Molecules in strong magnetic fields
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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
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