Molecules in Magnetic Fields

Transcription

1 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) 2017 Torre Normanna, Sicily, Italy September 10 23, 2017 Trygve Helgaker (University of Oslo) Molecules in Magnetic Fields ESQC / 25

2 Sections 1 Electronic Hamiltonian 2 London Orbitals 3 Paramagnetism and diamagnetism Trygve Helgaker (University of Oslo) Molecules in Magnetic Fields ESQC / 25

3 Electronic Hamiltonian Section 1 Electronic Hamiltonian Trygve Helgaker (University of Oslo) Molecules in Magnetic Fields ESQC / 25

4 Electronic Hamiltonian Particle in a Conservative Force Field 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 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 note: the Hamiltonian H is not unique! (Hamilton s equations) Example: a single particle of mass m in a conservative force field F (q) the Hamiltonian is constructed from the corresponding scalar potential: H(q, p) = p2 2m + V (q), V (q) F (q) = q Hamilton s equations of motion 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) = q q Hamilton s equations are first-order differential equations Newton s are second-order Trygve Helgaker (University of Oslo) Molecules in Magnetic Fields ESQC / 25

5 Electronic Hamiltonian Particle in a Conservative Force Field Quantization of a Particle in a 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 (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 operator 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 (University of Oslo) Molecules in Magnetic Fields ESQC / 25

6 Electronic Hamiltonian Particle in a Lorentz Force Field 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(r, t) and B(r, t) satisfy Maxwell s equations ( ): E = ρ/ε 0 Coulomb s law B ε 0 µ 0 E/ t = µ 0 J Ampère s law with Maxwell s correction B = 0 E + B/ t = 0 Faraday s law of induction where ρ(r, t) and J(r, t) are the charge and current densities, respectively Note: when ρ and J are known, Maxwell s equations can be solved for E and B but the particles are driven by the Lorentz force, so ρ and J are functions of E and B We here consider the motion of particles in a given (fixed) electromagnetic field Trygve Helgaker (University of Oslo) Molecules in Magnetic Fields ESQC / 25

7 Electronic Hamiltonian Particle in a Lorentz Force Field 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 = 0 (1) E + B t = 0 (2) B = 0 = B = A vector potential (3) 2 inserting Eq. (3) in Eq. (2) and introducing a scalar potential φ, we obtain ( E + A ) = 0 = 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). 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 / 25

8 Electronic Hamiltonian Particle in a Lorentz Force Field Gauge Transformations Consider the following gauge transformation of the potentials: φ = φ f } t A with f = f (q, t) gauge function of position and time = A + f Such a transformation of the potentials does not affect the physical fields: E = φ A t = φ + f t A t f t B = A = (A + f ) = B + f = B Conclusion: the scalar and vector potentials φ and A are not unique we are free to choose f (q, t) to make the potentials satisfy additional conditions typically, we require the vector potential to be divergenceless: A = 0 = (A + f ) = 0 = 2 f = A = E Coulomb gauge We shall always assume that the vector potential satisfies the Coulomb gauge: A = B, A = 0 Coulomb gauge note: A is still not uniquely determined, the following transformation being allowed: A = A + f, 2 f = 0 Trygve Helgaker (University of Oslo) Molecules in Magnetic Fields ESQC / 25

9 Electronic Hamiltonian Particle in a Lorentz Force Field 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 (University of Oslo) Molecules in Magnetic Fields ESQC / 25

10 Electronic Hamiltonian Electron Spin 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 ( 0 1 σ x = 1 0 ) ( 0 i, σ y = i 0 ) ( 1 0, σ z = 0 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 coupled only in the presence of an external magnetic field Trygve Helgaker (University of Oslo) Molecules in Magnetic Fields ESQC / 25

11 Electronic Hamiltonian Electron Spin 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 on the other hand, 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 (University of Oslo) Molecules in Magnetic Fields ESQC / 25

12 Electronic Hamiltonian Molecular Electronic Hamiltonian Molecular Electronic 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 = 0, 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 0 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 0 r Ki ik 4πɛ 0 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 (University of Oslo) Molecules in Magnetic Fields ESQC / 25

13 Electronic Hamiltonian Molecular Electronic Hamiltonian Magnetic Perturbations In atomic units, the molecular Hamiltonian is given by H = H 0 + 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 (University of Oslo) Molecules in Magnetic Fields ESQC / 25

14 London Orbitals Section 2 London Orbitals Trygve Helgaker (University of Oslo) Molecules in Magnetic Fields ESQC / 25

15 London Orbitals Hamiltonian in a Uniform Magnetic Field The nonrelativistic electronic Hamiltonian (implied summation over electrons): H = H 0 + A (r) p + B (r) s A (r)2 The vector potential of the uniform field 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 becomes: 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 becomes: L O = r O p 1 2 A2 O (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] as we shall see, a change of the origin is a gauge transformation Trygve Helgaker (University of Oslo) Molecules in Magnetic Fields ESQC / 25

16 London Orbitals Gauge-Origin Transformations Gauge Transformation of 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 that 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 wave function undergoes 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 (University of Oslo) Molecules in Magnetic Fields ESQC / 25

17 London Orbitals Gauge-Origin Transformations 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 = 0.2 a.u. in the y direction wave function with gauge origin at O = (0, 0, 0) (left) and G = (100, 0, 0) (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 (University of Oslo) Molecules in Magnetic Fields ESQC / 25

18 London Orbitals London Orbitals London 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 modelled 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 centred 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 (University of Oslo) Molecules in Magnetic Fields ESQC / 25

19 London Orbitals London Orbitals Dissociation With and Without 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 B B B 0.0 Without London orbitals, the FCI method is not size extensive in magnetic fields Trygve Helgaker (University of Oslo) Molecules in Magnetic Fields ESQC / 25

20 Paramagnetism and diamagnetism Section 3 Paramagnetism and diamagnetism Trygve Helgaker (University of Oslo) Molecules in Magnetic Fields ESQC / 25

21 Paramagnetism and diamagnetism Paramagnetism Hamiltonian for a molecule in a uniform magnetic field in the z direction: H = H BLz + Bsz B2 (x 2 + y 2 ) a paramagnetic, linear dependence on the magnetic field a diamagnetic, quadratic dependence on the magnetic field The linear paramagnetic Zeeman terms are easily understood: the angular momenta L z and s z set up a magnetic moment: m z = 1 Lz sz 2 this magnetic moment interacts with the field B in a dipolar fashion: Bm z = 1 BLz + Bsz 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 Trygve Helgaker (University of Oslo) Molecules in Magnetic Fields ESQC / 25

22 Paramagnetism and diamagnetism Diamagnetism Hamiltonian for a molecule in a uniform magnetic field in the z direction: H = H BLz + Bsz B2 (x 2 + y 2 ) 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(x 2 + y 2 ) 3 this induced magnetic moment interacts with B, raising the energy quadratically 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 Trygve Helgaker (University of Oslo) Molecules in Magnetic Fields ESQC / 25

23 Paramagnetism and diamagnetism Open-shell systems Open-shell systems the 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 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) Trygve Helgaker (University of Oslo) Molecules in Magnetic Fields ESQC / 25

24 Paramagnetism and diamagnetism Open-shell systems 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 Trygve Helgaker (University of Oslo) Molecules in Magnetic Fields ESQC / 25

25 Paramagnetism and diamagnetism Closed-shell systems 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) Trygve Helgaker (University of Oslo) Molecules in Magnetic Fields ESQC / 25