Time-independent molecular properties

Time-independent molecular properties 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) Time-independent molecular properties ESQC 2017 1 / 51

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 Trygve Helgaker (University of Oslo) Time-independent molecular properties ESQC 2017 2 / 51

Introduction Section 1 Introduction Trygve Helgaker (University of Oslo) Time-independent molecular properties ESQC 2017 3 / 51

Introduction Table of Contents 1 Introduction 2 Energy Functions Variational and nonvariational wave functions 3 Derivatives for Variational Wave Functions Molecular gradients Molecular Hessians 4 Derivatives for Nonvariational Wave Functions Nonvariational Wave Functions Lagrangian method 5 Examples of derivatives Hartree Fock Molecular Gradient FCI Lagrangian and Derivatives 6 Derivatives in second quantization Second-quantization Hamiltonian Molecular gradients 7 Geometrical Properties Bond Distances Harmonic and Anharmonic Constants Trygve Helgaker (University of Oslo) Time-independent molecular properties ESQC 2017 4 / 51

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 (University of Oslo) Time-independent molecular properties ESQC 2017 5 / 51

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 (University of Oslo) Time-independent molecular properties ESQC 2017 6 / 51

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 (University of Oslo) Time-independent molecular properties ESQC 2017 7 / 51

Energy Functions Section 2 Energy Functions Trygve Helgaker (University of Oslo) Time-independent molecular properties ESQC 2017 8 / 51

Energy Functions Table of Contents 1 Introduction 2 Energy Functions Variational and nonvariational wave functions 3 Derivatives for Variational Wave Functions Molecular gradients Molecular Hessians 4 Derivatives for Nonvariational Wave Functions Nonvariational Wave Functions Lagrangian method 5 Examples of derivatives Hartree Fock Molecular Gradient FCI Lagrangian and Derivatives 6 Derivatives in second quantization Second-quantization Hamiltonian Molecular gradients 7 Geometrical Properties Bond Distances Harmonic and Anharmonic Constants Trygve Helgaker (University of Oslo) Time-independent molecular properties ESQC 2017 9 / 51

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 (University of Oslo) Time-independent molecular properties ESQC 2017 10 / 51

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 (University of Oslo) Time-independent molecular properties ESQC 2017 11 / 51

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 wave functions whose energy does not fulfil the stationary 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 (University of Oslo) Time-independent molecular properties ESQC 2017 12 / 51

Derivatives for Variational Wave Functions Section 3 Derivatives for Variational Wave Functions Trygve Helgaker (University of Oslo) Time-independent molecular properties ESQC 2017 13 / 51

Derivatives for Variational Wave Functions Table of Contents 1 Introduction 2 Energy Functions Variational and nonvariational wave functions 3 Derivatives for Variational Wave Functions Molecular gradients Molecular Hessians 4 Derivatives for Nonvariational Wave Functions Nonvariational Wave Functions Lagrangian method 5 Examples of derivatives Hartree Fock Molecular Gradient FCI Lagrangian and Derivatives 6 Derivatives in second quantization Second-quantization Hamiltonian Molecular gradients 7 Geometrical Properties Bond Distances Harmonic and Anharmonic Constants Trygve Helgaker (University of Oslo) Time-independent molecular properties ESQC 2017 14 / 51

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 (University of Oslo) Time-independent molecular properties ESQC 2017 15 / 51

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 (University of Oslo) Time-independent molecular properties ESQC 2017 16 / 51

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 (University of Oslo) Time-independent molecular properties ESQC 2017 17 / 51

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 (University of Oslo) Time-independent molecular properties ESQC 2017 18 / 51

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 (University of Oslo) Time-independent molecular properties ESQC 2017 19 / 51

Derivatives for Nonvariational Wave Functions Section 4 Derivatives for Nonvariational Wave Functions Trygve Helgaker (University of Oslo) Time-independent molecular properties ESQC 2017 20 / 51

Derivatives for Nonvariational Wave Functions Table of Contents 1 Introduction 2 Energy Functions Variational and nonvariational wave functions 3 Derivatives for Variational Wave Functions Molecular gradients Molecular Hessians 4 Derivatives for Nonvariational Wave Functions Nonvariational Wave Functions Lagrangian method 5 Examples of derivatives Hartree Fock Molecular Gradient FCI Lagrangian and Derivatives 6 Derivatives in second quantization Second-quantization Hamiltonian Molecular gradients 7 Geometrical Properties Bond Distances Harmonic and Anharmonic Constants Trygve Helgaker (University of Oslo) Time-independent molecular properties ESQC 2017 21 / 51

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: We shall now consider its molecular gradient: E CI (x, c, κ) = 0 stationary c E CI (x, c, κ) 0 nonstationary κ 1 by straightforward differentiation of the CI energy 2 by differentiation of the CI Lagrangian In coupled-cluster theory, all parameters are nonvariationally determined Trygve Helgaker (University of Oslo) Time-independent molecular properties ESQC 2017 22 / 51

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 (University of Oslo) Time-independent molecular properties ESQC 2017 23 / 51

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 (University of Oslo) Time-independent molecular properties ESQC 2017 24 / 51

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 (University of Oslo) Time-independent molecular properties ESQC 2017 25 / 51

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 (University of Oslo) Time-independent molecular properties ESQC 2017 26 / 51

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 (University of Oslo) Time-independent molecular properties ESQC 2017 27 / 51

Examples of derivatives Section 5 Examples of derivatives Trygve Helgaker (University of Oslo) Time-independent molecular properties ESQC 2017 28 / 51

Examples of derivatives Table of Contents 1 Introduction 2 Energy Functions Variational and nonvariational wave functions 3 Derivatives for Variational Wave Functions Molecular gradients Molecular Hessians 4 Derivatives for Nonvariational Wave Functions Nonvariational Wave Functions Lagrangian method 5 Examples of derivatives Hartree Fock Molecular Gradient FCI Lagrangian and Derivatives 6 Derivatives in second quantization Second-quantization Hamiltonian Molecular gradients 7 Geometrical Properties Bond Distances Harmonic and Anharmonic Constants Trygve Helgaker (University of Oslo) Time-independent molecular properties ESQC 2017 29 / 51

Examples of derivatives Hartree Fock Molecular Gradient Trygve Helgaker (University of Oslo) Time-independent molecular properties ESQC 2017 30 / 51

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 where h pq(x) = ( φ p(r, x) 1 2 2 ) Z K K rk φ q(r, x) dr g pqrs(x) = φ p(r 1,x)φ q(r 1,x)φ r (r 2,x)φ s (r 2,x) r 12 dr 1 dr 2 the 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 Z K Z L R KL Trygve Helgaker (University of Oslo) Time-independent molecular properties ESQC 2017 30 / 51

Examples of derivatives Hartree Fock Molecular Gradient Hartree Fock Equations The HF energy is minimized subject to orthonormality constraints We therefore introduce the HF Lagrangian: L HF = E HF ε ( ) ij Sij δ ij ij S ij = φ i φ j = δij = 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 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 (University of Oslo) Time-independent molecular properties ESQC 2017 31 / 51

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 We then transform to the AO basis: de HF dx = µν Dµν AO hµν AO x + 1 2 µνρσ ijkl d ijkl g ijkl x ij dµνρσ AO gµνρσ AO x µν ε ij S ij x + Fnuc 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 (University of Oslo) Time-independent molecular properties ESQC 2017 32 / 51

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 (University of Oslo) Time-independent molecular properties ESQC 2017 33 / 51

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 (University of Oslo) Time-independent molecular properties ESQC 2017 34 / 51

Derivatives in second quantization Section 6 Derivatives in second quantization Trygve Helgaker (University of Oslo) Time-independent molecular properties ESQC 2017 35 / 51

Derivatives in second quantization Table of Contents 1 Introduction 2 Energy Functions Variational and nonvariational wave functions 3 Derivatives for Variational Wave Functions Molecular gradients Molecular Hessians 4 Derivatives for Nonvariational Wave Functions Nonvariational Wave Functions Lagrangian method 5 Examples of derivatives Hartree Fock Molecular Gradient FCI Lagrangian and Derivatives 6 Derivatives in second quantization Second-quantization Hamiltonian Molecular gradients 7 Geometrical Properties Bond Distances Harmonic and Anharmonic Constants Trygve Helgaker (University of Oslo) Time-independent molecular properties ESQC 2017 36 / 51

Derivatives in second quantization Second-quantization Hamiltonian Second-Quantization Hamiltonian In second quantization, the Hamiltonian operator is given by: H = pq h pqa pa q + pqrs g pqrsa pa r a sa q + h nuc 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, [a p, a q] + = δ pq 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 (University of Oslo) Time-independent molecular properties ESQC 2017 37 / 51

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 pa q + pqrs gpqrsa pa r a sa q + h nuc h pq = ψ p (r) h(r) ψ q(r) g pqrs = ψ p (r 1 )ψ r (r 2 ) r 1 12 ψq(r 1)ψ s(r 2 ) Trygve Helgaker (University of Oslo) Time-independent molecular properties ESQC 2017 38 / 51

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 (University of Oslo) Time-independent molecular properties ESQC 2017 39 / 51

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 h(1) pq 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 (University of Oslo) Time-independent molecular properties ESQC 2017 40 / 51

Geometrical Properties Section 7 Geometrical Properties Trygve Helgaker (University of Oslo) Time-independent molecular properties ESQC 2017 41 / 51

Geometrical Properties Table of Contents 1 Introduction 2 Energy Functions Variational and nonvariational wave functions 3 Derivatives for Variational Wave Functions Molecular gradients Molecular Hessians 4 Derivatives for Nonvariational Wave Functions Nonvariational Wave Functions Lagrangian method 5 Examples of derivatives Hartree Fock Molecular Gradient FCI Lagrangian and Derivatives 6 Derivatives in second quantization Second-quantization Hamiltonian Molecular gradients 7 Geometrical Properties Bond Distances Harmonic and Anharmonic Constants Trygve Helgaker (University of Oslo) Time-independent molecular properties ESQC 2017 42 / 51

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 (University of Oslo) Time-independent molecular properties ESQC 2017 43 / 51

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 (University of Oslo) Time-independent molecular properties ESQC 2017 44 / 51

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 (University of Oslo) Time-independent molecular properties ESQC 2017 45 / 51

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 (University of Oslo) Time-independent molecular properties ESQC 2017 46 / 51

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 (University of Oslo) Time-independent molecular properties ESQC 2017 47 / 51

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 (University of Oslo) Time-independent molecular properties ESQC 2017 48 / 51

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 (University of Oslo) Time-independent molecular properties ESQC 2017 49 / 51

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 (University of Oslo) Time-independent molecular properties ESQC 2017 50 / 51

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 (University of Oslo) Time-independent molecular properties ESQC 2017 51 / 51