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, Dage Sundholm, Andy Teale, Erik Tellgren Hylleraas Centre for Quantum Molecular Sciences, Department of Chemistry, University of Oslo, Norway Methods and Algorithms in Quantum Chemistry, Department of Chemistry, Aarhus University, Denmark, 13 15 December 2018 T. Helgaker (Hylleraas Centre, Oslo) Dia- and Parmagnetism in Atoms and Molecules MEAL 2018 1 / 21

Introduction Para- and diamagnetism When a magnetic field is applied to a molecule, one of two things can happen: the energy decreases and the molecule moves into the field: paramagnetism the energy increases and the molecule moves out of the field: diamagnetism The BH molecule with the magnetic field along the bond axis We associate paramagnetism with open shells and diamagnetism with closed shells nevertheless, some closed-shell paramagnetic molecules do exist We will discuss such matters, highlighting the interplay between dia- and paramagnetism we will also consider molecular paramagnetic bonding we will touch upon the stability of atoms and molecules in a strong field Helgaker (Hylleraas Centre, Oslo) Dia- and Paramagnetism in Atoms and Molecules MEAL 2018 2 / 21

Introduction Hamiltonian in a uniform magnetic field In a magnetic field B = A, the Hamiltonian operator is given by 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 0 + Bs z + 1 2 BLz + 1 8 B2 ( x 2 + y 2) 1 The paramagnetic Zeeman operators represent a dipolar interaction: 1 they reduce symmetry and split energy levels 2 energy is raised or lowered, depending on orientation 1s 1s α high-angular momentum states are favoured 2 The diamagnetic operator arises from precession: 1 it raises the energy of all systems 2 it squeezes all systems transversal and longitudinal sizes of He high-angular momentum states favoured 2p +1 1s β 2p 2p 0 2p -1 Helgaker (Hylleraas Centre, Oslo) Dia- and Paramagnetism in Atoms and Molecules MEAL 2018 3 / 21

Introduction Quadratic Zeeman effect Initial energy lowering by the Zeeman terms is counteracted by the diamagnetic term H = H 0 + Bs z + 1 2 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, 074110 (2015) Note: one atomic unit of magnetic field strength B 0 corresponds to B 0 = 2.35 10 5 T Helgaker (Hylleraas Centre, Oslo) Dia- and Paramagnetism in Atoms and Molecules MEAL 2018 4 / 21

Introduction Closed-shell diamagnetism In a closed-shell system, the ground-state energy should (naively) increase diamagnetically: 0 H 0 = 0 H 0 0 + 1 8 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 10 3 14 12 0.08 10 0.06 8 0.04 0.02 6 4 2 0 0.1 0.05 0 0.05 0.1 0 0.1 0.05 Helgaker (Hylleraas Centre, Oslo) Dia- and Paramagnetism in Atoms and Molecules MEAL 2018 5 / 21

Introduction Nevertheless, closed-shell paramagnetic molecules such as C 20 do exist Paramagnetism must in some way arise from field-induced changes in the ground state: 0 H 0 = 0 H 0 + 1 2 BLz + 1 8 B2 (x 2 + y 2 ) 0 Helgaker (Hylleraas Centre, Oslo) Dia- and Paramagnetism in Atoms and Molecules MEAL 2018 6 / 21

BH molecule Molecular orbitals of BH from the 1s 2 B 2s2 B 2p1 B AOs of B and 1s H AO of H 1s B }{{} core < 1σ 2s B + 1s H }{{} bonding orbital < 2p }{{} 0 < 2π ±1 }{{} lone pair HOMO LUMOs Lowest electronic states of BH MRCI, Miliordos and Mavridis, JCP 128, 144308 (2008) three lowest singlet states: 1 Σ + = 1s 2 1σ 2 2p 2 0 ground state 1 Π = 1s 2 1σ 2 2p 0 2π ±1 singly excited 1 = 1s 2 1σ 2 2π 2 ±1 doubly excited lowest triplet state: 3 Π = 1s 2 1σ 2 2p 0 2π ±1 singly excited E. Tellgren, T. Helgaker, A. Soncini, PCCP 11, 5489 (2009) S. Stopkowicz, A. Borgoo, J. Gauss, F. Hampe, W. Klopper, S. Stopkowicz, A. Teale, E. Tellgren, T. Helgaker Helgaker (Hylleraas Centre, Oslo) Dia- and Paramagnetism in Atoms and Molecules MEAL 2018 7 / 21

BH in a parallel magnetic field CCSD and EOM-CCSD calculations in the aug-cc-pvqz basis on BH in parallel field 1 = 1s 2 1σ 2 2π 1 2 1 Π = 1s 2 1σ 2 2p 0 2π 1 3 Π = 1s 2 1σ 2 2p 0 2π 1 1 Σ = 1s 2 1σ 2 2p 2 0 The states do not mix, having different eigenvalues of the angular momentum operator the triplet becomes the ground state at about 0.04B 0, being strongly stabilized in the field there is a (near) three-state crossing at about 0.25B 0 where 2p 0 and 2π 1 are degenerate The initial slope is determined by the orbital and spin angular momenta: ml, m S 1 2 Lz + Sz ml, m S = 1 2 m L + m S = 0, 3/2, 1/2, 1 all states except the singlet ground state are paramagnetic Helgaker (Hylleraas Centre, Oslo) Dia- and Paramagnetism in Atoms and Molecules MEAL 2018 8 / 21

BH in a non-parallel magnetic field In a non-parallel field, the singlet states belong to the same symmetry Σ = 1s 2 1σ 2 2p 2 0, Π = 1s 2 1σ 2 2p 0 2π 1, = 1s 2 1σ 2 2π 1 2 The orbital Zeeman operator 1 LzB couples states that differ by a single excitation 2 This coupling generates a three-state avoided crossing with the Π state as mediator -25.00 BH at 0-25.05-25.10 Etot/Eh -25.15-25.20-25.25 0-25.30 Σ Π -25.35 0.0 0.1 0.2 0.3 0.4 0.5 B/B0 basis: unc-aug-cc-pvqz Helgaker (Hylleraas Centre, Oslo) Dia- and Paramagnetism in Atoms and Molecules MEAL 2018 9 / 21

Emergence of paramagnetism in BH Helgaker (Hylleraas Centre, Oslo) Dia- and Paramagnetism in Atoms and Molecules MEAL 2018 10 / 21

BH in parallel and perpendicular fields BH behaves very differently in parallel (left) and perpendicular (right) fields The closed-shell ground state becomes paramagnetic in the perpendicular orientation unquenching of the angular momentum by an avoided crossing The lowest triplet becomes the ground state in relatively weak fields crossing before 0.04B 0 (0.08B 0) in the parallel (perpendicular) field orientation The magnetic field orients the molecule preferred perpendicular (parallel) orientation in the singlet (triplet) state Paramagnetism is a magnetic Jahn Teller effect open-shell paramagnetism: Jahn Teller effect closed-shell paramagnetism: pseudo Jahn Teller effect Helgaker (Hylleraas Centre, Oslo) Dia- and Paramagnetism in Atoms and Molecules MEAL 2018 11 / 21

Three lowest singlet states of BH Helgaker (Hylleraas Centre, Oslo) Dia- and Paramagnetism in Atoms and Molecules MEAL 2018 12 / 21

Modelling of three-state (avoided) crossing The three-state avoided crossing can easily be modelled for the diabatic states m L : 0 = 1s 2 1σ 2 2p 2 0, 1 = 1s 2 1σ 2 2p 0 2π 1, 2 = 1s 2 1σ 2 2π 1 2 The diabatic energies are modelled in the manner E ml (B) = E ml 1 2 m L cos(θ)b + 1 2 am L B2 Introducing the coupling parameter µ, we obtain the Hamiltonian E 0 (B) iµb sin(θ) 0 H(B, θ) = 1 iµb sin(θ) 2 E 0(B) + 1 2 E 2(B) iµb sin(θ) 0 iµb sin(θ) E 2 (B) This 5-parameter Hamiltonian accurately reproduces the CCSD energies at all angles: Note: the middle state is an active spectator, mediating the coupling Helgaker (Hylleraas Centre, Oslo) Dia- and Paramagnetism in Atoms and Molecules MEAL 2018 13 / 21

Evolution of BH orbitals and states in a non-parallel field As we increase the (non-parallel) field, the Σ state takes on the character of the state 2 = 1s 2 1σ 2 2π 1 2 1s 2 1σ 2 2p 2 0 1 = 1s 2 1σ 2 2p 0 2π 1 1s 2 1σ 2 2π 1 2p 0 0 = 1s 2 1σ 2 2p 2 0 1s 2 1σ 2 2π 1 2 HOMO in the 0 ground state at 0.00B 0, 0.26B 0 and 0.40B 0 HOMO in the 2 doubly excited state at 0.00B 0, 0.26B 0 and 0.40B 0 Note: nodal plane rotated by the field as the angular momentum is unquenched Helgaker (Hylleraas Centre, Oslo) Dia- and Paramagnetism in Atoms and Molecules MEAL 2018 14 / 21

A more complete picture... In reality, we should include a fourth singlet state Σ = 1s 2 1σ 2 2p 2 0, Π = 1s 2 1σ 2 2p 0 2π 1, Π + = 1s 2 1σ 2 2p 0 2π +1, = 1s 2 1σ 2 2π 1 2 CH + FCI, pvdz 0 degrees 5 degrees -37.6-37.6 0.0 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5 0.0 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5-37.65-37.65-37.7-37.75-37.8-37.85-37.9-37.95-38 -38.05-38.1 Sigma Pi-1 Pi+1 Delta-2 45 degrees -37.6 0.0 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5-37.65-37.7-37.75-37.8-37.85-37.9-37.95-38 -38.05-38.1 A A A A -37.7-37.75-37.8-37.85-37.9-37.95-38 -38.05-38.1 A A A A 90 degrees -37.6 0.0 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5-37.65-37.7-37.75-37.8-37.85-37.9-37.95-38 -38.05-38.1 A' A'' A' A' Helgaker (Hylleraas Centre, Oslo) Dia- and Paramagnetism in Atoms and Molecules MEAL 2018 15 / 21

Closed-shell diamagnetism by gauge transformations The vector potential A that satisfies B = A is nonunique but takes the general form A = A 0 + f, A 0 = B, A 0 = 0 (arbitrary gauge function f) The lowest-order energy of a closed-shell system in the field represented by A is given by: E (2) = 1 2 0 A2 0 }{{} diamagnetic term 0 A + A n n A + A 0 } E n E {{ 0 } paramagnetic term n>0 If the paramagnetic term vanishes, then the lowest-order energy is positive: (A + A) Ψ 0 = 0 = E (2) = 1 2 0 A2 0 > 0 Closed-shell diamagnetism therefore follows if a solution f can be found for the equation (A 0 + f) Ψ 0 + 1 2 ( 2 f)ψ 0 = 0 For real Ψ 0, Rebane (1960) gave an explicit solution by expansion in excited states: f(r) = n>0 n A 0 0 E n E 0 Ψn(r) Ψ 0 (r) but a solution is guaranteed only if Ψ 0 is nodeless or has the same nodes as Ψ n ground states with more than two electrons can have nodal surfaces Helgaker (Hylleraas Centre, Oslo) Dia- and Paramagnetism in Atoms and Molecules MEAL 2018 16 / 21

Paramagnetic bonding Paramagnetic bonding Consider a covalent bond such that of H 2 in a magnetic field In a sufficiently strong field, stabilization of beta spin reduces H 2 bond order 1σg 2 (αβ βα) (bond order one) (1σg1σu 1σu1σg)ββ (bond order zero) The 1σ u orbital is a distorted p orbital, which is stabilized in the perpendicular orientation 1σ u = 1s A 1s B 2p 1 (united atom) This paramagnetic stabilization increases at short separations, binding the molecule Induced currents in singlet (left) and triplet (right) H 2 (relative to free atoms) K. Lange, Erik Tellgren, M. Hoffman, T. Helgaker, Science 337, 327 (2012) Helgaker (Hylleraas Centre, Oslo) Dia- and Paramagnetism in Atoms and Molecules MEAL 2018 17 / 21

Paramagnetic bonding Methane molecule in a strong magnetic field The methane molecule is stable in magnetic field of strength B 0 The molecule becomes near planar with increasing field strength All bonds are paramagnetic, attempting to orient themselves perpendicular to the field M. Dimitrova, D. Sundholm, T. Saue, S. Stopkowicz, A. Teale, W. Klopper, P. Schwerdtfeger, T. Helgaker Helgaker (Hylleraas Centre, Oslo) Dia- and Paramagnetism in Atoms and Molecules MEAL 2018 18 / 21

Ionization in a magnetic field Ionization in a magnetic field I A magnetic field stabilizes atoms and molecules towards ionization I evgw@pbe0 ionization potentials (IPs) plotted against magnetic field strength B He = brown Ne = black F = violet O = blue N = cyan H = gray C = green B = yellow Be = orange Li = red I The ionized electron enters the lowest Landau level of angular momentum (m`, ms ) ELandau = n + 12 m` + 21 m` + ms + 12 B, n = 0, 1, 2,..., I The IP for ionization from orbital of GW quasienergy εm`,ms (B) is therefore given by IP = Em`,ms (B) + 21 m` + 12 m` + ms + 21 B I increased stability almost entirely due to increased diamagnetic Landau energy I increased diamagnetic energy of electrons in the atom gives rise to slight concavity I C. Holzer, A. Teale, F. Hampe, S. Stopkowicz, T. Helgaker, W. Klopper Helgaker (Hylleraas Centre, Oslo) Dia- and Paramagnetism in Atoms and Molecules MEAL 2018 19 / 21

Conclusions Conclusions the field couples the ground state to a low-lying paramagnetic doubly excited state the ground states takes on the paramagnetic character of the excited state Paramagnetic bonding antibonding orbitals are stabilized in a magnetic field Ionization potentials atoms are stabilized towards ionization by an increased Landau energy Helgaker (Hylleraas Centre, Oslo) Dia- and Paramagnetism in Atoms and Molecules MEAL 2018 20 / 21

Conclusions Acknowledgements I Co-workers: I I I I I I I Alex Borgoo, Erik Tellgren (University of Oslo, CAS) Ju rgen Gauss, Florian Hampe, Stella Stopkowicz (University of Mainz, CAS) Wim Klopper, Christof Holzer (Karlsruhe Institute of Technology, CAS) Trond Saue (University of Toulouse, CAS) Peter Schwerdtfeger (Massey University, CAS) Dage Sundholm, Maria Dimitrova (University of Helsinki, CAS) Andy Teale (University of Nottingham, CAS) I Funding: I Centre for Advanced Study (CAS) at the Norwegian Academy of Science and Letters I Norwegian Research Council for Centres of Excellence CTCC & Hylleraas I ERC Advanced Grant ABACUS Helgaker (Hylleraas Centre, Oslo) Dia- and Paramagnetism in Atoms and Molecules MEAL 2018 21 / 21