Density functional theory in magnetic fields

Density functional theory in magnetic fields U. E. Ekström, T. Helgaker, S. Kvaal, E. Sagvolden, E. Tellgren, A. M. Teale Centre for Theoretical and Computational Chemistry (CTCC), Department of Chemistry, University of Oslo, Norway School of Chemistry, University of Nottingham, Nottingham, UK Electronic Structure Theory for Strongly Correlated Systems celebrating Per Åke Malmqvist s 6th birthday and his career in quantum chemistry Splendid Hotel La Torre, Mondello, Palermo, Italy, May 3 June 1, 212 T. Helgaker (CTCC, University of Oslo) Calculation of the universal density functional Malmqvist Celebration Palermo 212 1 / 23

Overview 1 Density functional theory (DFT) 2 The universal density functional and its calculation by Lieb maximization 3 Regularization of DFT 4 The adiabatic connection with examples 5 Molecules in magnetic fields 6 Magnetic-field DFT (BDFT) and current DFT (CDFT) Helgaker et al. (CTCC, University of Oslo) Overview Malmqvist Celebration Palermo 212 2 / 23

The universal density functional Consider the ground-state energy with interaction strength λ: E λ [v] = inf Ψ T + i v(r i ) + λw Ψ the Rayleigh Ritz variation principle Ψ We may summarize density-functional theory in two variation principles: ( E λ [v] = inf Fλ [ρ] + (v ρ) ) the Hohenberg Kohn variation principle (1964) ρ ( F λ [ρ] = sup Eλ [v] (v ρ) ) the Lieb variation principle (1983) v these are alternative attempts at sharpening the same inequality into an equality F λ [ρ] E λ [v] (v ρ) E λ [v] F λ [ρ] + (v ρ) Fenchel s inequality The Hohenberg Kohn variation principle underlies all applications of DFT in chemistry its success hinges on the accurate modelling of F λ [ρ] The Lieb variation principle provides a tool for studying F λ [ρ] numerically we have implemented the Lieb variation principle at various levels ab initio theory accurate numerical studies of F λ [ρ] are a guide to insight our motivation is to benchmark approximate F λ [ρ] and to help develop new ones Previous work: studies by Colonna and Savin (1999) for few-electron atoms work by Wu and Yang (23) on Lieb maximizations Helgaker et al. (CTCC, University of Oslo) Lieb s universal density functional Malmqvist Celebration Palermo 212 3 / 23

Conjugate functionals E[v] F [ρ] The ground-state energy E[v] is concave in v by the Rayleigh Ritz variation principle it can therefore be exactly represented by its convex conjugate F [ρ]: E[v] F [ρ] FΡ sup v EvvΡ FΡ FΡ Ρ min v max Ev Ev Ev inf Ρ FΡvΡ vρ vρ Both variation principles represent a Legendre Fenchel transformation the essential point is the concavity of E[v] rather than the details of the Schrödinger equation roughly speaking, the derivatives of E[v] are F [ρ] are each other s inverse functions the density functional F [ρ] is more complicated that E[v] Helgaker et al. (CTCC, University of Oslo) Lieb s universal density functional Malmqvist Celebration Palermo 212 4 / 23

Lieb maximization For a given density ρ(r) and level of theory E λ [v], we wish to calculate F λ [ρ] = max v ( Eλ [v] (v ρ) ) The potential is parameterized as suggested by Wu and Yang 23: v c(r) = v ext(r) + (1 λ)v ref (r) + ct gt(r) t the physical, external potential v ext(r) the Fermi Amaldi reference potential to ensure correct asymptotic behaviour an expansion in Gaussians g t(r) with coefficients c t The maximization may be carried out by Newton-type methods: because of concavity, all maxima are global maxima 5 1 iterations with the exact Hessian to gradient norm 1 6 1 15 iterations with an approximate noninteracting Hessian Hessian regularized by a singular-value decomposition with cutoff 1 6 possible (unphysical) oscillations in the potential do not affect F λ [ρ] Convergence is mostly unproblematic but difficulties may arise for λ from a (near) piecewise linearity in c t from a poor description of E λ [v c] for large c t All code is implemented in DALTON for HF, MP2, CCD, CCSD, and CCSD(T) Helgaker et al. (CTCC, University of Oslo) Lieb maximization Malmqvist Celebration Palermo 212 5 / 23

The bundle method for near piecewise linear functions The goal function contains alternating regions of small and large curvature The cutting-plane method minimizes linearized functions (supporting hyperplanes) The bundle method combines Newton s method (Hessian information) with cutting planes Helgaker et al. (CTCC, University of Oslo) Lieb maximization Malmqvist Celebration Palermo 212 6 / 23

Moreau Yosida regularization The exact density functional is complicated: F λ [ρ] is convex, unbounded, everywhere discontinuous and not everywhere (sub)differentiable not all densities are v-representable or even N-representable ( F λ [ρ] = sup Eλ [v] (v ρ) ) ( = sup ψv T + λw ψ v + (v ρ v ρ) ) v v in practice, it may be impossible to satisfy ρ v = ρ in a small orbital basis maximization may then give a too large potential v and a too high F λ [ρ] This may be avoided by adding a penalty function to the ground-state energy: E µ λ [v] = E λ[v] µ 2 v 2 penalty for large potentials The Lieb variation principle now gives a bounded regularized universal density functional: F µ ( λ [ρ] = sup E µ λ [v] (v ρ)) v where the Moreau envelope of the density functional is given by ( ) F µ λ [ρ] = inf F λ [n] + 1 n 2µ ρ n 2 infimal convolution it represents the intrinsic energy of the most favourable nearby density n ρ Moreau Yosida density functional is simple: F µ λ [ρ] with is convex, bounded, everywhere continuous and differentiable for µ > all densities are N-representable and even v-representable Helgaker et al. (CTCC, University of Oslo) Lieb maximization Malmqvist Celebration Palermo 212 7 / 23

några minuter till. Moreau envelopes Helgaker et al. (CTCC, University of Oslo) Lieb maximization Malmqvist Celebration Palermo 212 8 / 23

Choice of regularization metric The Moreau Yosida regularization norms are related by convex conjugation: v 2 = v M v ρ 2 = ρ M 1 ρ convex conjugation Different M and µ lead to different regularizations: E µ λ [v] = E λ[v] µ 2 v M v extrinsic energy with damped potential ( ) F µ λ [ρ] = inf F λ [ρ + ] + 1 2µ M 1 intrinsic energy of favoured nearby density where contains no electrons and only changes the shape Two choices of metric: M = I (overlap) and M = T (kinetic) The kinetic metric connects two established regularizations the regularization of Heaton Burgess, Bulat and Yang (27) for OEPs: v T v = v(r) 2 dr small for smooth potentials the regularization of Zhao, Morrison and Parr (ZMP) (1994) for KS potentials: T 1 = (r 1 )r 1 12 (r 2) dr 1 dr 2 small for oscillatory densities v becomes smoother as density oscillations are removed Helgaker et al. (CTCC, University of Oslo) Lieb maximization Malmqvist Celebration Palermo 212 9 / 23

MY regularization for the neon atom with µ = 1 18 aug-cc-pvtz orbital basis; 35s even-tempered potential basis Μ Μ 1 5 5 1 5 5 5 5 1 2 2 2 4 4 4 6 6 6 8 8 8 1. TSVD cutoff 1^18 1. MY S Metric Μ 1^18 1. MY T Metric Μ 1^18 1.5 1.5 1.5 2. 2. 2. 2.5 2.5 2.5 3. 3. 3. TSVD cutoff 1^18 6.5 MY S Metric Μ 1^18 6.5 MY T Metric Μ 1^18 6.5 7. 7. 7. 7.5 7.5 7.5 8. 8. 8. 8.5 8.5 8.5.1.1.1 Helgaker et al. (CTCC, University of Oslo) Lieb maximization Malmqvist Celebration Palermo 212 1 / 23

MY regularization for the neon atom with µ = 1 9 aug-cc-pvtz orbital basis; 35s even-tempered potential basis Μ Μ 1 5 5 1 5 5 5 5 1 2 2 2 4 4 4 6 6 6 8 8 8 1. TSVD cutoff 1^9 1. MY S Metric Μ 1^9 1. MY T Metric Μ 1^9 1.5 1.5 1.5 2. 2. 2. 2.5 2.5 2.5 3. 3. 3. TSVD cutoff 1^9 6.5 MY S Metric Μ 1^9 6.5 MY T Metric Μ 1^9 6.5 7. 7. 7. 7.5 7.5 7.5 8. 8. 8. 8.5 8.5 8.5.1.1.1 Helgaker et al. (CTCC, University of Oslo) Lieb maximization Malmqvist Celebration Palermo 212 11 / 23

MY regularization for the neon atom with µ = 1 5 aug-cc-pvtz orbital basis; 35s even-tempered potential basis Μ Μ 1 5 5 1 5 5 5 5 1 2 2 2 4 4 4 6 6 6 8 8 8 1. TSVD cutoff 1^5 1. MY S Metric Μ 1^5 1. MY T Metric Μ 1^5 1.5 1.5 1.5 2. 2. 2. 2.5 2.5 2.5 3. 3. 3. TSVD cutoff 1^5 6.5 MY S Metric Μ 1^5 6.5 MY T Metric Μ 1^5 6.5 7. 7. 7. 7.5 7.5 7.5 8. 8. 8. 8.5 8.5 8.5.1.1.1 Helgaker et al. (CTCC, University of Oslo) Lieb maximization Malmqvist Celebration Palermo 212 12 / 23

Kohn Sham theory and the adiabatic connection We introduce Kohn Sham theory by expanding about λ = : F λ [ρ] = F [ρ] + λf [ρ] + E c,λ[ρ] = T s[ρ] + λ(j[ρ] K[ρ]) + E c,λ [ρ] The correlation energy is the only term that depends on λ in a nontrivial manner: E c,λ [ρ] = F λ [ρ] F [ρ] = Ψ λ V ee Ψ λ Ψ V ee Ψ = W λ [ρ] AC integrand The correlation energy obtained by integration over the AC integrand: 1 E c[ρ] = W λ [ρ] dλ CCSD and CCSD(T) AC curves for the water molecule: 9. 9.1 E c Ρ WΛ a.u. 9.2 9.3 9.4 T c Ρ CCSD 9.5 CCSDT..2.4.6.8 1. Λ Helgaker et al. (CTCC, University of Oslo) The adiabatic connection Malmqvist Celebration Palermo 212 13 / 23

H 2 dissociation: from dynamical to static correlation Near equilibrium, correlation is predominantly dynamic nearly linear curve since correlation is dominated by doubles (λ 2 in PT) Towards dissociation, static correlation becomes important increased curvature form higher-order contributions in PT At dissociation, correlation is fully static wave function adjusts immediately for λ (degenerate PT) first-order degenerate PT yields constant curve for λ >.5 1.4 bohr.1.15.2 5 bohr.25 1 bohr Helgaker et al. (CTCC, University of Oslo) The adiabatic connection Malmqvist Celebration Palermo 212 14 / 23

A two-parameter model for the H 2 molecule From a two-level CI model, we obtain the ground-state energy E CI (λ) = 1 2 E 1 2 E 2 + 4g 2 λ 2, E = h + gλ Differentiation gives a two-parameter model for the AC integrand good least-square fits (full lines) and fits to initial gradient and end point (dashed lines). Wc,Λ a.u..5.1.15 R.7 a.u. R 1.4 a.u. R 3. a.u..2 R 5. a.u..25 R 7. a.u. R 1. a.u...2.4.6.8 1. Only the initial slope (GL2) and the end point (perhaps at λ = ) are needed Helgaker et al. (CTCC, University of Oslo) The adiabatic connection Malmqvist Celebration Palermo 212 15 / 23

BLYP and FCI correlation curves for H 2 The BLYP functional treats correlation as dynamical at all bond distances it was designed for spin-unrestricted theory but is used here in a spin-restricted manner it hence ignores static correlation BLYP FCI BLYP FCI R 1.4 bohr R 3. bohr BLYP BLYP R 5. bohr FCI R 1. bohr FCI Helgaker et al. (CTCC, University of Oslo) The adiabatic connection Malmqvist Celebration Palermo 212 16 / 23

BLYP and FCI exchange correlation curves for H 2 The improved BLYP performance arises from an overestimation of exchange error cancellation between exchange and correlation reduces total error to about one third R 1.4 bohr R 3. bohr FCI exchange BLYP exchange BLYP correlation FCI correlation R 5. bohr R 1. bohr Helgaker et al. (CTCC, University of Oslo) The adiabatic connection Malmqvist Celebration Palermo 212 17 / 23

Molecules in magnetic fields A magnetic field B modifies the kinetic-energy operator: H(B) = T (B) + W + i v(r i ), T (B) = 1 2 i (σ π i ) 2 where σ are the Pauli spin matrices and π i the kinetic-momentum operator: π i = i i + A(r i ), A = 1 B (r O) 2 We have recently developed the London code for molecular calculations in strong fields complex wave functions and London atomic orbitals Hartree Fock, FCI and Kohn Sham models Our calculations have revealed many interesting features: transition from paramagnetism to diamagnetism for all molecules at a critical B c molecular bonding of triplet H 2 and singlet He 2 in strong fields EêkJmol -1 EêkJmol -1-1 5 Σ -55 H2 1Σ u He2-5525 HF -1 7 1s A HF 1s B FCI -555-1 9 Rêpm 1 2 3 1Σ g FCI Rêpm 1 2 3 What can we learn about the exchange correlation functional in magnetic fields? Helgaker et al. (CTCC, University of Oslo) DFT in magnetic fields Malmqvist Celebration Palermo 212 18 / 23

BDFT: magnetic-field DFT The simplest approach is to set up a separate DFT for each B: E [v, B] = inf ρ (F [ρ, B] + (v ρ)) magnetic-field DFT (BDFT) (Grayce & Harris 1994) The density functional now depends on the density and on the field strength F [ρ, B] = min Ψ Tπ(B) + W Ψ = Ts[ρ, B] + J[ρ] + Exc[ρ, B] Ψ ρ the noninteracting magnetic response is exactly taken care of by T s[ρ, B] AC correlation curves of H 2 in a perpendicular magnetic field E xc[ρ, B] differs from E xc[ρ] only at stretched geometries an earlier onset of static correlation in strong magnetic fields.68 W.66.64 H H R=1.4 au B= B=.25 au.45 W.4.35 H H R=5. au B= B=.25 au dw/db.62.3.6.25.58.2.56.2.4.6.8 1 λ -.1.15.2.4.6.8 1 λ Helgaker et al. (CTCC, University of Oslo) DFT in magnetic fields Malmqvist Celebration Palermo 212 19 / 23

CDFT: current DFT Alternatively, we can build DFT with (ρ, j) as variables, conjugate to (v, A) The current consists of paramagnetic and diamagnetic parts: j = j p + j d = j p + ρ A, j p = i 1 2 ψ (r) ψ(r) + c.c. Note: different physical systems may share the same ground-state wave function: (ρ, j) (v, A) (ρ, j p) Ψ up to convex combinations and gauge transformations Eschrig (21), Capelle & Vignale (21), Pan & Sahni (21) We may now set up current DFT (CDFT): ( E [v, A] = inf F [ρ, jp] + ( v 1 ρ,j 2 A2 ρ ) + ( A j )) the density functional depends on ρ and j p = j ρa: F [ρ, j p] = min Ψ T + W Ψ Ψ ρ, j p Vignale and Rasolt (1987,1988), Pan and Sahni (21) A Lieb variation principle can be set up to calculate F [ρ, j p] we have a pilot implementation... Helgaker et al. (CTCC, University of Oslo) DFT in magnetic fields Malmqvist Celebration Palermo 212 2 / 23

Kohn Sham CDFT theory A Kohn Sham decomposition of the CDFT functional yields (Vignale and Rasolt): F [ρ, j p] = T s[ρ, j p] + J[ρ] + E xc[ρ, ν], ν(r) = jp(r) ρ(r) all gauge dependence is located in the kinetic energy in the usual manner the exchange and correlation functionals are separately gauge invariant they depend on the vorticity rather than the paramagnetic current vorticity The Vignale Rasolt Geldart (VRG) current density functional (1987,1988) ( ) E xc[ρ, j p] = Exc [ρ] + kf (r s) χ L 24π 2 χ 1 (r s) ν(r) dr, L ( ) 3 1/3 ( ) 9π 1/3 r s =, k F (r s) = rs 1 4πρ(r) 4 several parametrizations of the diamagnetic susceptibility ratio χ L /χ L are available We have completed a KS-CDFT implementation to the london program VRG calculations on shieldings in water (Huz II basis) HF LDA VRG(1) VRG(2) VRG(3) O 328.2 333.8 334.9 338.4 341.9 H 3.78 31.14 31.12 31.9 31.5 Lee, Handy and Colwell, JCP 13, 195 (1995) truncated at r s = 1 Helgaker et al. (CTCC, University of Oslo) DFT in magnetic fields Malmqvist Celebration Palermo 212 21 / 23

6 6 1 Density 4 and vorticity in benzene 4.5 2 6 log 1 () 1 1 2 6.B/ B (in plane).1 4 2 4 2 2 3 2 2.5.15 4 1 4 4.1.2 2 6 6 4 2 2 4 6 2 3 6 6 4 2 2 4 6 2.15 4 4 4.2 6 6 4 2 2 4 6 6 6 4 2 2 4 6 6 log 1 () 6.B/ B (1 bohr above) 4 1.5 2 4.2.4 2 6 log 1 () 2.5 2 6.B/ B (1 bohr above).6 4 2 2 3 1.5 3.5 2 4 2.5 4 2 2.2.8.4.1.6.12 4 4.5 3 4.8.14 6 6 4 2 2 4 6 2 3.5 6 6 4 2 2 4 6 2 4 Helgaker et al. (CTCC, University of Oslo) 4 DFT in magnetic fields 4 Malmqvist Celebration Palermo 212.14 22 / 23.16.12

Summary and acknowledgments We have discussed Lieb s theory of DFT ground-state energy and universal density functional related by conjugation We have discussed Lieb maximizations Moreau Yosida regularization of energy and density functional We have discussed the adabatic connection ab-initio adiabatic-connection curves for benchmarking and modelling We have discussed DFT in magnetic fields B-dependent (BDFT) and current-dependent (CDFT) density functionals This work was supported by Norwegian Research Council European Research Council Helgaker et al. (CTCC, University of Oslo) Summary and acknowledgments Malmqvist Celebration Palermo 212 23 / 23