) and decreases (v ρ) the exchange correlation energy decreases slightly ctpss performs well but overestimates the overall energy variation the quality of the density is important for a correct diamagnetic behaviour Helgaker (Hylleraas Centre, University of Oslo) Quantum Chemistry in Magnetic Fields WATOC 27 August / 35

30 DFT in magnetic fields Kohn Sham decomposition of BH in a magnetic field Changes in the CCSD and ctpss Kohn Sham energy components with field strength BH CCSD BH ctpss E total T s v J E xc E total T s v J E xc BH is paramagnetic, with an overall quadratic energy decrease in the field the changes are driven by the decreasing kinetic energy T s(ρ, B) a resulting decompression of the density decreases J(ρ) and increases (v ρ) the exchange correlation energy increases strongly in the field (mostly exchange) ctpss performs well but error cancellation between exchange and correlation Helgaker (Hylleraas Centre, University of Oslo) Quantum Chemistry in Magnetic Fields WATOC 27 August / 35

31 DFT in magnetic fields Closed-shell paramagnetic molecules We now have confidence in meta-gga functionals in magnetic fields We are now in a position to study large molecules in magnetic fields calculations by Andy Teale s group using QUEST Helgaker (Hylleraas Centre, University of Oslo) Quantum Chemistry in Magnetic Fields WATOC 27 August / 35

32 Born Oppenheimer and relativity Born Oppenheimer approximation and relativity Born Oppenheimer approximation: Relativity: for dynamics diagonal nonadiabatic coupling matrix elements Λ mm(b) are needed [T nuc(b) + E el (k; R)] χ m(r) = (E + Λ mm(b))χ m(r) magnetic field experienced by the nuclei screened by electronic motion see work by Cederbaum and Schmelcher we have performed non-born Oppenheimer calculations on HD Adamowicz, Tellgren and Helgaker, CPL (2015) Adamowicz, Stanke, Tellgren and Helgaker, CPL (2017) important only for extreme magnetic fields B α 2 B B 0 relativistic effects arise also in weak fields important for molecular properties magnetic multipole interactions requires Gaunt orbit orbit interaction: W C = 1 1, W G = 1 π i π j 2 r i j ij 2 α2 + π r i j ij 2 α2 δ(r ij ) i j Helgaker (Hylleraas Centre, University of Oslo) Quantum Chemistry in Magnetic Fields WATOC 27 August / 35

33 Relevance of high-field studies Atomic and molecular white-dwarf spectra Many white-dwarf stars have a magnetic field of about one atomic unit about 10% are magnetic with field strength T atmosphere of atomic hydrogen and helium 25% have heavier elements such as C, Si, P, and S fed from planetary debris Spectra of the hydrogen atom from a magnetized white dwarf PG brightest strongly magnetic white dwarf star, rotates every 100 minutes, field strength 0.1B 0 field strength determined by comparing observed and predicted atomic lines in the visible spectrum In 2013, hydrogen molecules were detected in two nonmagnetized white dwarfs the B 1 Σ + u X1 Σ + g transitions in the UV spectrum of the white dwarf G29 38 high-resolution UV spectra are needed for magnetic white dwarfs for molecular detection Helgaker (Hylleraas Centre, University of Oslo) Quantum Chemistry in Magnetic Fields WATOC 27 August / 35

34 Relevance of high-field studies High-field analogues in semiconductors At field strength B 0, the magnetic cyclotrone energy of the electron is one hartree: B 0 = e3 m 2 e (4πɛ 0 ) 2 3 = T in semiconductors, effective mass and dielectric constant may reduce B 0 dramatically the donated P electron in Si:P behaves as a hydrogen electron with B 0 = 32.8 T High-field phenomena may be studied in the laboratory B. N. Murdin et al., Nat. Commun. 4, 1469 (2013) Si:P as a laboratory analogue for hydrogen on high magnetic field white dwarf stars The spectra reproduce the high-field theory for free hydrogen, with quadratic Zeeman splitting[...] They show the way for experiments on He and H 2 analogues, and for investigation of He 2, a bound molecule predicted under extreme conditions. Helgaker (Hylleraas Centre, University of Oslo) Quantum Chemistry in Magnetic Fields WATOC 27 August / 35

35 Conclusions Conclusions Fascinating chemistry in magnetic fields velocity-dependent magnetic forces rivalling attractive and repulsive Coulomb forces interplay of paramagnetic and diamagnetic effects with Coulomb interactions enormous complexity and diversity of electronic structure Special software has been developed: LONDON, BAGEL, QUEST complex algebra and GIAOs for gauge-origin invariance Density-functional theory requires special care Relevance meta-gga functionals show great promise increasing laboratory fields, systems with weakened Coulomb forces crystal impurities, degenerate stars Acknowledgements: Alex Borgoo, Ulf Ekström, Sarah Reimann, Espen Sagvolden, Erik Tellgren Stella Stopkowicz (Mainz) will speak about CC and EOM-CC in magnetic fields Andy Teale (Nottingham) ERC Advanced Grant ABACUS Norwegian Research Council for Centres of Excellence CTCC & Hylleraas Helgaker (Hylleraas Centre, University of Oslo) Quantum Chemistry in Magnetic Fields WATOC 27 August / 35