# Molecular Magnetic Properties

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1 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 Sciences, Indian Association for the Cultivation of Science, Kolkata, India December 19, 213 Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties IACS / 28

2 Molecular Magnetism Part 1: The electronic Hamiltonian Hamiltonian mechanics and quantization electromagnetic fields scalar and vector potentials electron spin Part 2: Molecules in an external magnetic field Hamiltonian in an external magnetic field gauge transformations and London orbitals magnetizabilities diamagnetism and paramagnetism Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties IACS / 28

3 Hamiltonian mechanics In classical Hamiltonian mechanics, a system of particles is described in terms their positions q i and conjugate momenta p i. For each such system, there exists a scalar Hamiltonian function H(q i, p i ) such that the classical equations of motion are given by: q i = H p i, ṗ i = H q i (Hamilton s equations of motion) Example: a single particle of mass m in a conservative force field F (q) the Hamiltonian function is constructed from the corresponding scalar potential: H(q, p) = p2 2m + V (q), V (q) F (q) = q Note: Hamilton s equations are equivalent to Newton s equations: q = H(q,p) = p } p m ṗ = H(q,p) V (q) m q = F (q) (Newton s equations of motion) = q q Newton s equations are second-order differential equations Hamilton s equations are first-order differential equations the Hamiltonian function is not unique! Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties IACS / 28

4 Quantization of a particle in conservative force field The Hamiltonian formulation is more general than the Newtonian formulation: it is invariant to coordinate transformations it provides a uniform description of matter and field it constitutes the springboard to quantum mechanics The Hamiltonian function (the total energy) of a particle in a conservative force field: H(q, p) = p2 2m + V (q) Standard rule for quantization (in Cartesian coordinates): carry out the substitutions p i, H i t multiply the resulting expression by the wave function Ψ(q) from the right: i Ψ(q) ] = [ 2 t 2m 2 + V (q) Ψ(q) This approach is sufficient for a treatment of electrons in an electrostatic field it is insufficient for nonconservative systems it is therefore inappropriate for systems in a general electromagnetic field Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties IACS / 28

5 The Lorentz force and Maxwell s equations In the presence of an electric field E and a magnetic field (magnetic induction) B, a classical particle of charge z experiences the Lorentz force: F = z (E + v B) since this force depends on the velocity v of the particle, it is not conservative The electric and magnetic fields E and B satisfy Maxwell s equations ( ): Note: E = ρ/ε B ε µ E/ t = µ J B = E + B/ t = Coulomb s law Ampère s law with Maxwell s correction Faraday s law of induction when the charge and current densities ρ(r, t) and J(r, t) are known, Maxwell s equations can be solved for E(r, t) and B(r, t) on the other hand, since the charges (particles) are driven by the Lorentz force, ρ(r, t) and J(r, t) are functions of E(r, t) and B(r, t) We shall consider the motion of particles in a given (fixed) electromagnetic field. Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties IACS / 28

6 Scalar and vector potentials The second, homogeneous pair of Maxwell s equations involves only E and B: 1 Eq. (1) is satisfied by introducing the vector potential A: B = (1) E + B t = (2) B = B = A vector potential (3) 2 inserting Eq. (3) in Eq. (2) and introducing a scalar potential φ, we obtain ( E + A ) = E + A t t = φ scalar potential The second pair of Maxwell s equations is thus automatically satisfied by writing E = φ A t B = A The potentials (φ, A) contain four rather than six components as in (E, B). They are obtained by solving the first, inhomogeneous pair of Maxwell s equations, which contains ρ and J. Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties IACS / 28

7 Gauge transformations The scalar and vector potentials φ and A are not unique. Consider the following transformation of the potentials: φ = φ f } t A f = f (q, t) gauge function of position and time = A + f This gauge transformation of the potentials does not affect the physical fields: E = φ A t = φ + f t A t f t B = (A + f ) = B + f = B = E We are free to choose f (q, t) so as to make φ and A satisfy additional conditions. In the Coulomb gauge, the gauge function is chosen to make vector potential divergenceless: A = Coulomb gauge Note: the Hamiltonian changes in the following manner upon a gauge transformation: H = H z f t However, the equations of motion are unaffected! Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties IACS / 28

8 The Hamiltonian in an electromagnetic field We must construct a Hamiltonian function such that Hamilton s equations are equivalent to Newton s equation with the Lorentz force: q i = H p i & ṗ i = H q i ma = z (E + v B) To this end, we introduce scalar and vector potentials φ and A such that E = φ A t, B = A In terms of these potentials, the classical Hamiltonian function becomes H = π2 + zφ, π = p za kinetic momentum 2m Quantization is then accomplished in the usual manner, by the substitutions p i, H i t The time-dependent Schrödinger equation for a particle in an electromagnetic field: i Ψ t = 1 ( i za) ( i za) Ψ + zφ Ψ 2m Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties IACS / 28

9 Electron spin The nonrelativistic Hamiltonian for an electron in an electromagnetic field is then given by: H = π2 2m eφ, π = i + ea However, this description ignores a fundamental property of the electron: spin. Spin was introduced by Pauli in 1927, to fit experimental observations: (σ π)2 π2 H = eφ = 2m 2m + e 2m B σ eφ where σ contains three operators, represented by the two-by-two Pauli spin matrices ( 1 σ x = 1 ) ( i, σ y = i ) ( 1, σ z = 1 ) The Schrödinger equation now becomes a two-component equation: ( ) π 2 e eφ + 2m 2m Bz e (Ψα ) ( ) (Bx iby ) 2m Ψα e 2m (Bx + iby ) π 2 = E e eφ 2m 2m Bz Ψ β Ψ β Note: the two components are only coupled in the presence of an external magnetic field Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties IACS / 28

10 Spin and relativity The introduction of spin by Pauli in 1927 may appear somewhat ad hoc By contrast, spin arises naturally from Dirac s relativistic treatment in 1928 is spin a relativistic effect? However, reduction of Dirac s equation to nonrelativistic form yields the Hamiltonian H = (σ π)2 2m π2 eφ = 2m + e 2m B σ eφ π2 2m eφ in this sense, spin is not a relativistic property of the electron but we note that, in the nonrelativistic limit, all magnetic fields disappear... We interpret σ by associating an intrinsic angular momentum (spin) with the electron: s = σ/2 Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties IACS / 28

11 Molecular Hamiltonian The nonrelativistic Hamiltonian for an electron in an electromagnetic field is therefore H = π2 2m + e B s eφ, π = p + ea, p = i m expanding π 2 and assuming the Coulomb gauge A =, we obtain π 2 Ψ = (p + ea) (p + ea) Ψ = p 2 Ψ + ep AΨ + ea pψ + e 2 A 2 Ψ = p 2 Ψ + e(p A)Ψ + 2eA pψ + e 2 A 2 Ψ = ( p 2 + 2eA p + e 2 A 2) Ψ in molecules, the dominant electromagnetic contribution is from the nuclear charges: φ = 1 Z K e 4πɛ K + φ r ext K Summing over all electrons and adding pairwise Coulomb interactions, we obtain H = 1 2m p2 i e2 Z K + e2 r 1 ij 4πɛ i r Ki ik 4πɛ i>j + e A i p i + e B i s i e φ i m m i i i + e2 A 2 i 2m i zero-order Hamiltonian first-order Hamiltonian second-order Hamiltonian Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties IACS / 28

12 Magnetic perturbations In atomic units, the molecular Hamiltonian is given by H = H + A(r i ) p i + B(r i ) s i i i i }{{}}{{} orbital paramagnetic spin paramagnetic There are two kinds of magnetic perturbation operators: φ(r i ) + 1 A 2 (r i ) 2 i }{{} diamagnetic the paramagnetic operator is linear and may lower or raise the energy the diamagnetic operator is quadratic and always raises the energy There are two kinds of paramagnetic operators: the orbital paramagnetic operator couples the field to the electron s orbital motion the spin paramagnetic operator couples the field to the electron s spin In the study of magnetic properties, we are interested in two types of perturbations: uniform external magnetic field B, with vector potential A ext(r) = 1 2 B r leads to Zeeman interactions nuclear magnetic moments M K, with vector potential A nuc(r) = α 2 K M K r K r 3 K leads to hyperfine interactions where α 1/137 is the fine-structure constant Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties IACS / 28

13 Review We have studied the nonrelativistic electronic Hamiltonian: H = H + H (1) + H (2) = H + A (r) p + B (r) s A (r)2 where the vector potential in the Coulomb gauge is not unique but satisfies the relations A (r) = B (r), A (r) = We shall consider uniform external field and the nuclear magnetic moments: A (r) = 1 2 B r O, A K (r) = α 2 M K r K rk 3, First- and second-order Rayleigh Schrödinger perturbation theory gives: E (1) = A p + B s E (2) = 1 A 2 2 n A p + B s n n A p + B s E n E We shall first consider external magnetic fields and next nuclear magnetic fields Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties IACS / 28

14 Molecular Magnetic Properties Part 1: The electronic Hamiltonian Hamiltonian mechanics and quantization electromagnetic fields scalar and vector potentials electron spin Part 2: Molecules in an external magnetic field Hamiltonian in an external magnetic field gauge transformations and London orbitals magnetizabilities diamagnetism and paramagnetism Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties IACS / 28

15 Review We have studied the nonrelativistic electronic Hamiltonian: H = H + H (1) + H (2) = H + A (r) p + B (r) s A (r)2 where the vector potential in the Coulomb gauge is not unique but satisfies the relations A (r) = B (r), A (r) = We shall consider uniform external field and the nuclear magnetic moments: A (r) = 1 2 B r O, A K (r) = α 2 M K r K rk 3, First- and second-order Rayleigh Schrödinger perturbation theory gives: E (1) = A p + B s E (2) = 1 A 2 2 n A p + B s n n A p + B s E n E We shall first consider external magnetic fields and next nuclear magnetic fields Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties IACS / 28

16 Hamiltonian in a uniform magnetic field The nonrelativistic electronic Hamiltonian (implied summation over electrons): H = H + A (r) p + B (r) s A (r)2 The vector potential of the uniform (static) fields B is given by: B = A = const A O (r) = 1 2 B (r O) = 1 2 B r O note: the gauge origin O is arbitrary! The orbital paramagnetic interaction: A O (r) p = 1 2 B (r O) p = 1 2 B (r O) p = 1 2 B L O where we have introduced the angular momentum relative to the gauge origin: The diamagnetic interaction: L O = r O p 1 2 A2 (r) = 1 8 (B r O) (B r O ) = 1 8 [ B 2 r 2 O (B r O) 2] The electronic Hamiltonian in a uniform magnetic field depends on the gauge origin: H = H B L O + B s [ B 2 r 2 O (B r O) 2] a change of the origin is a gauge transformation Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties IACS / 28

17 Gauge transformation of the Schrödinger equation What is the effect of a gauge transformation on the wave function? Consider a general gauge transformation for the electron (atomic units): A = A + f, φ = φ f t It can be shown this represents a unitary transformation of H i / t: ( H i ) ( = exp ( if ) H i ) exp (if ) t t In order that the Schrödinger equation is still satisfied ( H i ) Ψ t ( H i t ) Ψ, the new wave function must undergo a compensating unitary transformation: Ψ = exp ( if ) Ψ All observable properties such as the electron density are then unaffected: ρ = (Ψ ) Ψ = [Ψ exp( if )] [exp( if )Ψ] = Ψ Ψ = ρ Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties IACS / 28

18 Gauge-origin transformations Different choices of gauge origin in the external vector potential are related by gauge transformations: A O (r) = 1 B (r O) 2 A G (r) = A O (r) A O (G) = A O (r) + f, f (r) = A O (G) r The exact wave function transforms accordingly and gives gauge-invariant results: Ψ exact G = exp [ if (r)] Ψ exact O = exp [ia O (G) r] Ψ exact O rapid oscillations Illustration: H 2 on the z axis in a magnetic field B =.2 a.u. in the y direction wave function with gauge origin at O = (,, ) (left) and G = (1,, ) (right) Wave function, ψ Gauge transformed wave function, ψ" Re(ψ) Im(ψ) ψ Re(ψ") Im(ψ") ψ" Space coordinate, x (along the bond) Space coordinate, x (along the bond) Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties IACS / 28

19 London atomic orbitals The exact wave function transforms in the following manner: Ψ exact G = exp [ i 1 2 B (G O) r] Ψ exact O this behaviour cannot easily be modeled by standard atomic orbitals Let us build this behaviour directly into the atomic orbitals: ω lm (r K, B, G) = exp [ i 1 2 B (G K) r] χ lm (r K ) χ lm (r K ) is a normal atomic orbital centered at K and quantum numbers lm ω lm (r K, B, G) is a field-dependent orbital at K with field B and gauge origin G Each AO now responds in a physically sound manner to an applied magnetic field indeed, all AOs are now correct to first order in B, for any gauge origin G the calculations become rigorously gauge-origin independent uniform (good) quality follows, independent of molecule size These are the London orbitals after Fritz London (1937) Questions: also known as GIAOs (gauge-origin independent AOs or gauge-origin including AOs) are London orbitals needed in atoms? why not attach the phase factor to the total wave function instead? Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties IACS / 28

20 London orbitals Let us consider the FCI dissociation of H 2 in a magnetic field full lines: with London atomic orbitals dashed lines: without London atomic orbitals 1..5 B B B. Without London orbitals, the FCI method is not size extensive in magnetic fields Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties IACS / 28

21 Molecular magnetic moments and magnetizabilities Expand the molecular electronic energy in the external magnetic field B: E (B) = E B T M perm 1 2 BT ξb + We have here introduced the permanent magnetic moment and the magnetizability: M perm = de db permanent magnetic moment B= describes the first-order change in the energy but vanishes for closed-shell systems ξ = d2 E db 2 molecular a) magnetizabilityb) x B= describes the second-order energy and the first-order induced magnetic 1 moment.6 The permanent magnetic moment vanishes for closed-shell systems:.4 c.c. ˆΩ.2 imaginary c.c. The magnetizability describes the curvature at zero magnetic field: c) d) a) b) x left: diamagnetic dependence on the field (ξ < ); right: paramagnetic dependence on the field (ξ > ) c) Trygve Helgaker (CTCC, University of Oslo) d) Magnetic Properties.2 IACS / 28.2

22 Basis-set convergence of Hartree Fock magnetizabilities London orbitals are correct to first-order in the external magnetic field For this reason, basis-set convergence is usually improved RHF magnetizabilities of benzene: basis set χ xx χ yy χ zz London STO-3G G cc-pvdz aug-cc-pvdz origin CM STO-3G G cc-pvdz aug-cc-pvdz origin H STO-3G G cc-pvdz aug-cc-pvdz Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties IACS / 28

23 Normal distributions of errors for magnetizabilities Normal distributions of magnetizability errors for 27 molecules in the aug-cc-pcvqz basis relative to CCSD(T)/aug-cc-pCV[TQ]Z values (Lutnæs et al., JCP 131, (29)) Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties IACS / 28

24 Mean absolute errors for magnetizabilities Mean relative errors (MREs, %) in magnetizabilities of 27 molecules relative to the CCSD(T)/aug-cc-pCV[TQ]Z values. The DFT results are grouped by functional type. The heights of the bars correspond to the largest MRE in each category. (Lutnæs et al., JCP 131, (29)) Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties IACS / 28

25 Molecular diamagnetism and paramagnetism The Hamiltonian has paramagnetic and diamagnetic parts: H = H BLz + Bsz B2 (x 2 + y 2 ) para- and diamagnetic parts Second-order perturbation theory gives the magnetizability expression: ξ zz = d2 E db 2 = 1 4 x 2 + y L z n n L z a) dia- andb) paramagnetic parts 2 x 1 E n n E 3 14 Most closed-shell molecules are diamagnetic with.8 ξ zz < (below left) their energy increases in an applied magnetic field the induced currents oppose the field Some closed-shell systems are paramagnetic with ξ zz > (below right) their energy decreases in a magnetic field relaxation of the wave function lowers the energy, dominating the diamagnetic term RHF calculations of the field dependence of the energy for two closed-shell systems: a) c) b) 14 x 1 3 d) Trygve Helgaker (CTCC, University of c) Oslo) Magnetic Properties d) IACS / 28

26 Closed-shell paramagnetism explained The Hamiltonian has a linear and quadratic dependence on the magnetic field: H = H BLz + Bsz B2 (x 2 + y 2 ) linear and quadratic B terms Consider the effect of B on the ground state and on the first excited state: the quadratic term raises the energy of both states, creating diabatic energy curves depending on the states, diabatic curve crossings may occur at a given field strength the linear term couples the two states, creating adiabatic states and avoiding crossings A sufficiently strong coupling gives a paramagnetic ground-state with a double minimum Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties IACS / 28

27 C 2 in a perpendicular magnetic field All systems become diamagnetic in sufficiently strong fields: The transition occurs at a characteristic stabilizing critical field strength B c B c.1 for C 2 (ring conformation) above B c is inversely proportional to the area of the molecule normal to the field we estimate that B c should be observable for C 72 H 72 We may in principle separate such molecules by applying a field gradient Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties IACS / 28

28 Induced electron rotation The magnetic field induces a rotation of the electrons about the field direction: the derivative of the energy with the field is half the kinetic angular-momentum operator E (B) = 1 2 r π, π = p + A Paramagnetic closed-shell molecules (here BH): E(B x ) Energy Energy!25.11!25.11!25.12!25.12!25.13!25.13 E(B x )!25.14!25.14 L x (B x ) L (B ) x x!25.15!25.15!25.16! Angular momentum 1 Angular momentum 1.5.5!.5! L x / / r r!! C C x nuc nuc.4.4 Nuclear shielding Nuclear shielding in Boron Boron Hydrog Hydrogen.2.2! there is no rotation at the field-free energy maximum: B = the onset of paramagnetic rotation Orbital (against energies the field) reduces the energy HOMO!LUMO for B > gap the strongest paramagnetic rotation occurs at the energy inflexion.5 point the rotation comes to a halt.1 at the stabilizing field strength: B = B c the onset of diamagnetic rotation in the opposite direction increases.48 the energy for B > B c.46 Diamagnetic closed-shell molecules: induced diamagnetic rotation!.1 always increases LUMOthe energy HOMO!(B x )! Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties.4 IACS / 28! gap (B x ).44 "(B x ) Singlet excitation en

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