Derivatives and Properties

Derivatives and Properties Trygve Helgaker Hylleraas Centre for Quantum Molecular Sciences, Department of Chemistry, University of Oslo, Norway Summer School: Modern Wavefunction Methods in Electronic Structure Theory Wissenschaftspark Gelsenkirchen, Gelsenkirchen, Germany September 30 October 5, 2018 Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 1 / 99

Sections 1 Introduction 2 Energy Functions 3 Derivatives for Variational Wave Functions 4 Derivatives for Nonvariational Wave Functions 5 Examples of derivatives 6 Derivatives in second quantization 7 Geometrical Properties 8 Electronic Hamiltonian 9 London Orbitals and Gauge-Origin Transformations 10 Zeeman and Hyperfine Interactions 11 First-Order Magnetic Properties 12 Molecular Magnetizabilities 13 High-Resolution NMR Spectra Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 2 / 99

Introduction Section 1 Introduction Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 3 / 99

Introduction Time-Independent Molecular Properties When a molecular system is perturbed, its total energy changes E(µ) = E (0) + E (1) µ + 1 2 E(2) µ 2 + The expansion coefficients are characteristic of the molecule and its quantum state we refer to these coefficients as molecular properties When the perturbation is static, the properties may be calculated by differentiation E (1) = de dµ µ=0 E (2) = d2 E dµ 2 µ=0 such properties are said to be time independent We do not here consider time-dependent molecular properties if periodic, these can be calculated in similar way using the quasi-energy Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 4 / 99

Introduction Examples of Derivatives Responses to geometrical perturbations forces and force constants spectroscopic constants Responses to external electromagnetic fields permanent and induced moments polarizabilities and magnetizabilities optical activity Responses to external magnetic fields and nuclear magnetic moments NMR shielding and indirect spin spin coupling constants EPR hyperfine coupling constants and g values Responses to nuclear quadrupole moments nuclear field gradients, quadrupole coupling constants Responses to molecular rotation spin rotation constants and molecular g values Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 5 / 99

Introduction Numerical vs. analytical differentiation Numerical differentiation (finite differences and polynomial fitting) often simple to implement (at least for real singlet perturbations) difficulties related to numerical accuracy and computational efficiency Analytical differentiation (derivatives calculated from analytical expressions) considerable programming effort required greater speed, precision, and convenience Implementations of analytical techniques first-order properties (dipole moments and gradients) second-order properties (polarizabilities and Hessians, NMR parameters) Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 6 / 99

Energy Functions Section 2 Energy Functions Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 7 / 99

Energy Functions Energy and energy functions Electronic Energy Function The electronic energy function contains the Hamiltonian and the wave function: It depends on two distinct sets of parameters: E(x, λ) = λ H(x) λ x: external (perturbation) parameters (geometry, external field) λ: electronic (wave-function) parameters (MOs, cluster amplitudes) The Hamiltonian (here in second quantization) H(x) = h pq(x)e pq + 1 g 2 pqrs(x)e pqrs + h nuc(x) pq pqrs depends explicitly on the external parameters: h pq(x) = φ p(x) h(x) φ q(x) The wave function λ depends implicitly on the external parameters λ(x). Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 8 / 99

Energy Functions Energy and energy functions Electronic Energy and its Derivatives The electronic energy E(x) is obtained by optimizing the energy function E(x, λ) with respect to λ for each value of x: E(x) = E(x, λ ) note: the optimization is not necessarily a variational minimization Our task is to calculate derivatives of E(x) with respect to x: de(x) E(x, λ ) E(x, λ) = + λ dx x λ }{{} λ=λ x λ=λ }{{} explicit dependence implicit dependence the implicit as well as explicit dependence must be accounted for The quantity λ/ x is the wave-function response it tells us how the electronic structure changes when the system is perturbed To proceed, we need to make a distinction between variationally determined wave functions nonvariationally determined wave functions Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 9 / 99

Energy Functions Variational and nonvariational wave functions Variational and Nonvariational Wave Functions Variational wave functions the optimized energy fulfils the stationary (variational) condition: E(x, λ) = 0 (for all x) λ the Hartree Fock energy in an unconstrained exponential parameterization HF = exp( κ) 0, κ = κ the energy of the full CI (FCI) wave function FCI = i c i i as an expectation value: E FCI (x, c) = 0 c Nonvariational wave functions the optimized energy does not fulfil the stationary (variational) condition: E(x, λ) 0 λ the Hartree Fock and Kohn Sham energies in a constrained LCAO parameterization (orthonormality) HF = 1 N! det φ 1, φ 2,... φ N, φ p(r; x) = µ Cµpχµ(r; x), φp φ q = δpq the truncated CI energy with respect to orbital rotations: E CI (x, c, κ) = 0, c E CI (x, c, κ) 0 κ Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 10 / 99

Derivatives for Variational Wave Functions Section 3 Derivatives for Variational Wave Functions Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 11 / 99

Derivatives for Variational Wave Functions Molecular gradients Molecular Gradients for Variational Wave Functions Applying the chain rule, we obtain for the total derivative of the energy: de(x) dx = E(x, λ) x + E(x, λ) λ λ x the first term accounts for the explicit dependence on x the last term accounts for the implicit dependence on x We now invoke the stationary condition: E(x, λ) λ The molecular gradient then simplifies to = 0 (zero electronic gradient for all x) de(x) dx = E(x, λ) x examples: HF/KS and MCSCF molecular gradients (exponential parameterization) For variational wave functions, we do not need the response of the wave function λ/ x to calculate the molecular gradient de/dx. Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 12 / 99

Derivatives for Variational Wave Functions Molecular gradients Hellmann Feynman Theorem Assume that the (stationary) energy is an expectation value: E(x, λ) = λ H(x) λ The gradient is then given by the expectation-value expression: de(x) dx = E(x, λ) x = λ H x λ Relationship to first-order perturbation theory: E (1) = 0 H (1) 0 the Hellmann Feynman theorem The Hellmann Feynman theorem was originally stated for geometrical distortions: de Z K r ik = λ dr K r 3 λ + Z I Z K R IK i ik R 3 I K IK Classical interpretation: integration over the force operator Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 13 / 99

Derivatives for Variational Wave Functions Molecular Hessians Molecular Hessians for Variational Wave Functions Differentiating the molecular gradient, we obtain the molecular Hessian: d 2 E(x) dx 2 = d dx E(x, λ) x ( = x + λ x ) E(x, λ) λ x = 2 E(x, λ) x 2 + 2 E(x, λ) λ x λ x we need the first-order response λ/ x to calculate the Hessian but we do not need the second-order response 2 λ/ x 2 for stationary energies To determine the response, we differentiate the stationary condition: E(x, λ) λ = 0 (all x) = d dx These are the first-order response equations: E(x, λ) λ = 2 E(x, λ) x λ 2 E λ λ }{{ 2 x = 2 E x λ }}{{} electronic right-hand Hessian side = 0 + 2 E(x, λ) λ λ 2 x = 0 Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 14 / 99

Derivatives for Variational Wave Functions Molecular Hessians Response Equations The molecular Hessian for stationary energies: The response equations: d 2 E dx 2 = 2 E x 2 + 2 E λ x λ x electronic Hessian 2 E λ 2 λ x = 2 E λ x perturbed electronic gradient the electronic Hessian is a Hermitian matrix, independent of the perturbation its dimensions are usually large and it cannot be constructed explicitly the response equations are typically solved by iterative techniques key step: multiplication of the Hessian with a trial vector Analogy with Hooke s law: force constant kx = F force the wave function relaxes by an amount proportional to the perturbation Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 15 / 99

Derivatives for Variational Wave Functions Molecular Hessians 2n + 1 Rule For molecular gradients and Hessians, we have the expressions de dx = E x d 2 E dx 2 = 2 E x 2 + 2 E λ x λ x In general, we have the 2n + 1 rule: zero-order response needed first-order response needed For variational wave functions, the derivatives of the wave function to order n determine the derivatives of the energy to order 2n + 1. Examples: wave-function responses needed to fourth order: energy E (0) E (1) E (2) E (3) E (4) wave function λ (0) λ (0) λ (0), λ (1) λ (0), λ (1) λ (0), λ (1), λ (2) Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 16 / 99

Derivatives for Nonvariational Wave Functions Section 4 Derivatives for Nonvariational Wave Functions Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 17 / 99

Derivatives for Nonvariational Wave Functions Nonvariational Wave Functions Nonvariational Wave Functions The 2n + 1 rule simplifies property evaluation for variational wave functions What about the nonvariational wave functions? any energy may be made stationary by Lagrange s method of undetermined multipliers the 2n + 1 rule is therefore of general interest Example: the CI energy the CI energy function is given by: E CI (x, c, κ) CI parameters c orbital-rotation parameters κ it is nonstationary with respect to the orbital-rotation parameters: E CI (x, c, κ) = 0 stationary c E CI (x, c, κ) 0 nonstationary κ We shall now consider its molecular gradient: 1 by differentiation of the CI energy 2 by differentiation of the CI Lagrangian In coupled-cluster theory, all parameters are nonvariationally determined Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 18 / 99

Derivatives for Nonvariational Wave Functions Nonvariational Wave Functions CI Molecular Gradients the Straightforward Way Straightforward differentiation of E CI (x, c, κ) gives the expression de CI dx = E CI x = E CI x + E CI c + E CI κ c x + E CI κ κ x κ x κ contribution does not vanish it appears that we need the first-order response of the orbitals The HF orbitals used in CI theory fulfil the following conditions at all geometries: E HF κ = 0 HF stationary conditions we obtain the orbital responses by differentiating this equation with respect to x: 2 E HF κ 2 κ x = 2 E HF x κ 1st-order response equations one such set of equations must be solved for each perturbation Calculated in this manner, the CI gradient becomes expensive Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 19 / 99

Derivatives for Nonvariational Wave Functions Lagrangian method Lagrange s Method of Undetermined Multipliers To calculate the CI energy, we minimize E CI with respect to c and κ: min E CI(x, c, κ) c,κ subject to the constraints Use Lagrange s method of undetermined multipliers: E HF (x, κ) κ construct the CI Lagrangian by adding these constraints with multipliers to the energy: ( ) EHF (x, κ) L CI (x, c, κ, κ) = E CI (x, c, κ) + κ 0 κ adjust the Lagrange multipliers κ such that the Lagrangian becomes stationary: = 0 L CI c L CI κ L CI κ = 0 = E CI c = 0 = E CI κ = 0 = E HF κ = 0 CI conditions + κ 2 E HF κ 2 = 0 linear set of equations for κ = 0 HF conditions note the duality between κ and κ Note that E CI = L CI when the Lagrangian is stationary we now have a stationary CI energy expression L CI Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 20 / 99

Derivatives for Nonvariational Wave Functions Lagrangian method CI Molecular Gradients the Easy Way The CI Lagrangian is given by L CI = E CI + κ E HF κ stationary with respect to all variables Since the Lagrangian is stationary, we may invoke the 2n + 1 rule: de CI dx = dl CI dx = L CI x zero-order response equations = E CI x κ 2 E HF κ 2 This result should be contrasted with the original expression first-order response equations de CI dx = E CI x 2 E HF κ 2 + κ 2 E HF κ x = E CI κ + E CI κ κ x κ x = 2 E HF κ x We have greatly reduced the number of response equations to be solved Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 21 / 99

Derivatives for Nonvariational Wave Functions Lagrangian method Lagrange s Method Summarized (1) Establish the energy function E(x, λ) and identify conditions on the variables (2) Set up the Lagrangian energy function: L(x, λ, λ) }{{} Lagrangian e(x, λ) = 0 = E(x, λ) }{{} + λ (e(x, λ) 0) }{{} energy function constraints (3) Satisfy the stationary conditions for the variables and their multipliers: L = e(x, λ) = 0 λ L λ = E λ + λ e λ = 0 condition for λ determines λ condition for λ determines λ note the duality between λ and λ! (4) Calculate derivatives from the stationary Lagrangian The Lagrangian approach is generally applicable: it gives the Hylleraas functional when applied to a perturbation expression it may be generalized to time-dependent properties Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 22 / 99

Derivatives for Nonvariational Wave Functions Lagrangian method 2n + 1 and 2n + 2 Rules For variational wave functions, we have the 2n + 1 rule: λ (n) determines the energy to order 2n + 1. The Lagrangian technique extends this rule to nonvariational wave functions For the new variables the multipliers the stronger 2n + 2 rule applies: λ (n) determines the energy to order 2n + 2. Responses required to order 10: E (n) 0 1 2 3 4 5 6 7 8 9 10 λ (k) 0 0 1 1 2 2 3 3 4 4 5 λ (k) 0 0 0 1 1 2 2 3 3 4 4 Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 23 / 99

Examples of derivatives Section 5 Examples of derivatives Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 24 / 99

Examples of derivatives Hartree Fock Molecular Gradient Hartree Fock Energy The MOs are expanded in atom-fixed AOs φ p(r; x) = Cµpχµ(r; x) µ The HF energy may be written in the general form E HF = pq Dpqhpq + 1 2 pqrs dpqrsgpqrs + K>L Z K Z L R KL where the one- and two-electron integrals are given by ( h pq(x) = φ p(r, x) 1 2 2 ) Z K K rk φ q(r, x) dr φp(r1, x)φ q(r 1, x)φ r (r 2, x)φ s(r 2, x) g pqrs(x) = dr 1 dr 2 r 12 note: all integrals depend explicitly on the geometry In closed-shell restricted HF (RHF) theory, the energy is given by E RHF = 2 h ii + ( ) 2giijj g ijji + i ij K>L summations over doubly occupied orbitals Z K Z L R KL Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 25 / 99

Examples of derivatives Hartree Fock Molecular Gradient Hartree Fock Equations The HF energy is minimized subject to orthonormality constraints S ij = φ i φ j = δij We therefore introduce the HF Lagrangian: L HF = E HF ε ( ) ij Sij δ ij ij = D ij h ij + 1 ij 2 d ijkl g ijkl ε ( ) ij Sij δ ij + ijkl ij K>L The stationary conditions on the Lagrangian become: Z K Z L R KL L HF ε ij = S ij δ ij = 0 Note: L HF C µi = E HF C µi ε S kl kl = 0 kl C µi the multiplier conditions are the orthonormality constraints the MO stationary conditions are the Roothaan Hall equations E HF C µi = ε S kl kl = F AO C = S AO Cε kl C µi Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 26 / 99

Examples of derivatives Hartree Fock Molecular Gradient Hartree Fock Molecular Gradient From the 2n + 1 rule, we obtain the RHF molecular gradient: de HF = dl HF = L HF = E HF dx dx x x S ij ε ij x ij In terms of MO integrals and density-matrix elements, we obtain the expression de HF dx = ij D ij h ij x + 1 2 ijkl d ijkl g ijkl x We then express the gradient in terms of AO integrals: ij ε ij S ij x + Fnuc de HF dx = µν Dµν AO hµν AO x + 1 2 µνρσ dµνρσ AO gµνρσ AO x µν ε AO Sµν AO µν + F nuc x density matrices transformed to AO basis derivative integrals added directly to gradient elements Important points: the gradient does not involve MO differentiation because of the 2n + 1 rule the time-consuming step is integral differentiation Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 27 / 99

Examples of derivatives FCI Lagrangian and Derivatives FCI Energy Consider a normalized CI wave function: c = n=0 cn n, ct c = 1, m n = δ mn The basis functions n are the normalized CI eigenstates of the unperturbed problem: m H n = δ mne n, E 0 E 1 E 2 We assume that the ground-state energy function depends on two external parameters: c H(x, y) c = cm m H(x, y) n cn, mn n We construct a variational CI Lagrangian: L(x, y, c, µ) = ( c m m H(x, y) n c n µ mn n The stationary conditions are given by c 2 n = 1 ) cn 2 1 L(x, y, c, µ) = 0 c n = 2 n H(x, y) c 2µc n = 0 = H(x, y)c = E 0 (x, y)c L(x, y, c, µ) = 0 µ = cn 2 1 = 0 n = ct c = 1 the first condition is the CI eigenvalue problem with ground-state energy E 0(x, y) = µ the second condition is the CI normalization condition Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 28 / 99

Examples of derivatives FCI Lagrangian and Derivatives FCI Molecular Gradient and Hessian Using the CI Lagrangian, we calculate CI energy derivative in the usual way: de dx = L x, d 2 E dxdy = 2 L x y + n 2 L x c n c n x, 2 L c n c n m c n x = 2 L x c m By inverting the electronic Hessian, we obtain the more compact expression: d 2 E dxdy = 2 L x y 2 [ L 2 ] 1 L 2 L x c mn m c m c n c n y We next evaluate the various partial derivatives at x = y = 0 where c = 0 : L x = 0 H x 0, 2 L x y = 0 2 H x y 0, 2 L c m c n = 2 m H E 0 n = 2(E n E 0 )δ mn 2 L = 2 n H x c x 0 n Inserted above, we recover Rayleigh Schrödinger perturbation theory to second order: de dx = 0 H d 2 E x 0, dxdy = 0 2 H x y 0 2 0 H x n n H y 0 E n n E 0 Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 29 / 99

Derivatives in second quantization Section 6 Derivatives in second quantization Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 30 / 99

Derivatives in second quantization Second-quantization Hamiltonian Second-Quantization Hamiltonian In second quantization, the Hamiltonian operator is given by: H = pq h pqa p aq + g pqrsa p a r asaq + hnuc pqrs h pq = φ p (r) h(r) φq(r) g pqrs = φ p (r 1)φ r (r 2) r 1 12 φq(r 1)φ s(r 2 ) Its construction assumes an orthonormal basis of MOs φ p: [a p, a q] + = 0, [a p, a q ]+ = 0, [ap, a q ]+ = δpq The MOs are expanded in AOs, which often depend explicitly on the perturbation such basis sets are said to be perturbation-dependent: φ p(r) = µ C pµ χ µ(r, x) we must make sure that the MOs remain orthonormal for all x this introduces complications as we take derivatives with respect to x Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 31 / 99

Derivatives in second quantization Second-quantization Hamiltonian MOs and Hamiltonian at Distorted geometries 1. Orthonormal MOs at the reference geometry: 2. Geometrical distortion x = x 0 + x: φ(x 0 ) = C (0) χ(x 0 ) S(x 0 ) = φ(x 0 ) φ (x 0 ) = I φ(x) = C (0) χ(x) S(x) = φ(x) φ (x) I note: this basis is nonorthogonal and not useful for setting up the Hamiltonian. 3. Orthonormalize the basis set (e.g., by Löwdin orthonormalization): ψ(x) = S 1/2 (x)φ(x) S(x) = S 1/2 (x)s(x)s 1/2 (x) = I 4. From these orthonormalized MOs (OMOs) ψ p, construct Hamiltonian in the usual manner H = h pq pqa p aq + pqrs gpqrsa p a r asaq + hnuc h pq = ψ p (r) h(r) ψq(r) g pqrs = ψ p (r 1)ψ r (r 2) r 1 12 ψq(r 1)ψ s(r 2 ) Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 32 / 99

Derivatives in second quantization Second-quantization Hamiltonian Hamiltonian at all Geometries The Hamiltonian is now well defined at all geometries: H(x) = h pq(x)e pq(x) + 1 g 2 pqrs(x)e pqrs(x) + h nuc(x) pq pqrs The OMO integrals are given by h pq(x) = h mn(x)[s 1/2 ] mp(x)[s 1/2 ] nq(x) mn in terms of the usual MO integrals h mn(x) = µν C mµc (0) nν (0) hµν AO (x), Smn(x) = C mµc (0) nν (0) Sµν AO (x) µν and similarly for the two-electron integrals. What about the geometry dependence of the excitation operators? this may be neglected when calculating derivatives since, for all geometries, [ ] a p(x), a q(x) = S pq(x) = δ pq + Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 33 / 99

Derivatives in second quantization Molecular gradients HF Molecular Gradients in Second Quantization The molecular gradient now follows from the Hellmann Feynman theorem: E (1) = 0 H (1) 0 = D pq h (1) pq + 1 d 2 pqrs g pqrs (1) + h nuc (1) pq pqrs We need the derivatives of the OMO integrals: h pq (1) = [ hmn(s 1/2 ) mp(s 1/2 ] (1) (1) ) nq = h pq 1 2 mn m S (1) pmh (0) mq 1 2 The gradient may therefore be written in the form E (1) = D pqh (1) pq + 1 d 2 pqrsg pqrs (1) F pqs pq (1) + h nuc, (1) pq pqrs pq m h pms (0) mq (1) where the generalized Fock matrix is given by: F pq = n D pnh qn + d pnrsg qnrs nrs For RHF theory, this result is equivalent to that derived in first quantization Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 34 / 99

Geometrical Properties Section 7 Geometrical Properties Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 35 / 99

Geometrical Properties Geometrical Derivatives In the Born Oppenheimer approximation, the nuclei move on the electronic potential-energy surface E(x), which is a function of the nuclear geometry: E(x) = E 0 + E (1) x + 1 2 E(2) x 2 + expansion around the reference geometry The derivatives of this surface are therefore important: The geometrical derivatives are E (1) = de dx E (2) = d2 E dx 2 molecular gradient molecular Hessian used for locating and characterizing critical points related to spectroscopic constants, vibrational frequencies, and intensities Usually, only a few terms are needed in the expansions in some cases low-order expansions are inadequate or useless Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 36 / 99

Geometrical Properties Uses of Geometrical Derivatives To explore molecular potential-energy surfaces (3N 6 dimensions) localization and characterization of stationary points localization of avoided crossings and conical intersections calculation of reaction paths and reaction-path Hamiltonians application to direct dynamics To calculate spectroscopic constants molecular structure quadratic force constants and harmonic frequencies cubic and quartic force constants; fundamental frequencies partition functions dipole gradients and vibrational infrared intensities polarizability gradients and Raman intensities Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 37 / 99

Geometrical Properties Bond distances Bond Distances Mean and mean abs. errors for 28 distances at the all-el. cc-pvx Z level (pm): 1-1 pvdz MP4 CCSD(T) MP2 CCSD MP3 DZ TZ QZ CCSD 1.2 0.6 0.8 CCSD(T) 1.7 0.2 0.2-2 pvtz pvqz CISD HF Bonds shorten with increasing basis: HF: DZ TZ 0.8 pm; TZ QZ 0.1 pm corr.: DZ TZ 1.6 pm; TZ QZ 0.1 0.2 pm Bonds lengthen with improvements in the N-electron model: singles < doubles < triples < There is considerable scope for error cancellation: CISD/DZ, MP3/DZ Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 38 / 99

Geometrical Properties Bond distances Bond Distances R e of BH, CO, N 2, HF, and F 2 (pm) CCSD(T) cc-pcvdz 2.1 CCSD(T) cc-pcvtz 0.2 CCSD(T) cc-pcvqz 0.1 CCSD(T) cc-pcv5z 0.1-6 6-6 6-6 6-6 6 CCSD cc-pcvdz 1.2 CCSD cc-pcvtz 0.6 CCSD cc-pcvqz 0.9 CCSD cc-pcv5z 1. -6 6-6 6-6 6-6 6 MP2 cc-pcvdz 1.5 MP2 cc-pcvtz 0.9 MP2 cc-pcvqz 0.8 MP2 cc-pcv5z 0.8-6 6-6 6-6 6-6 6 SCF cc-pcvdz 2.5 SCF cc-pcvtz 3.4 SCF cc-pcvqz 3.5 SCF cc-pcv5z 3.5-6 6-6 6-6 6-6 6 Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 39 / 99

Geometrical Properties Bond distances Contributions to Equilibrium Bond Distances (pm) RHF SD T Q 5 rel. adia. theory exp. err. HF 89.70 1.67 0.29 0.02 0.00 0.01 0.0 91.69 91.69 0.00 N 2 106.54 2.40 0.67 0.14 0.03 0.00 0.0 109.78 109.77 0.01 F 2 132.64 6.04 2.02 0.44 0.03 0.05 0.0 141.22 141.27 0.05 CO 110.18 1.87 0.75 0.04 0.00 0.00 0.0 112.84 112.84 0.00 We have agreement with experiment to within 0.01 pm except for F 2 Hartree Fock theory underestimates bond distances by up to 8.6 pm (for F 2 ) All correlation contributions are positive approximately linear convergence, slowest for F 2 triples contribute up to 2.0, quadruples up to 0.4, and quintuples 0.03 pm sextuples are needed for convergence to within 0.01 pm Relativistic corrections are small except for F 2 (0.05 pm) of the same magnitude and direction as the quintuples Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 40 / 99

Geometrical Properties Harmonic and anharmonic constants Harmonic Constants ω e of BH, CO, N 2, HF, and F 2 (cm 1 ) CCSD(T) cc-pcvdz 42 CCSD(T) cc-pcvtz 14 CCSD(T) cc-pcvqz 9 CCSD(T) cc-pcv5z 10-250 250-250 250-250 250-250 250 CCSD cc-pcvdz 34 CCSD cc-pcvtz 64 CCSD cc-pcvqz 71 CCSD cc-pcv5z 72-250 250-250 250-250 250-250 250 MP2 cc-pcvdz 68 MP2 cc-pcvtz 81 MP2 cc-pcvqz 73 MP2 cc-pcv5z 71-250 250-250 250-250 250-250 250 SCF cc-pcvdz 269 SCF cc-pcvtz 288 SCF cc-pcvqz 287 SCF cc-pcv5z 287-250 250-250 250-250 250-250 250 Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 41 / 99

Geometrical Properties Harmonic and anharmonic constants Contributions to Harmonic Frequencies ω e (cm 1 ) RHF SD T Q 5 rel. adia. theory exp. err. HF 4473.8 277.4 50.2 4.1 0.1 3.5 0.4 4138.9 4138.3 0.1 N 2 2730.3 275.8 72.4 18.8 3.9 1.4 0.0 2358.0 2358.6 0.6 F 2 1266.9 236.1 95.3 15.3 0.8 0.5 0.0 918.9 916.6 2.3 CO 2426.7 177.4 71.7 7.2 0.0 1.3 0.0 2169.1 2169.8 0.7 We have agreement with experiment to within 1 cm 1 except for F 2 Hartree Fock theory overestimates harmonic frequencies by up to 38% (in F 2 ). All correlation contributions are large and negative triples contribute up to 95 cm 1, quadruples 20 cm 1, and quintuples 4 cm 1 sextuples are sometimes needed for convergence to within 1 cm 1 The relativistic corrections are of the order of 1 cm 1 of the same magnitude and direction as the quadruples or quintuples Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 42 / 99

Geometrical Properties Harmonic and anharmonic constants Higher-Order Connected Contributions to ω e in N 2 (cm 1 ) There are substantial higher-order corrections: 371.9 84.6 4.1 13.8 23.5 4.7 0.8 HF CCSD FC CCSD T FC CCSD T CCSDT CCSDTQ CCSDTQ5 connected triples relaxation contributes 9.7 cm 1 (total triples 70.5 cm 1 ) connected quadruples contribute 18.8 cm 1 connected quintuples contribute 3.9 cm 1 Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 43 / 99

Geometrical Properties Harmonic and anharmonic constants Anharmonic Constants ω e x e of BH, CO, N 2, HF, and F 2 (cm 1 ) CCSD(T) cc-pcvdz 1 CCSD(T) cc-pcvtz 1 CCSD(T) cc-pcvqz 0 CCSD(T) cc-pcv5z 0-6 6-6 6-6 6-6 6 CCSD cc-pcvdz 2 CCSD cc-pcvtz 2 CCSD cc-pcvqz 2 CCSD cc-pcv5z 1-6 6-6 6-6 6-6 6 MP2 cc-pcvdz 3 MP2 cc-pcvtz 3 MP2 cc-pcvqz 3 MP2 cc-pcv5z 3-6 6-6 6-6 6-6 6 SCF cc-pcvdz 4 SCF cc-pcvtz 5 SCF cc-pcvqz 4 SCF cc-pcv5z 4-6 6-6 6-6 6-6 6 Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 44 / 99

Geometrical Properties Harmonic and anharmonic constants Sections 1 Introduction 2 Energy Functions 3 Derivatives for Variational Wave Functions 4 Derivatives for Nonvariational Wave Functions 5 Examples of derivatives 6 Derivatives in second quantization 7 Geometrical Properties 8 Electronic Hamiltonian 9 London Orbitals and Gauge-Origin Transformations 10 Zeeman and Hyperfine Interactions 11 First-Order Magnetic Properties 12 Molecular Magnetizabilities 13 High-Resolution NMR Spectra Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 45 / 99

Electronic Hamiltonian Section 8 Electronic Hamiltonian Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 46 / 99

Electronic Hamiltonian 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 (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 47 / 99

Electronic Hamiltonian 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 (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 48 / 99

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 (1861 1868): 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 (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 49 / 99

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 (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 50 / 99

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 (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 51 / 99

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 (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 52 / 99

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 (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 53 / 99

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 (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 54 / 99

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 (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 55 / 99

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 (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 56 / 99

London Orbitals and Gauge-Origin Transformations Section 9 London Orbitals and Gauge-Origin Transformations Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 57 / 99

London Orbitals and Gauge-Origin Transformations Hamiltonian in a Uniform Magnetic Field The nonrelativistic electronic Hamiltonian (implied summation over electrons): H = H 0 + A (r) p + B (r) s + 1 2 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 0 + 1 2 B L O + B s + 1 8 [ B 2 r 2 O (B r O) 2] as we shall see, a change of the origin is a gauge transformation Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 58 / 99

London Orbitals and Gauge-Origin Transformations 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 (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 59 / 99

London Orbitals and Gauge-Origin Transformations 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) 0.5 0.5 0.4 0.4 0.3 0.3 Wave function, ψ 0.2 0.1 0 0.1 0.2 Gauge transformed wave function, ψ" 0.2 0.1 0 0.1 0.2 0.3 0.4 Re(ψ) Im(ψ) ψ 2 0.3 0.4 Re(ψ") Im(ψ") ψ" 2 0.5 1.5 1 0.5 0 0.5 1 1.5 Space coordinate, x (along the bond) 0.5 1.5 1 0.5 0 0.5 1 1.5 Space coordinate, x (along the bond) Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 60 / 99

London Orbitals and Gauge-Origin Transformations London Orbitals London Orbitals The exact wave function transforms in the following manner: Ψ exact G (r) = exp [ i 1 2 B (G O) r] Ψ exact O (r) 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 (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 61 / 99

London Orbitals and Gauge-Origin Transformations 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 1.0 0.5 B 2.5 2 4 6 8 10 0.5 B 1.0 1.0 B 0.0 Without London orbitals, the FCI method is not size extensive in magnetic fields Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 62 / 99

Zeeman and Hyperfine Interactions Section 10 Zeeman and Hyperfine Interactions Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 63 / 99

Zeeman and Hyperfine Interactions Hamiltonian in Magnetic Field In atomic units, the molecular Hamiltonian is given by H = H 0 + A(r) p + B(r) s + 1 }{{}}{{} 2 A2 (r) }{{} orbital paramagnetic spin paramagnetic diamagnetic There are two kinds of magnetic perturbation operators: paramagnetic (may lower or raise energy) and diamagnetic (always raises energy) There are two kinds of paramagnetic operators: the orbital paramagnetic and spin paramagnetic First- and second-order Rayleigh Schrödinger perturbation theory gives: E (1) = 0 A p + B s 0 E (2) = 1 0 A 2 0 2 n 0 A p + B s n n A p + B s 0 E n E 0 In the study of magnetic properties, we are interested in two types of perturbations: externally applied uniform magnetic fields B fields generated internally by nuclear magnetic moments M K Both fields are weak well described by perturbation (response) theory Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 64 / 99

Zeeman and Hyperfine Interactions Paramagnetic Operators Orbital Paramagnetic Interactions: A p Vector potentials corresponding to uniform fields and nuclear magnetic moments: A O = 1 2 B r O, A K = α 2 M K r K rk 3, α 1/137 the external field is typically about 10 4 a.u. (NMR experiments) the nuclear vector potential is exceedingly small (about 10 8 a.u.) since: α 2 = c 2 10 4 a.u., M K = γ K I K 10 4 a.u. We obtain the following orbital paramagnetic operators: A O p = 1 2 B r O p = A K p = α 2 M K r K p r 3 K 1 2 B r O p = = α 2 M K r K p r 3 K 1 2 B L O orbital Zeeman = α 2 M K LK r 3 K orbital hyperfine interactions depend on angular momenta L O and L K relative to O and R K, respectively orbital hyperfine interaction expressed in terms of the paramagnetic spin orbit operator: A K p = M K h PSO K, hpso K = α 2 L K rk 3 These are imaginary singlet operators they have zero expectation values of closed-shell states they generate complex wave functions Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 65 / 99

Zeeman and Hyperfine Interactions Paramagnetic Operators Spin Paramagnetic Interactions: B s The spin interaction with the external uniform field B is trivial: B s spin Zeeman interaction should be compared with the orbital Zeeman interaction 1 2 B L O (different prefactor!) Taking the curl of A K, we obtain the nuclear magnetic field: B K = A K = 8πα2 3 δ(r K )M K + α 2 3r K (r K M K ) rk 2 M K rk 5 the first term is a contact interaction and contributes only at the nucleus the second term is a classical dipole field and contributes at a distance This magnetic field B K thus gives rise to two spin hyperfine interactions: B K s = M K (h FC K + hsd K ), h FC K h SD = 8πα2 3 δ(r K ) s Fermi contact (FC) K = α2 3r K rk T r K 2 I 3 r K 5 s spin dipole (SD) the FC operator contributes when the electron passes through the nucleus the SD operator is a classical dipolar interaction, decaying as r 3 K These are real triplet operators, which change the spin of the wave function they have zero expectation values of closed-shell states they couple closed-shell states to triplet states Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 66 / 99

Zeeman and Hyperfine Interactions Hamiltonian with Zeeman and Hyperfine Operators Perturbation Theory with Zeeman and Hyperfine Operators Hamiltonian with a uniform external field and with nuclear magnetic moments: H = H 0 + H (1) + H (2) = H 0 + H z (1) + H (1) hf + 1 2 A2 Zeeman interactions with the external magnetic field B: H (1) z = 1 2 B L O + B s 10 4 hyperfine interactions with the nuclear magnetic moments M K : H (1) hf = K M K h PSO K + K ( M K h FC K ) + hsd K 10 8 h PSO K = α 2 L K rk 3, h FC K = 8πα2 3 δ(r K ) s, h SD K = 3r α2 K rk T r K 2 I 3 rk 5 s Second-order Rayleigh Schrödinger perturbation theory: E (1) = 0 H z (1) + H (1) hf 0 E (2) = 1 0 A 2 0 H (1) z + H (1) 0 hf n n H (1) z + H (1) hf 0 2 E n n E 0 Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 67 / 99

Zeeman and Hyperfine Interactions Hamiltonian with Zeeman and Hyperfine Operators Zeeman and Hyperfine Interactions Zeeman SO SS, SO, OO SO PSO FC+SD hyperfine FC+SD PSO Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 68 / 99

Zeeman and Hyperfine Interactions Diamagnetic Operators Diamagnetic Operators: 1 2 A2 From A O and A K, we obtain three diamagnetic operators: A = A O + A K = A 2 = A O A O + 2A O A K + A K A K Their explicit forms and typical magnitudes (atomic units) are given by A O A O = 1 4 (B r O) (B r O ) 10 8 A O A K = α2 2 These are all real singlet operators (B r O ) (M K r K ) r 3 K A K A L = α 4 (M K r K ) (M L r L ) r 3 K r 3 L 10 12 10 16 their expectation values contribute to second-order magnetic properties they are all exceedingly small but nonetheless all observable Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 69 / 99

First-Order Magnetic Properties Section 11 First-Order Magnetic Properties Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 70 / 99

First-Order Magnetic Properties Taylor Expansion of Energy Expand the energy in the presence of an external magnetic field B and nuclear magnetic moments M K around zero field and zero moments: E (B, M) = E 0 + perm. magnetic moments {}}{ B T E (10) + + 1 2 BT E (20) B + 1 2 }{{} K magnetizability hyperfine coupling {}}{ K B T E (11) K M K } {{ } shieldings + 1 M T K E(01) K + 1 M T K 2 E(02) KL M L + KL }{{} spin spin couplings First-order terms vanish for closed-shell systems because of symmetry they shall be considered only briefly here Second-order terms are important for many molecular properties magnetizabilities nuclear shieldings constants of NMR nuclear spin spin coupling constants of NMR electronic g tensors of EPR (not dealt with here) Higher-order terms are negligible since the perturbations are tiny: 1) the magnetic induction B is weak ( 10 4 a.u.) 2) the nuclear magnetic moments M K couple weakly (µ 0 µ N 10 8 a.u.) Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 71 / 99

First-Order Magnetic Properties First-Order Magnetic Properties First-Order Molecular Properties The first-order properties are expectation values of H (1) Permanent magnetic moment M = 0 H z (1) 0 = 0 1 2 L O + s 0 permanent magnetic moment dominates the magnetism of molecules the molecule reorients itself and enters the field such molecules are therefore paramagnetic Hyperfine coupling constants A K = 0 H (1) hf 0 = 8πα2 3 0 δ (rk ) s 0 MK + measure spin density at the nucleus important in electron paramagnetic resonance (EPR) recall: there are three hyperfine mechanisms: FC, SD and PSO Note: there are no first-order Zeeman or hyperfine couplings for closed-shell molecules all expectation values vanish for imaginary operators and triplet operators: c.c. ˆΩ imaginary c.c. c.c. ˆΩ triplet c.c. 0 Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 72 / 99

First-Order Magnetic Properties First-Order Magnetic Properties Atoms in a Magnetic Field Lowest states of the fluorine atom (left) and neon atom (right) in a magnetic field ( H = H 0 + Bs z + 1 2 BLz + 1 8 B2 x 2 + y 2) the ground-state neon atom is closed shell, with zero initial slope all other states are open shells, with a nonzero initial slope one atomic unit magnetic field strength is B 0 = 2.35 10 5 T CCSD(T) calculations in uncontracted aug-cc-pcvqz basis (atomic units) Stopkowicz, Gauss, Lange, Tellgren, and Helgaker, JCP 143, 074110 (2015) Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 73 / 99

Molecular Magnetizabilities Section 12 Molecular Magnetizabilities Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 74 / 99

Molecular Magnetizabilities Molecular Magnetizabilities Expand the molecular electronic energy in the external magnetic field: E (B) = E 0 B T M 1 2 BT ξb + The magnetizability describes the second-order energy: 0.1 ξ = d2 E db 2 = 0 2 H B 2 0 + 2 0 H B n n H 0.08 B 0 0.06 E n n E 0 = 1 0 ro T 4 r OI 3 r O ro T 0 + 1 0 L O n n L T O 0 0.04 0.02 2 E }{{} n n E 0 } 0 0.1 0.05 {{ 0 0.05 } 0.1 diamagnetic term paramagnetic term The magnetizability describes the curvature at zero magnetic field: a) a) c) b) x 10 3 0 14 b) d) 14 12 10 8 6 4 2 x 10 3 0 0.1 0.05 0 0.1 0.08 0.06 0.04 0.02 0 0.1 0.05 0 0.05 0.1 0.002 12 0.004 0.00610 0.008 8 0.01 0.0126 0.014 4 0.016 0.0182 0.02 0 0.1 0.05 0 0.05 0.05 0.1 0.1 0 0.02 0.01 0 0.01 0.02 0.03 0 0.05 0.1 0.15 0.2 0.2 left: diamagnetic dependence on the field (ξ < 0); right: paramagnetic dependence on the field (ξ > 0) c) d) Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 75 / 99

Molecular Magnetizabilities Zeeman and Hyperfine Interactions Zeeman SO SS, SO, OO SO PSO FC+SD hyperfine FC+SD PSO Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 76 / 99

Molecular Magnetizabilities 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 8.1 8.1 23.0 6-31G 8.2 8.2 23.1 cc-pvdz 8.1 8.1 22.3 aug-cc-pvdz 8.0 8.0 22.4 origin CM STO-3G 35.8 35.8 48.1 6-31G 31.6 31.6 39.4 cc-pvdz 15.4 15.4 26.9 aug-cc-pvdz 9.9 9.9 25.2 origin H STO-3G 35.8 176.3 116.7 6-31G 31.6 144.8 88.0 cc-pvdz 15.4 48.0 41.6 aug-cc-pvdz 9.9 20.9 33.9 Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 77 / 99

Molecular Magnetizabilities 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, 144104 (2009)) Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 78 / 99

Molecular Magnetizabilities C 20 in a Perpendicular Magnetic Field All systems become diamagnetic in sufficiently strong fields: 756.685 756.690 756.695 756.700 756.705 756.710 Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 79 / 99

High-Resolution NMR Spectra Section 13 High-Resolution NMR Spectra Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 80 / 99

High-Resolution NMR Spectra NMR Spin Hamiltonian High-Resolution NMR Spin Hamiltonian Consider a molecule in an external magnetic field B along the z axis and with nuclear spins I K related to the nuclear magnetic moments M K as: M K = γ K I K 10 4 a.u. where γ K is the magnetogyric ratio of the nucleus. Assuming free molecular rotation, the nuclear magnetic energy levels can be reproduced by the following high-resolution NMR spin Hamiltonian: where we have introduced H NMR = γ K (1 σ K )BI K z + γ K γ L 2 K KL I K I L K K>L }{{}}{{} nuclear Zeeman interaction nuclear spin spin interaction the nuclear shielding constants σ K the (reduced) indirect nuclear spin spin coupling constants K KL This is an effective nuclear spin Hamiltonian: it reproduces NMR spectra without considering the electrons explicitly the spin parameters σ K and K KL are adjusted to fit the observed spectra we shall consider their evaluation from molecular electronic-structure theory Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 81 / 99

High-Resolution NMR Spectra NMR Spin Hamiltonian Simulated 200 MHz NMR spectra of Vinyllithium 12 C 2 H 3 6 Li Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 82 / 99

NMR Shielding Constants Section 14 NMR Shielding Constants Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 83 / 99

NMR Shielding Constants Nuclear Shielding Constants Nuclear Shielding Constants Expansion of closed-shell energy in an external field B and nuclear magnetic moments M K : Here E (11) K E (B, M) = E 0 + 1 2 BT E (20) B + 1 2 K B T E (11) K M K + 1 2 KL M T K E(02) KL M L + describes the coupling between the applied field and the nuclear moments: in the absence of electrons (i.e., in vacuum), this coupling is identical to I 3 : Hz nuc = B M K the purely nuclear Zeeman interaction K in the presence of electrons (i.e., in a molecule), the coupling is modified slightly: E (11) K = I 3 + σ K the nuclear shielding tensor Shielding constants arise from a hyperfine interaction between the electrons and the nuclei and Zeeman interaction between the electrons and the field they are of the order of α 2 5 10 5 and are measured in ppm The nuclear Zeeman interaction does not enter the electronic problem compare with the nuclear nuclear Coulomb repulsion Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 84 / 99

NMR Shielding Constants Nuclear Shielding Constants Zeeman and Hyperfine Interactions Zeeman SO SS, SO, OO SO PSO FC+SD hyperfine FC+SD PSO Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 85 / 99

NMR Shielding Constants Nuclear Shielding Constants Ramsey s Expression for Nuclear Shielding Tensors Ramsey s expression for nuclear shielding tensors of a closed-shell system: σ K = d2 E el = 0 2 H dbdm K B M 0 2 0 H B n n H M 0 K K E n n E 0 = α2 ro T 0 r K I 3 r O rk T 2 rk 3 0 α 0 LO n n r 3 2 K LT K 0 E n n E 0 }{{}}{{} diamagnetic term paramagnetic term The (usually) dominant diamagnetic term arises from differentiation of the operator: A O A K = 1 2 α2 r 3 K (B r O) (M K r K ) As for the magnetizability, there is no spin contribution for singlet states: S 0 0 singlet state For 1 S systems (closed-shell atoms), the paramagnetic term vanishes completely and the shielding is given by (assuming gauge origin at the nucleus): σ Lamb = 1 1 3 α2 S r 1 K 1 S Lamb formula Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 86 / 99

NMR Shielding Constants Nuclear Shielding Constants Benchmark Calculations of BH Shieldings (ppm) σ( 11 B) σ( 11 B) σ( 1 H) σ( 1 H) HF 261.3 690.1 24.21 14.15 MP2 220.7 629.9 24.12 14.24 CCSD 166.6 549.4 24.74 13.53 CCSD(T) 171.5 555.2 24.62 13.69 CCSDT 171.8 557.3 24.59 13.72 CCSDTQ 170.1 554.7 24.60 13.70 CISD 182.4 572.9 24.49 13.87 CISDT 191.7 587.0 24.35 14.06 CISDTQ 170.2 554.9 24.60 13.70 FCI 170.1 554.7 24.60 13.70 TZP+ basis, R BH = 123.24 pm, all electrons correlated J. Gauss and K. Ruud, Int. J. Quantum Chem. S29 (1995) 437 M. Kállay and J. Gauss, J. Chem. Phys. 120 (2004) 6841 Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 87 / 99

NMR Shielding Constants Nuclear Shielding Constants Coupled-Cluster Convergence of Shielding Constants in CO (ppm) CCSD CCSD(T) CCSDT CCSDTQ CCSDTQ5 FCI σ( 13 C) 32.23 35.91 35.66 36.10 36.14 36.15 σ( 13 C) 361.30 356.10 356.47 355.85 355.80 355.79 σ( 17 O) 13.93 13.03 13.16 12.81 12.91 12.91 σ( 17 O) 636.01 634.55 634.75 634.22 634.52 634.35 All calculations in the cc-pvdz basis and with a frozen core. Kállay and Gauss, J. Chem. Phys. 120 (2004) 6841. Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 88 / 99

NMR Shielding Constants Nuclear Shielding Constants Calculated and Experimental Shielding Constants (ppm) HF CAS MP2 CCSD CCSD(T) exp. HF F 413.6 419.6 424.2 418.1 418.6 410 ± 6 (300K) H 28.4 28.5 28.9 29.1 29.2 28.5 ± 0.2 (300K) H 2O O 328.1 335.3 346.1 336.9 337.9 323.6 ± 6 (300K) H 30.7 30.2 30.7 30.9 30.9 30.05 ± 0.02 NH 3 N 262.3 269.6 276.5 269.7 270.7 264.5 H 31.7 31.0 31.4 31.6 31.6 31.2 ± 1.0 CH 4 C 194.8 200.4 201.0 198.7 198.9 198.7 H 31.7 31.2 31.4 31.5 31.6 30.61 F 2 F 167.9 136.6 170.0 171.1 186.5 192.8 N 2 N 112.4 53.0 41.6 63.9 58.1 61.6 ± 0.2 (300K) CO C 25.5 8.2 10.6 0.8 5.6 3.0 ± 0.9 (eq) O 87.7 38.9 46.5 56.0 52.9 56.8 ± 6 (eq) For references and details, see Chem. Rev. 99 (1999) 293. for exp. CO and H 2 O values, see Wasylishen and Bryce, JCP 117 (2002) 10061 Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 89 / 99

NMR Shielding Constants Nuclear Shielding Constants MAE (Exp.) MAE (Emp. Eq.) NMR: Mean Absolute Errors Relative to Experiment Mean absolute errors relative to experimental (blue) and empirical equilibrium values (red) 30 25 MAE / ppm 20 15 10 5 MAE (Exp.) MAE (Emp. Eq.) 0 Kohn Sham calculations give shielding constants of uneven quality errors increase when vibrational corrections are applied Teale et al. JCP 138, 024111 (2013) Trygve Helgaker (Hylleraas, University of Oslo) Derivatives and Properties MWM Summer School 2018 90 / 99