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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 European Summer School in Quantum Chemistry (ESQC) 2019 Torre Normanna, Sicily, Italy September 8{21, 2019 Trygve Helgaker (University of Oslo) Molecules in Magnetic Fields ESQC 2019 1 / 25 Sections 1 Electronic Hamiltonian 2 London Orbitals 3 Paramagnetism and diamagnetism Trygve Helgaker (University of Oslo) Molecules in Magnetic Fields ESQC 2019 2 / 25 Electronic Hamiltonian Section 1 Electronic Hamiltonian Trygve Helgaker (University of Oslo) Molecules in Magnetic Fields ESQC 2019 3 / 25 Electronic Hamiltonian Particle in a Conservative Force Field Hamiltonian Mechanics I In classical Hamiltonian mechanics, a system of particles is described in terms their positions qi and conjugate momenta pi . I For each system, there exists a scalar Hamiltonian function H(qi ; pi ) such that the classical equations of motion are given by: @H @H q_i = ; p_i = (Hamilton's equations) @pi − @qi I note: the Hamiltonian H is not unique! I Example: a single particle of mass m in a conservative force field F (q) I the Hamiltonian is constructed from the corresponding scalar potential: p2 @V (q) H(q; p) = + V (q); F (q) = 2m − @q I Hamilton's equations of motion are equivalent to Newton's equations: @H(q;p) ) q_ = = p @p m = mq¨ = F (q) (Newton's equations) p_ = @H(q;p) = @V (q) ) − @q − @q I Hamilton's equations are first-order differential equations { Newton's are second-order Trygve Helgaker (University of Oslo) Molecules in Magnetic Fields ESQC 2019 4 / 25 Electronic Hamiltonian Particle in a Conservative Force Field Quantization of a Particle in a Conservative Force Field I The Hamiltonian formulation is more general than the Newtonian formulation: I it is invariant to coordinate transformations I it provides a uniform description of matter and field I it constitutes the springboard to quantum mechanics I The Hamiltonian function (total energy) of a particle in a conservative force field: p2 H(q; p) = + V (q) 2m I Standard rule for quantization (in Cartesian coordinates): I carry out the operator substitutions @ p i~r; H i~ ! − ! @t I multiply the resulting expression by the wave functionΨ( q) from the right: 2 @Ψ(q) ~ 2 i~ = + V (q) Ψ(q) @t − 2m r I This approach is sufficient for a treatment of electrons in an electrostatic field I it is insufficient for nonconservative systems I it is therefore inappropriate for systems in a general electromagnetic field Trygve Helgaker (University of Oslo) Molecules in Magnetic Fields ESQC 2019 5 / 25 Electronic Hamiltonian Particle in a Lorentz Force Field Lorentz Force and Maxwell's Equations I 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) × I since this force depends on the velocity v of the particle, it is not conservative I The electric and magnetic fields E(r; t) and B(r; t) satisfy Maxwell's equations (1861{1868): r E = ρ/ε Coulomb's law · 0 r B " µ @E=@t = µ J Amp`ere'slaw with Maxwell's correction × − 0 0 0 r B =0 · r E + @B=@t = 0 Faraday's law of induction × where ρ(r; t) and J(r; t) are the charge and current densities, respectively I Note: I when ρ and J are known, Maxwell's equations can be solved for E and B I but the particles are driven by the Lorentz force, so ρ and J are functions of E and B I We here consider the motion of particles in a given (fixed) electromagnetic field Trygve Helgaker (University of Oslo) Molecules in Magnetic Fields ESQC 2019 6 / 25 Electronic Hamiltonian Particle in a Lorentz Force Field Scalar and Vector Potentials I The second, homogeneous pair of Maxwell's equations involves only E and B: r · B =0 (1) @B r × E + = 0 (2) @t 1 Eq. (1) is satisfied by introducing the vector potential A: r B = 0= B = r A vector potential (3) · ) × 2 inserting Eq. (3) in Eq. (2) and introducing a scalar potential φ, we obtain @A @A r E + = 0 = E + = rφ scalar potential × @t ) @t − I The second pair of Maxwell's equations is thus automatically satisfied by writing @A E = −rφ − @t B = r × A I The potentials( φ, A) contain four rather than six components as in( E; B). I φ and A are obtained by solving the inhomogeneous pair of Maxwell's equations, containing ρ and J Trygve Helgaker (University of Oslo) Molecules in Magnetic Fields ESQC 2019 7 / 25 Electronic Hamiltonian Particle in a Lorentz Force Field Gauge Transformations I Consider the following gauge transformation of the potentials: @ φ0 = φ f − @t with f = f (q; t) gauge function of position and time A0 = A + rf I Such a transformation of the potentials does not affect the physical fields: @A0 @f @A @rf E0 = −rφ0 − = −rφ + r − − = E @t @t @t @t B0 = r × A0 = r × (A + rf ) = B + r × rf = B I Conclusion: the scalar and vector potentials φ and A are not unique I we are free to choose f (q; t) to make the potentials satisfy additional conditions I typically, we require the vector potential to be divergenceless: r A0 = 0= r (A + rf ) = 0= 2f = A Coulomb gauge · ) · )r −∇ · I We shall always assume that the vector potential satisfies the Coulomb gauge: r × A = B; r · A = 0 Coulomb gauge I note: A is still not uniquely determined, the following transformation being allowed: A0 = A + rf ; r2f = 0 Trygve Helgaker (University of Oslo) Molecules in Magnetic Fields ESQC 2019 8 / 25 Electronic Hamiltonian Particle in a Lorentz Force Field Hamiltonian in an Electromagnetic Field I We must construct a Hamiltonian function such that Hamilton's equations are equivalent to Newton's equation with the Lorentz force: @H @H q_i = &_ pi = ma = z (E + v B) @pi − @qi () × I To this end, we introduce scalar and vector potentials φ and A such that @A E = rφ ; B = r A − − @t × I In terms of these potentials, the classical Hamiltonian function becomes π2 H = + zφ, π = p zA kinetic momentum 2m − I Quantization is then accomplished in the usual manner, by the substitutions @ p i~r; H i~ ! − ! @t I The time-dependent Schr¨odingerequation for a particle in an electromagnetic field: @Ψ 1 i~ = ( i~r zA) ( i~r zA) Ψ + zφ Ψ @t 2m − − · − − Trygve Helgaker (University of Oslo) Molecules in Magnetic Fields ESQC 2019 9 / 25 Electronic Hamiltonian Electron Spin Electron Spin I The nonrelativistic Hamiltonian for an electron in an electromagnetic field is then given by: π2 H = eφ, π = i~r + eA 2m − − I However, this description ignores a fundamental property of the electron: spin. I Spin was introduced by Pauli in 1927, to fit experimental observations: (σ π)2 π2 e~ H = · eφ = + B σ eφ 2m − 2m 2m · − where σ contains three operators, represented by the two-by-two Pauli spin matrices 0 1 0 i 1 0 σ = ; σ = ; σ = x 1 0 y i −0 z 0 1 − I The Schr¨odingerequation now becomes a two-component equation: π2 e e ! eφ + ~ Bz ~ (Bx iBy ) Ψα Ψα 2m − 2m 2m − = E e π2 e Ψ Ψ ~ (Bx + iBy ) eφ ~ Bz β β 2m 2m − − 2m I Note: the two components are coupled only in the presence of an external magnetic field Trygve Helgaker (University of Oslo) Molecules in Magnetic Fields ESQC 2019 10 / 25 Electronic Hamiltonian Electron Spin Spin and Relativity I The introduction of spin by Pauli in 1927 may appear somewhat ad hoc I By contrast, spin arises naturally from Dirac's relativistic treatment in 1928 I is spin a relativistic effect? I However, reduction of Dirac's equation to nonrelativistic form yields the Hamiltonian (σ π)2 π2 e~ π2 H = · eφ = + B σ eφ = eφ 2m − 2m 2m · − 6 2m − I in this sense, spin is not a relativistic property of the electron I on the other hand, in the nonrelativistic limit, all magnetic fields disappear. I We interpret σ by associating an intrinsic angular momentum (spin) with the electron: s = ~σ=2 Trygve Helgaker (University of Oslo) Molecules in Magnetic Fields ESQC 2019 11 / 25 Electronic Hamiltonian Molecular Electronic Hamiltonian Molecular Electronic Hamiltonian I The nonrelativistic Hamiltonian for an electron in an electromagnetic field is therefore π2 e H = + B s eφ, π = p + eA; p = i~r 2m m · − − I expanding π2 and assuming the Coulomb gauge r A = 0, we obtain · π2Ψ=( p + eA) (p + eA)Ψ= p2Ψ + ep AΨ + eA pΨ + e2A2Ψ · · · = p2Ψ + e(p A)Ψ + 2eA pΨ + e2A2Ψ= p2 + 2eA p + e2A2 Ψ · · · I in molecules, the dominant electromagnetic contribution is from the nuclear charges: 1 P ZK e φ = + φext − 4π0 K rK I Summing over all electrons and adding pairwise Coulomb interactions, we obtain X 1 e2 X Z e2 X H = p2 K + r −1 zero-order Hamiltonian 2m i − 4π r 4π ij i 0 Ki iK 0 i>j e X e X X + Ai pi + Bi si e φi first-order Hamiltonian m · m · − i i i e2 X + A2 second-order Hamiltonian 2m i i Trygve Helgaker (University of Oslo) Molecules in Magnetic Fields ESQC 2019 12 / 25 Electronic Hamiltonian Molecular Electronic Hamiltonian Magnetic Perturbations I In atomic units, the molecular Hamiltonian is given by X X X 1 X 2 H = H0 + A(ri ) pi + B(ri ) si φ(ri ) + A (ri ) · · − 2 i i i i | {z } | {z } | {z } orbital paramagnetic spin paramagnetic diamagnetic I There are two kinds of magnetic perturbation operators: I the paramagnetic operator is linear and may lower or raise the energy I the diamagnetic operator is quadratic and always raises the energy I There are two kinds of paramagnetic operators: I the orbital paramagnetic operator couples the field to the electron's orbital motion I the spin paramagnetic operator couples the field to the electron's spin I In the study of magnetic properties, we are interested in two types of perturbations: I uniform external magnetic
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