density functional theory in magnetic fields - universitetet i...

Density functional theory in magnetic fields

U. E. Ekstrom†, 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 Systemscelebrating Per Ake Malmqvist’s 60th birthday and his career in quantum chemistry

Splendid Hotel La Torre, Mondello, Palermo, Italy, May 30–June 1, 2012

T. Helgaker (CTCC, University of Oslo) Calculation of the universal density functional Malmqvist Celebration Palermo 2012 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 2012 2 / 23

The universal density functionalI Consider the ground-state energy with interaction strength λ:

Eλ[v ] = infΨ

⟨Ψ|T +

∑i v(ri ) + λW |Ψ

⟩the Rayleigh–Ritz variation principle

I We may summarize density-functional theory in two variation principles:

Eλ[v ] = infρ

(Fλ[ρ] + (v |ρ)

)the Hohenberg–Kohn variation principle (1964)

Fλ[ρ] = supv

(Eλ[v ]− (v |ρ)

)the Lieb variation principle (1983)

I these are alternative attempts at sharpening the same inequality into an equality

Fλ[ρ] ≥ Eλ[v ]− (v |ρ) ⇔ Eλ[v ] ≤ Fλ[ρ] + (v |ρ) Fenchel’s inequality

I The Hohenberg–Kohn variation principle underlies all applications of DFT in chemistry

I its success hinges on the accurate modelling of Fλ[ρ]

I The Lieb variation principle provides a tool for studying Fλ[ρ] numerically

I we have implemented the Lieb variation principle at various levels ab initio theoryI accurate numerical studies of Fλ[ρ] are a guide to insightI our motivation is to benchmark approximate Fλ[ρ] and to help develop new ones

I Previous work:

I studies by Colonna and Savin (1999) for few-electron atomsI work by Wu and Yang (2003) on Lieb maximizations

Helgaker et al. (CTCC, University of Oslo) Lieb’s universal density functional Malmqvist Celebration Palermo 2012 3 / 23

Conjugate functionals E [v ]↔ F [ρ]

I The ground-state energy E [v ] is concave in v by the Rayleigh–Ritz variation principle

I it can therefore be exactly represented by its convex conjugate F [ρ]: E [v ]↔ F [ρ]

E@vD

HvÈΡL

F@ΡD

vmax

F@ΡD=supvHE@vD-HvÈΡLL¥

F@ΡD

HvÈΡL

E@vD

Ρmin

E@vD = infΡHF@ΡD+HvÈΡLL

I Both variation principles represent a Legendre–Fenchel transformation

I the essential point is the concavity of E [v ] rather than the details of the Schrodinger equationI roughly speaking, the derivatives of E [v ] are F [ρ] are each other’s inverse functionsI 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 2012 4 / 23

Lieb maximization

I For a given density ρ(r) and level of theory Eλ[v ], we wish to calculate

Fλ[ρ] = maxv(Eλ[v ]− (v |ρ)

)I The potential is parameterized as suggested by Wu and Yang 2003:

vc(r) = vext(r) + (1− λ)vref(r) +∑

tct gt(r)

I the physical, external potential vext(r)I the Fermi–Amaldi reference potential to ensure correct asymptotic behaviourI an expansion in Gaussians gt(r) with coefficients ct

I The maximization may be carried out by Newton-type methods:

I because of concavity, all maxima are global maximaI 5–10 iterations with the exact Hessian to gradient norm 10−6

I 10–15 iterations with an approximate noninteracting HessianI Hessian regularized by a singular-value decomposition with cutoff 10−6

I possible (unphysical) oscillations in the potential do not affect Fλ[ρ]

I Convergence is mostly unproblematic but difficulties may arise for λ ≈ 0

I from a (near) piecewise linearity in ctI from a poor description of Eλ[vc] for large ct

I 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 2012 5 / 23

The bundle method for near piecewise linear functionsI The goal function contains alternating regions of small and large curvatureConvex opt

The goal function contains regions of small curvature.

Convex opt

But Newton-like methods require curvature information.I The cutting-plane method minimizes linearized functions (supporting hyperplanes)Convex optimization with Bundle Methods

All convex functions are bounded from below by their linearizations.

Convex optimization with Bundle Methods

The Cutting Plane methods minimize the linearized function.

Convex optimization with Bundle Methods

All data from all visited points is used.I The bundle method combines Newton’s method (Hessian information) with cutting planesConvex optimization with Bundle Methods

In smooth regions Hessian information can be used: bundlemethods. Quadratic Programming problem.Helgaker et al. (CTCC, University of Oslo) Lieb maximization Malmqvist Celebration Palermo 2012 6 / 23

Moreau–Yosida regularization

I The exact density functional is complicated:

I Fλ[ρ] is convex, unbounded, everywhere discontinuous and not everywhere (sub)differentiableI not all densities are v -representable or even N-representable

Fλ[ρ] = supv

(Eλ[v ]− (v |ρ)

)= sup

v

(〈ψv |T + λW |ψv 〉+ (v |ρv − ρ)

)I in practice, it may be impossible to satisfy ρv = ρ in a small orbital basisI maximization may then give a too large potential v and a too high Fλ[ρ]

I This may be avoided by adding a penalty function to the ground-state energy:

Eµλ [v ] = Eλ[v ]− µ2‖v‖2 ← penalty for large potentials

I The Lieb variation principle now gives a bounded regularized universal density functional:

Fµλ [ρ] = supv

(Eµλ [v ]− (v |ρ)

)where the Moreau envelope of the density functional is given by

Fµλ [ρ] = infn

(Fλ[n] + 1

2µ‖ρ− n‖2

∗

)← infimal convolution

I it represents the intrinsic energy of the most favourable nearby density n ≈ ρ

I Moreau–Yosida density functional is simple:

I Fµλ [ρ] with is convex, bounded, everywhere continuous and differentiable for µ > 0

I all densities are N-representable and even v -representable


Moreau envelopes

envelopes.pdf är en illustration av Moreau och konvex-envelopen.finite-basis-lieb.pdf är sammanfattning av problemen i en finit basiscoarse.pdf är en idé för hur man kan göra coarse-graining med Moreau, även i gaussiskabasset.

Letar efter texten om Moreau för DFT, några minuter till.

finite-basis….pdf (189 KB) coarse.pdf (162 KB)

From: Ulf Egil Ekström <[email protected]>Subject: Moreau

Date: April 23, 2012 1:01:31 PM GMT+02:00To: Trygve Ulf Helgaker <[email protected]>

3 Attachments, 373 KB


Choice of regularization metric

I The Moreau–Yosida regularization norms are related by convex conjugation:

‖v‖2 = 〈v |M|v〉 ↔ ‖ρ‖2∗ = 〈ρ|M−1|ρ〉 ← convex conjugation

I Different M and µ lead to different regularizations:

Eµλ [v ] = Eλ[v ]− µ2〈v |M|v〉 ← extrinsic energy with damped potential

Fµλ [ρ] = inf∆

(Fλ[ρ+ ∆] + 1

2µ〈∆|M−1|∆〉

)← intrinsic energy of favoured nearby density

I where ∆ contains no electrons and only changes the shape

I Two choices of metric: M = I (overlap) and M = T (kinetic)

I The kinetic metric connects two established regularizations

I the regularization of Heaton–Burgess, Bulat and Yang (2007) for OEPs:

〈v |T|v〉 =

∫|∇v(r)|2 dr ← small for smooth potentials

I the regularization of Zhao, Morrison and Parr (ZMP) (1994) for KS potentials:

〈∆|T−1|∆〉 =

∫∫∆(r1)r−1

12 ∆(r2)dr1dr2 ← small for oscillatory densities

I v becomes smoother as density oscillations are removed


MY regularization for the neon atom with µ = 10−18

aug-cc-pVTZ orbital basis; 35s even-tempered potential basis

-10 -5 5 10

-8

-6

-4

-2

TSVD cutoff=10^-18

-10 -5 5 10

-8

-6

-4

-2

MY S Metric Μ=10^-18

-10 -5 5 10

-8

-6

-4

-2

MY T Metric Μ=10^-18

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-3.0

-2.5

-2.0

-1.5

-1.0TSVD cutoff=10^-18

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-3.0

-2.5

-2.0

-1.5

-1.0MY S Metric Μ=10^-18

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-3.0

-2.5

-2.0

-1.5

-1.0MY T Metric Μ=10^-18

-0.10 -0.05 0.00 0.05 0.10

-8.5

-8.0

-7.5

-7.0


-0.10 -0.05 0.00 0.05 0.10

-8.5

-8.0

-7.5

-7.0


-0.10 -0.05 0.00 0.05 0.10

-8.5

-8.0

-7.5

-7.0





-10 -5 5 10

-8

-6

-4

-2

TSVD cutoff=10^-9

-10 -5 5 10

-8

-6

-4

-2


-10 -5 5 10

-8

-6

-4

-2


0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-3.0

-2.5

-2.0

-1.5


0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-3.0

-2.5

-2.0

-1.5


0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-3.0

-2.5

-2.0

-1.5


-0.10 -0.05 0.00 0.05 0.10

-8.5

-8.0

-7.5

-7.0


-0.10 -0.05 0.00 0.05 0.10

-8.5

-8.0

-7.5

-7.0


-0.10 -0.05 0.00 0.05 0.10

-8.5

-8.0

-7.5

-7.0





-10 -5 5 10

-8

-6

-4

-2

TSVD cutoff=10^-5

-10 -5 5 10

-8

-6

-4

-2


-10 -5 5 10

-8

-6

-4

-2


0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-3.0

-2.5

-2.0

-1.5


0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-3.0

-2.5

-2.0

-1.5


0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-3.0

-2.5

-2.0

-1.5


-0.10 -0.05 0.00 0.05 0.10

-8.5

-8.0

-7.5

-7.0


-0.10 -0.05 0.00 0.05 0.10

-8.5

-8.0

-7.5

-7.0


-0.10 -0.05 0.00 0.05 0.10

-8.5

-8.0

-7.5

-7.0



Kohn–Sham theory and the adiabatic connectionI We introduce Kohn–Sham theory by expanding about λ = 0:

Fλ[ρ] = F0[ρ] + λF ′0[ρ] + Ec,λ[ρ] = Ts[ρ] + λ(J[ρ]− K [ρ]) + Ec,λ[ρ]

I The correlation energy is the only term that depends on λ in a nontrivial manner:

E ′c,λ[ρ] = F ′λ[ρ]− F ′0[ρ] = 〈Ψλ|Vee|Ψλ〉 − 〈Ψ0|Vee|Ψ0〉 = Wλ[ρ] ← AC integrand

I The correlation energy obtained by integration over the AC integrand:

Ec[ρ] =

∫ 1

0Wλ[ρ] dλ

I CCSD and CCSD(T) AC curves for the water molecule:

0.0 0.2 0.4 0.6 0.8 1.0

-9.5

-9.4

-9.3

-9.2

-9.1

-9.0

Λ

WΛ

�a.

u.

CCSD

CCSDHTL

Ec@ΡD

-Tc@ΡD

Helgaker et al. (CTCC, University of Oslo) The adiabatic connection Malmqvist Celebration Palermo 2012 13 / 23

H2 dissociation: from dynamical to static correlationI Near equilibrium, correlation is predominantly dynamic

I nearly linear curve since correlation is dominated by doubles (λ2 in PT)

I Towards dissociation, static correlation becomes importantI increased curvature form higher-order contributions in PT

I At dissociation, correlation is fully staticI wave function adjusts immediately for λ 6= 0 (degenerate PT)I first-order degenerate PT yields constant curve for λ > 0

0.0 0.2 0.4 0.6 0.8 1.0

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

1.4 bohr

5 bohr

10 bohr


A two-parameter model for the H2 moleculeI From a two-level CI model, we obtain the ground-state energy

ECI(λ) = 12 ∆E − 1

2

√∆E 2 + 4g2λ2, ∆E = h + gλ

I Differentiation gives a two-parameter model for the AC integrandI good least-square fits (full lines) and fits to initial gradient and end point (dashed lines)

0.0 0.2 0.4 0.6 0.8 1.0

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

Λ

Wc,

Λ�

a.u.

R = 0.7 a.u.

R = 1.4 a.u.

R = 3.0 a.u.

R = 5.0 a.u.

R = 7.0 a.u.

R = 10.0 a.u.

I Only the initial slope (GL2) and the end point (perhaps at λ =∞) are needed


BLYP and FCI correlation curves for H2

I The BLYP functional treats correlation as dynamical at all bond distancesI it was designed for spin-unrestricted theory but is used here in a spin-restricted mannerI it hence ignores static correlation

R = 1.4 bohr

BLYP

FCI

R = 3.0 bohr

BLYP

FCI

R = 5.0 bohr

BLYP

FCI

R = 10.0 bohr

BLYP

FCI


BLYP and FCI exchange–correlation curves for H2

I The improved BLYP performance arises from an overestimation of exchangeI error cancellation between exchange and correlation reduces total error to about one third

R = 1.4 bohr

FCI

BLYP

HF

R = 3.0 bohr

FCI

BLYP

HF

R = 5.0 bohr

FCI correlation

BLYP correlation

BLYP exchange

FCI exchange

R = 10.0 bohr


Molecules in magnetic fieldsI A magnetic field B modifies the kinetic-energy operator:

H(B) = T (B) + W +∑

iv(ri ), T (B) = 1

2

∑i(σ · πi )

2

where σ are the Pauli spin matrices and πi the kinetic-momentum operator:

πi = −i∇i + A(ri ), A = 12

B× (r −O)

I We have recently developed the London code for molecular calculations in strong fieldsI complex wave functions and London atomic orbitalsI Hartree–Fock, FCI and Kohn–Sham models

I Our calculations have revealed many interesting features:I transition from paramagnetism to diamagnetism for all molecules at a critical BcI molecular bonding of triplet H2 and singlet He2 in strong fields

100 200 300Rêpm

-5550

-5525

-5500

EêkJmol-1

FCI

HF

H2

100 200 300Rêpm

-10900

-10700

-10500

EêkJmol-1

FCI

HF

He2

1sA 1sB

1Σu*H´L

1Σu*HÞL

1ΣgH´L

1ΣgHÞL

I 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 2012 18 / 23

BDFT: magnetic-field DFTI The simplest approach is to set up a separate DFT for each B:

E0[v ,B] = infρ (F [ρ,B] + (v |ρ))

I magnetic-field DFT (BDFT) (Grayce & Harris 1994)

I The density functional now depends on the density and on the field strength

F [ρ,B] = minΨ7→ρ

〈Ψ|Tπ(B) + W |Ψ〉 = Ts[ρ,B] + J[ρ] + Exc[ρ,B]

I the noninteracting magnetic response is exactly taken care of by Ts[ρ,B]

I AC correlation curves of H2 in a perpendicular magnetic fieldI Exc[ρ,B] differs from Exc[ρ] only at stretched geometriesI an earlier onset of static correlation in strong magnetic fields

0.56

0.58

0.6

0.62

0.64

0.66

0.68

0 0.2 0.4 0.6 0.8 1

λ

WB=0B=0.25 au

H HR=1.4 au

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 0.2 0.4 0.6 0.8 1

B=0B=0.25 au

H HR=5.0 au

λ

W

0

-0.1

dW/dB


CDFT: current DFT

I Alternatively, we can build DFT with (ρ, j) as variables, conjugate to (v ,A)

I The current consists of paramagnetic and diamagnetic parts:

j = jp + jd = jp + ρA, jp = −i 12ψ†(r)∇ψ(r) + c.c.

I Note: different physical systems may share the same ground-state wave function:

(ρ, j) ↔ (v ,A)→ (ρ, jp) ↔ Ψ ← up to convex combinations and gauge transformations

I Eschrig (2001), Capelle & Vignale (2001), Pan & Sahni (2010)

I We may now set up current DFT (CDFT):

E0[v ,A] = infρ,j

(F [ρ, jp] +

(v − 1

2A2|ρ

)+(A|j))

I the density functional depends on ρ and jp = j− ρA:

F [ρ, jp] = minΨ7→ρ, jp

〈Ψ|T + W |Ψ〉

I Vignale and Rasolt (1987,1988), Pan and Sahni (2010)

I A Lieb variation principle can be set up to calculate F [ρ, jp]

I we have a pilot implementation...


Kohn–Sham CDFT theoryI A Kohn–Sham decomposition of the CDFT functional yields (Vignale and Rasolt):

F [ρ, jp] = Ts[ρ, jp] + J[ρ] + Exc[ρ, ν], ν(r) = ∇×jp(r)

ρ(r)← vorticity

I all gauge dependence is located in the kinetic energy in the usual mannerI the exchange and correlation functionals are separately gauge invariantI they depend on the vorticity rather than the paramagnetic current

I The Vignale–Rasolt–Geldart (VRG) current density functional (1987,1988)

Exc[ρ, jp] = E0xc[ρ] +

∫kF(rs)

24π2

(χL

χ0L

− 1

)(rs) ν(r) dr,

rs =

(3

4πρ(r)

)1/3

, kF(rs) =

(9π

4

)1/3

r−1s

I several parametrizations of the diamagnetic susceptibility ratio χL/χ0L are available

I We have completed a KS-CDFT implementation to the london programI VRG calculations on shieldings in water (Huz–II basis)

HF LDA VRG(10) VRG(20) VRG(30)

O 328.2 333.8 334.9 338.4 341.9H 30.78 31.14 31.12 31.09 31.05

I Lee, Handy and Colwell, JCP 103, 10095 (1995) truncated at rs = 10


Density and vorticity in benzene

log10(l)

−6 −4 −2 0 2 4 6−6

−4

−2

0

2

4

6

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

i.B/|B| (1 bohr above)

−6 −4 −2 0 2 4 6−6

−4

−2

0

2

4

6

−0.16

−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

log10(l)

−6 −4 −2 0 2 4 6−6

−4

−2

0

2

4

6

−4

−3

−2

−1

0

1

i.B/|B| (in plane)

−6 −4 −2 0 2 4 6−6

−4

−2

0

2

4

6

−0.2

−0.15

−0.1

−0.05

0

log10(l)

−6 −4 −2 0 2 4 6−6

−4

−2

0

2

4

6

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

i.B/|B| (1 bohr above)

−6 −4 −2 0 2 4 6−6

−4

−2

0

2

4

6

−0.16

−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

log10(l)

−6 −4 −2 0 2 4 6−6

−4

−2

0

2

4

6

−4

−3

−2

−1

0

1

i.B/|B| (in plane)

−6 −4 −2 0 2 4 6−6

−4

−2

0

2

4

6

−0.2

−0.15

−0.1

−0.05

0


Summary and acknowledgments

I We have discussed Lieb’s theory of DFTI ground-state energy and universal density functional related by conjugation

I We have discussed Lieb maximizationsI Moreau–Yosida regularization of energy and density functional

I We have discussed the adabatic connectionI ab-initio adiabatic-connection curves for benchmarking and modelling

I We have discussed DFT in magnetic fieldsI B-dependent (BDFT) and current-dependent (CDFT) density functionals

I This work was supported byI Norwegian Research CouncilI European Research Council

Helgaker et al. (CTCC, University of Oslo) Summary and acknowledgments Malmqvist Celebration Palermo 2012 23 / 23

density functional theory in magnetic fields - universitetet i...

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