Density-functional theory in strong magnetic fields
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal,
U. Ekström, E. Sagvolden)
Center for Theoretical and Computational Chemistry, Department of Chemistry,
University of Oslo, Norway
CMA-CTCC workshop on computational quantum mechanics,
December 19-20, 2012
DFT in strong fields. . .
2/35
Outline
1
Review of standard DFT
DFT and the Hohenberg–Kohn theorem
Constrained-search formulation of DFT
Lieb’s formulation of DFT
2
Magnetic fields and DFT
BDFT
Paramagnetic CDFT
3
Practical use of the Lieb formulation
The Adiabatic Connection
Some initial BDFT calculations
4
Vorticity and approximate CDFT functionals
The VRG functional(s)
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
2/35
Overview
1
Review of standard DFT
DFT and the Hohenberg–Kohn theorem
Constrained-search formulation of DFT
Lieb’s formulation of DFT
2
Magnetic fields and DFT
BDFT
Paramagnetic CDFT
3
Practical use of the Lieb formulation
The Adiabatic Connection
Some initial BDFT calculations
4
Vorticity and approximate CDFT functionals
The VRG functional(s)
DFT in strong fields. . .
3/35
Review of standard DFT
DFT and the Hohenberg–Kohn theorem
The HK mapping
Restrict attention to N-particle Hamiltonians of the form
H[v ] = 12 p 2 + W + v
(1)
P
where W = i<j rij−1 is fixed and v is an arbitrary one-body
potential.
There exists a one-to-one mapping between potentials and
non-degenerate ground states1
AN = {ρ|ρ comes from g.s. ψ of some H[v ]}
VN = {v |H[v ] has a non-deg. g.s.}
Bijection between VN and AN
ρ ∈ AN determines v and H[v ], which in turn determines ψ
and all properties
Restriction to non-deg. g.s. is not essential
1
P. Hohenberg and W. Kohn, Phys. Rev. 136:B864 (1964)
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
3/35
DFT in strong fields. . .
3/35
Review of standard DFT
DFT and the Hohenberg–Kohn theorem
The HK mapping
Restrict attention to N-particle Hamiltonians of the form
H[v ] = 12 p 2 + W + v
(1)
P
where W = i<j rij−1 is fixed and v is an arbitrary one-body
potential.
There exists a one-to-one mapping between potentials and
non-degenerate ground states1
AN = {ρ|ρ comes from g.s. ψ of some H[v ]}
VN = {v |H[v ] has a non-deg. g.s.}
Bijection between VN and AN
ρ ∈ AN determines v and H[v ], which in turn determines ψ
and all properties
Restriction to non-deg. g.s. is not essential
1
P. Hohenberg and W. Kohn, Phys. Rev. 136:B864 (1964)
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
3/35
DFT in strong fields. . .
3/35
Review of standard DFT
DFT and the Hohenberg–Kohn theorem
The HK mapping
Restrict attention to N-particle Hamiltonians of the form
H[v ] = 12 p 2 + W + v
(1)
P
where W = i<j rij−1 is fixed and v is an arbitrary one-body
potential.
There exists a one-to-one mapping between potentials and
non-degenerate ground states1
AN = {ρ|ρ comes from g.s. ψ of some H[v ]}
VN = {v |H[v ] has a non-deg. g.s.}
Bijection between VN and AN
ρ ∈ AN determines v and H[v ], which in turn determines ψ
and all properties
Restriction to non-deg. g.s. is not essential
1
P. Hohenberg and W. Kohn, Phys. Rev. 136:B864 (1964)
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
3/35
DFT in strong fields. . .
3/35
Review of standard DFT
DFT and the Hohenberg–Kohn theorem
The HK mapping
Restrict attention to N-particle Hamiltonians of the form
H[v ] = 12 p 2 + W + v
(1)
P
where W = i<j rij−1 is fixed and v is an arbitrary one-body
potential.
There exists a one-to-one mapping between potentials and
non-degenerate ground states1
AN = {ρ|ρ comes from g.s. ψ of some H[v ]}
VN = {v |H[v ] has a non-deg. g.s.}
Bijection between VN and AN
ρ ∈ AN determines v and H[v ], which in turn determines ψ
and all properties
Restriction to non-deg. g.s. is not essential
1
P. Hohenberg and W. Kohn, Phys. Rev. 136:B864 (1964)
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
3/35
DFT in strong fields. . .
3/35
Review of standard DFT
DFT and the Hohenberg–Kohn theorem
The HK mapping
Restrict attention to N-particle Hamiltonians of the form
H[v ] = 12 p 2 + W + v
(1)
P
where W = i<j rij−1 is fixed and v is an arbitrary one-body
potential.
There exists a one-to-one mapping between potentials and
non-degenerate ground states1
AN = {ρ|ρ comes from g.s. ψ of some H[v ]}
VN = {v |H[v ] has a non-deg. g.s.}
Bijection between VN and AN
ρ ∈ AN determines v and H[v ], which in turn determines ψ
and all properties
Restriction to non-deg. g.s. is not essential
1
P. Hohenberg and W. Kohn, Phys. Rev. 136:B864 (1964)
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
3/35
DFT in strong fields. . .
4/35
Review of standard DFT
DFT and the Hohenberg–Kohn theorem
The HK functional
Let E [v ] be the energy of the g.s. of H[v ]
E : VN → R
Denote the direct interaction between potential and density by
Z
(ρ|v ) = ρ(r) v (r) dr
(2)
Define the intrinsic energy as
FHK [ρ? ] = E [v ? ] − (ρ? |v ? ) = hψ ? | 12 p 2 + W |ψ ? i
where ρ? determines the potential v ?
FHK : AN → R
FHK is universal, i.e. independent of v
Problem: in general not easy to tell whether ρ ∈ AN or
ρ∈
/ AN for a given ρ
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
4/35
(3)
DFT in strong fields. . .
4/35
Review of standard DFT
DFT and the Hohenberg–Kohn theorem
The HK functional
Let E [v ] be the energy of the g.s. of H[v ]
E : VN → R
Denote the direct interaction between potential and density by
Z
(ρ|v ) = ρ(r) v (r) dr
(2)
Define the intrinsic energy as
FHK [ρ? ] = E [v ? ] − (ρ? |v ? ) = hψ ? | 12 p 2 + W |ψ ? i
where ρ? determines the potential v ?
FHK : AN → R
FHK is universal, i.e. independent of v
Problem: in general not easy to tell whether ρ ∈ AN or
ρ∈
/ AN for a given ρ
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
4/35
(3)
DFT in strong fields. . .
4/35
Review of standard DFT
DFT and the Hohenberg–Kohn theorem
The HK functional
Let E [v ] be the energy of the g.s. of H[v ]
E : VN → R
Denote the direct interaction between potential and density by
Z
(ρ|v ) = ρ(r) v (r) dr
(2)
Define the intrinsic energy as
FHK [ρ? ] = E [v ? ] − (ρ? |v ? ) = hψ ? | 12 p 2 + W |ψ ? i
where ρ? determines the potential v ?
FHK : AN → R
FHK is universal, i.e. independent of v
Problem: in general not easy to tell whether ρ ∈ AN or
ρ∈
/ AN for a given ρ
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
4/35
(3)
DFT in strong fields. . .
4/35
Review of standard DFT
DFT and the Hohenberg–Kohn theorem
The HK functional
Let E [v ] be the energy of the g.s. of H[v ]
E : VN → R
Denote the direct interaction between potential and density by
Z
(ρ|v ) = ρ(r) v (r) dr
(2)
Define the intrinsic energy as
FHK [ρ? ] = E [v ? ] − (ρ? |v ? ) = hψ ? | 12 p 2 + W |ψ ? i
where ρ? determines the potential v ?
FHK : AN → R
FHK is universal, i.e. independent of v
Problem: in general not easy to tell whether ρ ∈ AN or
ρ∈
/ AN for a given ρ
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
4/35
(3)
DFT in strong fields. . .
5/35
Review of standard DFT
Constrained-search formulation of DFT
Constrained-search
Let IN = {ρ|ρ comes from valid wave fun. ψ}
The constrained-search2 is a very transparent formulation of
DFT:
E [v ] = inf hψ|H[v ]|ψi = inf hψ| 12 p 2 + W + v |ψi
ψ
ψ
1 2
= inf (ρ|v ) + inf hψ| 2 p + W |ψi
(4)
ψ7→ρ
ρ∈IN
Universal intrinsic energy:
Fcs [ρ] = inf hψ| 12 p 2 + W |ψi
(5)
ψ7→ρ
Fcs : IN → R; arbitrary potentials can be allowed
Easy generalization: replace pure states ψ with mixed states
above
2
M. Levy Proc. Natl. Acad. Sci. USA 76:6062 (1979)
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
5/35
DFT in strong fields. . .
5/35
Review of standard DFT
Constrained-search formulation of DFT
Constrained-search
Let IN = {ρ|ρ comes from valid wave fun. ψ}
The constrained-search2 is a very transparent formulation of
DFT:
E [v ] = inf hψ|H[v ]|ψi = inf hψ| 12 p 2 + W + v |ψi
ψ
ψ
1 2
= inf (ρ|v ) + inf hψ| 2 p + W |ψi
(4)
ψ7→ρ
ρ∈IN
Universal intrinsic energy:
Fcs [ρ] = inf hψ| 12 p 2 + W |ψi
(5)
ψ7→ρ
Fcs : IN → R; arbitrary potentials can be allowed
Easy generalization: replace pure states ψ with mixed states
above
2
M. Levy Proc. Natl. Acad. Sci. USA 76:6062 (1979)
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
5/35
DFT in strong fields. . .
5/35
Review of standard DFT
Constrained-search formulation of DFT
Constrained-search
Let IN = {ρ|ρ comes from valid wave fun. ψ}
The constrained-search2 is a very transparent formulation of
DFT:
E [v ] = inf hψ|H[v ]|ψi = inf hψ| 12 p 2 + W + v |ψi
ψ
ψ
1 2
= inf (ρ|v ) + inf hψ| 2 p + W |ψi
(4)
ψ7→ρ
ρ∈IN
Universal intrinsic energy:
Fcs [ρ] = inf hψ| 12 p 2 + W |ψi
(5)
ψ7→ρ
Fcs : IN → R; arbitrary potentials can be allowed
Easy generalization: replace pure states ψ with mixed states
above
2
M. Levy Proc. Natl. Acad. Sci. USA 76:6062 (1979)
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
5/35
DFT in strong fields. . .
5/35
Review of standard DFT
Constrained-search formulation of DFT
Constrained-search
Let IN = {ρ|ρ comes from valid wave fun. ψ}
The constrained-search2 is a very transparent formulation of
DFT:
E [v ] = inf hψ|H[v ]|ψi = inf hψ| 12 p 2 + W + v |ψi
ψ
ψ
1 2
= inf (ρ|v ) + inf hψ| 2 p + W |ψi
(4)
ψ7→ρ
ρ∈IN
Universal intrinsic energy:
Fcs [ρ] = inf hψ| 12 p 2 + W |ψi
(5)
ψ7→ρ
Fcs : IN → R; arbitrary potentials can be allowed
Easy generalization: replace pure states ψ with mixed states
above
2
M. Levy Proc. Natl. Acad. Sci. USA 76:6062 (1979)
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
5/35
DFT in strong fields. . .
6/35
Review of standard DFT
Constrained-search formulation of DFT
Constrained-search
The HK formulation shows that ρ ∈ AN determines all
properties.
The constrained-search formulation shows that ρ ∈ IN
determines the intrinsic energy.
ρ ∈ IN is easier to work with than ρ ∈ AN
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
6/35
DFT in strong fields. . .
6/35
Review of standard DFT
Constrained-search formulation of DFT
Constrained-search
The HK formulation shows that ρ ∈ AN determines all
properties.
The constrained-search formulation shows that ρ ∈ IN
determines the intrinsic energy.
ρ ∈ IN is easier to work with than ρ ∈ AN
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
6/35
DFT in strong fields. . .
6/35
Review of standard DFT
Constrained-search formulation of DFT
Constrained-search
The HK formulation shows that ρ ∈ AN determines all
properties.
The constrained-search formulation shows that ρ ∈ IN
determines the intrinsic energy.
ρ ∈ IN is easier to work with than ρ ∈ AN
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
6/35
DFT in strong fields. . .
7/35
Review of standard DFT
Lieb’s formulation of DFT
Convexity and DFT
A function is convex if linear interpolation is always an
overestimate,
F [µρ1 + (1 − µ)ρ2 ] ≤ µF [ρ1 ] + (1 − µ)F [ρ2 ]
(6)
for 0 ≤ µ ≤ 1.
A function is concave if linear interpolation is always an
underestimate,
E [µv1 + (1 − µ)v2 ] ≤ µE [v1 ] + (1 − µ)E [v2 ]
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
7/35
(7)
DFT in strong fields. . .
7/35
Review of standard DFT
Lieb’s formulation of DFT
Convexity and DFT
A function is convex if linear interpolation is always an
overestimate,
F [µρ1 + (1 − µ)ρ2 ] ≤ µF [ρ1 ] + (1 − µ)F [ρ2 ]
(6)
for 0 ≤ µ ≤ 1.
A function is concave if linear interpolation is always an
underestimate,
E [µv1 + (1 − µ)v2 ] ≤ µE [v1 ] + (1 − µ)E [v2 ]
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
7/35
(7)
Convexity and concavity
8
6
4
convex
neither
concave
y
2
0
−2
−4
−6
−3
−2
−1
0
x
1
2
3
DFT in strong fields. . .
9/35
Review of standard DFT
Lieb’s formulation of DFT
Convexity and DFT
E [v ] is a concave functional
Fcs [ρ] (mixed state-version) is a convex functional
Results from convex analysis are then available.3
Legendre–Fenchel transformations:4
E [v ] = inf ((ρ|v ) + FL [ρ])
(8)
FL [ρ] = sup (E [v ] − (ρ|v ))
(9)
ρ
v
ρ is now an element of a Banach space, v is an element of the
dual space; for very unphysical densities, FL [ρ] = +∞
Lieb suggested: ρ ∈ L1 (R3 ) ∩ L3 (R3 ) and
v ∈ L3/2 (R3 ) + L∞ (R3 ); other choices conceivable too
3
4
E. H. Lieb Int. J. Quantum Chem. 24:243 (1983)
Technical point: lower/upper semicontinuity
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
9/35
DFT in strong fields. . .
9/35
Review of standard DFT
Lieb’s formulation of DFT
Convexity and DFT
E [v ] is a concave functional
Fcs [ρ] (mixed state-version) is a convex functional
Results from convex analysis are then available.3
Legendre–Fenchel transformations:4
E [v ] = inf ((ρ|v ) + FL [ρ])
(8)
FL [ρ] = sup (E [v ] − (ρ|v ))
(9)
ρ
v
ρ is now an element of a Banach space, v is an element of the
dual space; for very unphysical densities, FL [ρ] = +∞
Lieb suggested: ρ ∈ L1 (R3 ) ∩ L3 (R3 ) and
v ∈ L3/2 (R3 ) + L∞ (R3 ); other choices conceivable too
3
4
E. H. Lieb Int. J. Quantum Chem. 24:243 (1983)
Technical point: lower/upper semicontinuity
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
9/35
DFT in strong fields. . .
9/35
Review of standard DFT
Lieb’s formulation of DFT
Convexity and DFT
E [v ] is a concave functional
Fcs [ρ] (mixed state-version) is a convex functional
Results from convex analysis are then available.3
Legendre–Fenchel transformations:4
E [v ] = inf ((ρ|v ) + FL [ρ])
(8)
FL [ρ] = sup (E [v ] − (ρ|v ))
(9)
ρ
v
ρ is now an element of a Banach space, v is an element of the
dual space; for very unphysical densities, FL [ρ] = +∞
Lieb suggested: ρ ∈ L1 (R3 ) ∩ L3 (R3 ) and
v ∈ L3/2 (R3 ) + L∞ (R3 ); other choices conceivable too
3
4
E. H. Lieb Int. J. Quantum Chem. 24:243 (1983)
Technical point: lower/upper semicontinuity
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
9/35
DFT in strong fields. . .
9/35
Review of standard DFT
Lieb’s formulation of DFT
Convexity and DFT
E [v ] is a concave functional
Fcs [ρ] (mixed state-version) is a convex functional
Results from convex analysis are then available.3
Legendre–Fenchel transformations:4
E [v ] = inf ((ρ|v ) + FL [ρ])
(8)
FL [ρ] = sup (E [v ] − (ρ|v ))
(9)
ρ
v
ρ is now an element of a Banach space, v is an element of the
dual space; for very unphysical densities, FL [ρ] = +∞
Lieb suggested: ρ ∈ L1 (R3 ) ∩ L3 (R3 ) and
v ∈ L3/2 (R3 ) + L∞ (R3 ); other choices conceivable too
3
4
E. H. Lieb Int. J. Quantum Chem. 24:243 (1983)
Technical point: lower/upper semicontinuity
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
9/35
DFT in strong fields. . .
9/35
Review of standard DFT
Lieb’s formulation of DFT
Convexity and DFT
E [v ] is a concave functional
Fcs [ρ] (mixed state-version) is a convex functional
Results from convex analysis are then available.3
Legendre–Fenchel transformations:4
E [v ] = inf ((ρ|v ) + FL [ρ])
(8)
FL [ρ] = sup (E [v ] − (ρ|v ))
(9)
ρ
v
ρ is now an element of a Banach space, v is an element of the
dual space; for very unphysical densities, FL [ρ] = +∞
Lieb suggested: ρ ∈ L1 (R3 ) ∩ L3 (R3 ) and
v ∈ L3/2 (R3 ) + L∞ (R3 ); other choices conceivable too
3
4
E. H. Lieb Int. J. Quantum Chem. 24:243 (1983)
Technical point: lower/upper semicontinuity
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
9/35
DFT in strong fields. . .
10/35
Review of standard DFT
Lieb’s formulation of DFT
Convexity and DFT
Usefulness of convex formulation:
Solutions of convex optimization problems are either unique or
form a convex set
Compare restrictions on ρ:
HK: ρ is g.s. density of some v , i.e. ρ ∈ AN
CS: ρ comes from a valid pure/mixed state, i.e. ρ ∈ IN
Lieb: ρ ∈ L1 (R3 ) ∩ L3 (R3 )
Would like to identify functional derivatives: v = −δF [ρ]/δρ,
ρ = δE [v ]/δv
Differentiability of FHK [ρ]? Differentiability Fcs [ρ]?
In a convex formulation, differentiability may be set aside.
Instead, the notion of sub-gradients and sub-differentials of a
convex functional is available.
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
10/35
DFT in strong fields. . .
10/35
Review of standard DFT
Lieb’s formulation of DFT
Convexity and DFT
Usefulness of convex formulation:
Solutions of convex optimization problems are either unique or
form a convex set
Compare restrictions on ρ:
HK: ρ is g.s. density of some v , i.e. ρ ∈ AN
CS: ρ comes from a valid pure/mixed state, i.e. ρ ∈ IN
Lieb: ρ ∈ L1 (R3 ) ∩ L3 (R3 )
Would like to identify functional derivatives: v = −δF [ρ]/δρ,
ρ = δE [v ]/δv
Differentiability of FHK [ρ]? Differentiability Fcs [ρ]?
In a convex formulation, differentiability may be set aside.
Instead, the notion of sub-gradients and sub-differentials of a
convex functional is available.
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
10/35
DFT in strong fields. . .
10/35
Review of standard DFT
Lieb’s formulation of DFT
Convexity and DFT
Usefulness of convex formulation:
Solutions of convex optimization problems are either unique or
form a convex set
Compare restrictions on ρ:
HK: ρ is g.s. density of some v , i.e. ρ ∈ AN
CS: ρ comes from a valid pure/mixed state, i.e. ρ ∈ IN
Lieb: ρ ∈ L1 (R3 ) ∩ L3 (R3 )
Would like to identify functional derivatives: v = −δF [ρ]/δρ,
ρ = δE [v ]/δv
Differentiability of FHK [ρ]? Differentiability Fcs [ρ]?
In a convex formulation, differentiability may be set aside.
Instead, the notion of sub-gradients and sub-differentials of a
convex functional is available.
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
10/35
DFT in strong fields. . .
10/35
Review of standard DFT
Lieb’s formulation of DFT
Convexity and DFT
Usefulness of convex formulation:
Solutions of convex optimization problems are either unique or
form a convex set
Compare restrictions on ρ:
HK: ρ is g.s. density of some v , i.e. ρ ∈ AN
CS: ρ comes from a valid pure/mixed state, i.e. ρ ∈ IN
Lieb: ρ ∈ L1 (R3 ) ∩ L3 (R3 )
Would like to identify functional derivatives: v = −δF [ρ]/δρ,
ρ = δE [v ]/δv
Differentiability of FHK [ρ]? Differentiability Fcs [ρ]?
In a convex formulation, differentiability may be set aside.
Instead, the notion of sub-gradients and sub-differentials of a
convex functional is available.
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
10/35
Overview
1
Review of standard DFT
DFT and the Hohenberg–Kohn theorem
Constrained-search formulation of DFT
Lieb’s formulation of DFT
2
Magnetic fields and DFT
BDFT
Paramagnetic CDFT
3
Practical use of the Lieb formulation
The Adiabatic Connection
Some initial BDFT calculations
4
Vorticity and approximate CDFT functionals
The VRG functional(s)
DFT in strong fields. . .
11/35
Magnetic fields and DFT
Generalizations of DFT
When generalizing DFT, one has to choose which properties to
retain:
framework
DFT
TD-DFT
SDFT
Paramagnetic CDFT
Physical CDFT
dens.-pot. map
yes
yes
no
no
maybe
CS
yes
no?
yes
yes
no (not universal)
convex
yes
no?
yes
yes
no
Blue answers are recent results.5
Other aspects: TD-DFT derivation from action functional,
non-zero temperature, partial Legendre–Fenchel transforms, . . .
5
Tellgren, Kvaal, Sagvolden, Ekström, Teale & Helgaker, Phys. Rev. A
86:062506 (2012)
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
11/35
DFT in strong fields. . .
12/35
Magnetic fields and DFT
BDFT
Magnetic-field DFT (BDFT)
Consider now Hamiltonians
H[v , A] = 12 (p + A)2 + v + W
(10)
The Hamiltonians under consideration now differ by more
than a multiplicative potential.
One approach6 : fix a gauge A = A[B], imagine a family of
Density Functional Theories parametrized by the magnetic
field B
For each particular choice of B, one obtains a HK-type
mapping, a constrained-search, and even a convex formulation
involving only ρ and v .
E [v ; B] is concave in v
F [ρ; B] is convex in ρ, “half-universal” (i.e. independent of v
but not B and A)
6
C. J. Grayce and R. A. Harris, Phys. Rev. A 50:3089 (1994)
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
12/35
DFT in strong fields. . .
13/35
Magnetic fields and DFT
Paramagnetic CDFT
Paramagnetic Current-Density Functional Theory
Energy of trial ψ:
hψ|H[v , A]|ψi = hψ| 12 p 2 + 21 {p, A} + 21 A2 + v + W |ψi
= (ρψ |v + 21 A2 ) + (jp;ψ |A) + hψ| 12 p 2 + W |ψi
Paramagnetic and physical current density:
jp (r0 ) = hψ| 21 {−i∇1 , δ(r0 − r1 )}|ψi
(11)
j(r0 ) = hψ| 12 {−i∇1 + A, δ(r0 − r1 )}|ψi
(12)
jp is (a) gauge-dependent and (b) determined by ψ
j is (a) gauge-invariant and (b) not determined by ψ alone
Paramagnetic CDFT uses (ρ, jp ) as the density variables7
7
G. Vignale and M. Rasolt, Phys. Rev. Lett. 59:2360 (1987)
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
13/35
DFT in strong fields. . .
14/35
Magnetic fields and DFT
Paramagnetic CDFT
Paramagnetic Current-Density Functional Theory
Constrained-search remains straightforward in Vignale &
Rasolt’s formulation:
E [v , A] = inf hψ|H[v , A]|ψi
ψ
= inf hψ| 12 p 2 + 12 {p, A} + 12 A2 + v + W |ψi
ψ
1 2
1 2
= inf (ρ|v + 2 A ) + (jp |A) + inf hψ| 2 p + W |ψi
ψ7→ρ,jp
ρ,jp
The intrinsic energy is universal (independent of (v , A)):
FVR [ρ, jp ] = inf hψ| 12 p 2 + W |ψi
ψ7→ρ,jp
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
14/35
(13)
DFT in strong fields. . .
15/35
Magnetic fields and DFT
Paramagnetic CDFT
Paramagnetic CDFT and convexity
FVR [ρ, jp ] (mixed state version) is convex in (ρ, jp )
E [v , A] is not concave
densities and potentials are not paired linearly → not the form
of a Legendre–Fenchel transformation
Solution: a change of variables to
A0 = A
(14)
u = v + 12 A2
(15)
Ē [u, A] = E [v + 12 A2 , A]
(16)
The resulting functional
is concave in (u, A)
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
15/35
DFT in strong fields. . .
15/35
Magnetic fields and DFT
Paramagnetic CDFT
Paramagnetic CDFT and convexity
FVR [ρ, jp ] (mixed state version) is convex in (ρ, jp )
E [v , A] is not concave
densities and potentials are not paired linearly → not the form
of a Legendre–Fenchel transformation
Solution: a change of variables to
A0 = A
(14)
u = v + 12 A2
(15)
Ē [u, A] = E [v + 12 A2 , A]
(16)
The resulting functional
is concave in (u, A)
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
15/35
DFT in strong fields. . .
15/35
Magnetic fields and DFT
Paramagnetic CDFT
Paramagnetic CDFT and convexity
FVR [ρ, jp ] (mixed state version) is convex in (ρ, jp )
E [v , A] is not concave
densities and potentials are not paired linearly → not the form
of a Legendre–Fenchel transformation
Solution: a change of variables to
A0 = A
(14)
u = v + 12 A2
(15)
Ē [u, A] = E [v + 12 A2 , A]
(16)
The resulting functional
is concave in (u, A)
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
15/35
DFT in strong fields. . .
15/35
Magnetic fields and DFT
Paramagnetic CDFT
Paramagnetic CDFT and convexity
FVR [ρ, jp ] (mixed state version) is convex in (ρ, jp )
E [v , A] is not concave
densities and potentials are not paired linearly → not the form
of a Legendre–Fenchel transformation
Solution: a change of variables to
A0 = A
(14)
u = v + 12 A2
(15)
Ē [u, A] = E [v + 12 A2 , A]
(16)
The resulting functional
is concave in (u, A)
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
15/35
DFT in strong fields. . .
16/35
Magnetic fields and DFT
Paramagnetic CDFT
Paramagnetic CDFT and convexity
Legendre–Fenchel transformations:
Ē [u, A] = inf ((ρ|u) + (jp |A) + FVR [ρ, jp ]) ,
ρ,jp
FVR [ρ, jp ] = sup Ē [u, A] − (ρ|u) − (jp |A)
(17)
(18)
u,A
Care is required when identifying dual Banach spaces for
densities and potentials.
Gauge transformations can make A grow arbitrarily fast as
r → ∞, (jp |A) may be lowered without bound through gauge
transformations, etc.
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
16/35
Overview
1
Review of standard DFT
DFT and the Hohenberg–Kohn theorem
Constrained-search formulation of DFT
Lieb’s formulation of DFT
2
Magnetic fields and DFT
BDFT
Paramagnetic CDFT
3
Practical use of the Lieb formulation
The Adiabatic Connection
Some initial BDFT calculations
4
Vorticity and approximate CDFT functionals
The VRG functional(s)
DFT in strong fields. . .
17/35
Practical use of the Lieb formulation
The Adiabatic Connection
AC
Consider Hamiltonians with scaled electron repulsion,
H[v , A] = 12 (p + A)2 + v + λW ,
(19)
with 0 ≤ λ ≤ 1.
Can define a family of concave functionals Ēλ [u, A] . . .
. . . and convex functionals FVR,λ [ρ, jp ]
λ = 1 (full electron repulsion) is the case of actual interest
λ = 0 (no repulsion) corresponds to the fictional Kohn–Sham
system
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
17/35
DFT in strong fields. . .
17/35
Practical use of the Lieb formulation
The Adiabatic Connection
AC
Consider Hamiltonians with scaled electron repulsion,
H[v , A] = 12 (p + A)2 + v + λW ,
(19)
with 0 ≤ λ ≤ 1.
Can define a family of concave functionals Ēλ [u, A] . . .
. . . and convex functionals FVR,λ [ρ, jp ]
λ = 1 (full electron repulsion) is the case of actual interest
λ = 0 (no repulsion) corresponds to the fictional Kohn–Sham
system
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
17/35
DFT in strong fields. . .
17/35
Practical use of the Lieb formulation
The Adiabatic Connection
AC
Consider Hamiltonians with scaled electron repulsion,
H[v , A] = 12 (p + A)2 + v + λW ,
(19)
with 0 ≤ λ ≤ 1.
Can define a family of concave functionals Ēλ [u, A] . . .
. . . and convex functionals FVR,λ [ρ, jp ]
λ = 1 (full electron repulsion) is the case of actual interest
λ = 0 (no repulsion) corresponds to the fictional Kohn–Sham
system
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
17/35
DFT in strong fields. . .
17/35
Practical use of the Lieb formulation
The Adiabatic Connection
AC
Consider Hamiltonians with scaled electron repulsion,
H[v , A] = 12 (p + A)2 + v + λW ,
(19)
with 0 ≤ λ ≤ 1.
Can define a family of concave functionals Ēλ [u, A] . . .
. . . and convex functionals FVR,λ [ρ, jp ]
λ = 1 (full electron repulsion) is the case of actual interest
λ = 0 (no repulsion) corresponds to the fictional Kohn–Sham
system
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
17/35
DFT in strong fields. . .
18/35
Practical use of the Lieb formulation
The Adiabatic Connection
The Adiabatic Connection
FVR,0 [ρ, jp ] = inf hψ| 12 p 2 |ψi
(20)
ψ7→ρ,jp
is essentially the Kohn–Sham kinetic energy8 , denoted Ts [ρ, jp ]
Remaining intrinsic energy of actual (λ = 1) system:
Z 1
d
FVR,1 [ρ, jp ] − FVR,0 [ρ, jp ] =
FVR,λ [ρ, jp ] dλ
0 dλ
(21)
Z 1
Z 1
λ
λ
=
hψρ,jp |W |ψρ,jp idλ =
W (λ)dλ
0
0
λ minimizes hψ| 1 p 2 + λW |ψi subject to ψ 7→ ρ, j
where ψρ,j
p
2
p
When analyzing and constructing approximate functionals, it
is useful to model the integrand W (λ)
8
If the Fermi level is non-degenerate, there is no distinction.
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
18/35
DFT in strong fields. . .
18/35
Practical use of the Lieb formulation
The Adiabatic Connection
The Adiabatic Connection
FVR,0 [ρ, jp ] = inf hψ| 12 p 2 |ψi
(20)
ψ7→ρ,jp
is essentially the Kohn–Sham kinetic energy8 , denoted Ts [ρ, jp ]
Remaining intrinsic energy of actual (λ = 1) system:
Z 1
d
FVR,1 [ρ, jp ] − FVR,0 [ρ, jp ] =
FVR,λ [ρ, jp ] dλ
0 dλ
(21)
Z 1
Z 1
λ
λ
=
hψρ,jp |W |ψρ,jp idλ =
W (λ)dλ
0
0
λ minimizes hψ| 1 p 2 + λW |ψi subject to ψ 7→ ρ, j
where ψρ,j
p
2
p
When analyzing and constructing approximate functionals, it
is useful to model the integrand W (λ)
8
If the Fermi level is non-degenerate, there is no distinction.
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
18/35
DFT in strong fields. . .
19/35
Practical use of the Lieb formulation
The Adiabatic Connection
The Adiabatic Connection
The constraint ψ 7→ ρ, jp is difficult to enforce directly in a
computer implementation.
A Lieb-type convex formulation allows an alternative
treatment.
For each λ, one may obtain an approximation to
FVR,λ [ρ, jp ] = sup Ēλ [u, A] − (ρ|u) − (jp |A)
(22)
u,A
The exact energy Ēλ [u, A] is replaced by, say, a FCI energy
evaluated in some finite orbital basis
The optimization over all (u, A) in a large space is replaced by
optimization over a space spanned by practically convenient
basis functions.
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
19/35
DFT in strong fields. . .
19/35
Practical use of the Lieb formulation
The Adiabatic Connection
The Adiabatic Connection
The constraint ψ 7→ ρ, jp is difficult to enforce directly in a
computer implementation.
A Lieb-type convex formulation allows an alternative
treatment.
For each λ, one may obtain an approximation to
FVR,λ [ρ, jp ] = sup Ēλ [u, A] − (ρ|u) − (jp |A)
(22)
u,A
The exact energy Ēλ [u, A] is replaced by, say, a FCI energy
evaluated in some finite orbital basis
The optimization over all (u, A) in a large space is replaced by
optimization over a space spanned by practically convenient
basis functions.
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
19/35
DFT in strong fields. . .
19/35
Practical use of the Lieb formulation
The Adiabatic Connection
The Adiabatic Connection
The constraint ψ 7→ ρ, jp is difficult to enforce directly in a
computer implementation.
A Lieb-type convex formulation allows an alternative
treatment.
For each λ, one may obtain an approximation to
FVR,λ [ρ, jp ] = sup Ēλ [u, A] − (ρ|u) − (jp |A)
(22)
u,A
The exact energy Ēλ [u, A] is replaced by, say, a FCI energy
evaluated in some finite orbital basis
The optimization over all (u, A) in a large space is replaced by
optimization over a space spanned by practically convenient
basis functions.
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
19/35
DFT in strong fields. . .
19/35
Practical use of the Lieb formulation
The Adiabatic Connection
The Adiabatic Connection
The constraint ψ 7→ ρ, jp is difficult to enforce directly in a
computer implementation.
A Lieb-type convex formulation allows an alternative
treatment.
For each λ, one may obtain an approximation to
FVR,λ [ρ, jp ] = sup Ēλ [u, A] − (ρ|u) − (jp |A)
(22)
u,A
The exact energy Ēλ [u, A] is replaced by, say, a FCI energy
evaluated in some finite orbital basis
The optimization over all (u, A) in a large space is replaced by
optimization over a space spanned by practically convenient
basis functions.
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
19/35
DFT in strong fields. . .
19/35
Practical use of the Lieb formulation
The Adiabatic Connection
The Adiabatic Connection
The constraint ψ 7→ ρ, jp is difficult to enforce directly in a
computer implementation.
A Lieb-type convex formulation allows an alternative
treatment.
For each λ, one may obtain an approximation to
FVR,λ [ρ, jp ] = sup Ēλ [u, A] − (ρ|u) − (jp |A)
(22)
u,A
The exact energy Ēλ [u, A] is replaced by, say, a FCI energy
evaluated in some finite orbital basis
The optimization over all (u, A) in a large space is replaced by
optimization over a space spanned by practically convenient
basis functions.
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
19/35
DFT in strong fields. . .
20/35
Practical use of the Lieb formulation
The Adiabatic Connection
The Adiabatic Connection
The resulting implementation is computationally demanding,
but theoretically very well founded as it becomes an instance
of a convex optimization problem
Previous work by A. Teale et al. in CTCC have compared AC
curves from high-quality ab inition calculations against AC
curves corresponding to approximate DFT functionals
We now wish to study BDFT and CDFT in the same way.
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
20/35
DFT in strong fields. . .
20/35
Practical use of the Lieb formulation
The Adiabatic Connection
The Adiabatic Connection
The resulting implementation is computationally demanding,
but theoretically very well founded as it becomes an instance
of a convex optimization problem
Previous work by A. Teale et al. in CTCC have compared AC
curves from high-quality ab inition calculations against AC
curves corresponding to approximate DFT functionals
We now wish to study BDFT and CDFT in the same way.
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
20/35
He2 molecule
3.8
B=0
B=0.5
3.78
3.76
3.74
W
3.72
3.7
3.68
3.66
3.64
3.62
0
0.1
0.2
0.3
0.4
0.5
lambda
0.6
0.7
0.8
0.9
1
LiH molecule
4.06
B=0
B = 0.1 au
B = 0.5 au
4.05
χxc = 1/2 d2 Wxc/dB2 [au]
W
xc
[hartree]
4.04
0.5
4.03
4.02
4.01
4
0.45
0.4
0.35
3.99
0.3
3.98
3.97
0.25
0.1
0.2
0.3
0.4
0.5
λ
0.6
0.7
0.8
0.9
0.1
0.2
0.3
0.4
0.5
λ
0.6
0.7
0.8
0.9
Strategy: construct simple analytical model functions Wxc (λ, B)
2W xc and χxc (λ) = ∂∂B
, try to relate to practical approximate
2
B=0
density functionals
Overview
1
Review of standard DFT
DFT and the Hohenberg–Kohn theorem
Constrained-search formulation of DFT
Lieb’s formulation of DFT
2
Magnetic fields and DFT
BDFT
Paramagnetic CDFT
3
Practical use of the Lieb formulation
The Adiabatic Connection
Some initial BDFT calculations
4
Vorticity and approximate CDFT functionals
The VRG functional(s)
DFT in strong fields. . .
23/35
Vorticity and approximate CDFT functionals
Gauge-invariance and jp
The paramagnetic current is not gauge-invariant.
FVR [ρ, jp ] = Ts [ρ, jp ] + FHartree [ρ] + F xc [ρ, jp ]
(23)
All gauge-dependence on the RHS should be contained in Ts .
Vignale and Rasolt therefore introduced the vorticity
ν =∇×
jp
j − ρA
j
=∇×
=∇× −B
ρ
ρ
ρ
(24)
Tempting to stipulate that
F xc [ρ, jp ] = F xc [ρ, ν]
(25)
to ensure gauge-invariance.
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
23/35
DFT in strong fields. . .
24/35
Vorticity and approximate CDFT functionals
CDFT: what is vorticity?
In fluid mechanics, vorticity is the curl of the velocity field
∇×
j
ρv
= ∇×
= ∇×v (26)
ρ
ρ
vector indicating volume element’s tendency to rotate and its
rotation axis
In CDFT, vorticity is slightly different
ν =∇×
jp
j − ρA
j
=∇×
=∇× −B
ρ
ρ
ρ
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
24/35
(27)
DFT in strong fields. . .
24/35
Vorticity and approximate CDFT functionals
CDFT: what is vorticity?
In fluid mechanics, vorticity is the curl of the velocity field
∇×
j
ρv
= ∇×
= ∇×v (26)
ρ
ρ
vector indicating volume element’s tendency to rotate and its
rotation axis
In CDFT, vorticity is slightly different
ν =∇×
jp
j − ρA
j
=∇×
=∇× −B
ρ
ρ
ρ
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
24/35
(27)
DFT in strong fields. . .
24/35
Vorticity and approximate CDFT functionals
CDFT: what is vorticity?
In fluid mechanics, vorticity is the curl of the velocity field
∇×
j
ρv
= ∇×
= ∇×v (26)
ρ
ρ
vector indicating volume element’s tendency to rotate and its
rotation axis
In CDFT, vorticity is slightly different
ν =∇×
jp
j − ρA
j
=∇×
=∇× −B
ρ
ρ
ρ
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
24/35
(27)
BH molecule, HF/u-aug-cc-pVTZ, field along z (out of the screen)
log (ρ) (in plane), B = 0.05
10
4
curl(j/ρ) proj. on B (in plane)
1
4
0.08
3
3
0.06
0
2
2
0.04
1
0.02
−1
y
1
−2
0
0
0
−0.02
−1
−3
−1
−0.04
−2
−4 −2
−0.06
−3
−5
−4
−4
−3
−2
−1
0
1
x
2
3
4
5
−3
−4
−4
−0.08
−0.1
−3
−2
−1
0
1
2
3
4
5
log (ρ) (in plane), B = 0.05
10
4
curl(j/ρ) proj. on B (in plane)
1
4
0.08
3
3
0.06
0
2
2
0.04
1
0.02
−1
y
1
−2
0
0
0
−0.02
−1
−3
−1
−0.04
−2
−4 −2
−0.06
−3
−5
−4
−4
−3
−2
−1
0
1
2
3
4
5
−3
−4
−4
x
−0.08
−0.1
−3
−2
−1
0
1
2
3
4
5
log10(ρ) (in plane), B = 0.10
4
curl(j/ρ) proj. on B (in plane)
1
4
3
3
0.1
0
2
2
0.05
−1
1
y
1
0
−2
0
0
−1
−3 −1
−0.05
−2
−4 −2
−0.1
−3
−3
−5
−4
−4
−3
−2
−1
0
1
x
2
3
4
5
−0.15
−4
−4
−3
−2
−1
0
1
2
3
4
5
log (ρ) (in plane), B = 0.05
10
4
curl(j/ρ) proj. on B (in plane)
1
4
0.08
3
3
0.06
0
2
2
0.04
1
0.02
−1
y
1
−2
0
0
0
−0.02
−1
−3
−1
−0.04
−2
−4 −2
−0.06
−3
−5
−4
−4
−3
−2
−1
0
1
2
3
4
5
−3
−4
−4
x
−0.08
−0.1
−3
−2
−1
0
1
2
3
4
5
log10(ρ) (in plane), B = 0.15
4
curl(j/ρ) proj. on B (in plane)
1
4
0.15
3
3
0
0.1
2
2
−1
1
0.05
y
1
−2
0
0
0
−1
−3
−0.05
−1
−2
−0.1
−4 −2
−3
−0.15
−3
−5
−4
−4
−3
−2
−1
0
1
x
2
3
4
5
−4
−4
−0.2
−3
−2
−1
0
1
2
3
4
5
log (ρ) (in plane), B = 0.05
10
4
curl(j/ρ) proj. on B (in plane)
1
4
0.08
3
3
0.06
0
2
2
0.04
1
0.02
−1
y
1
−2
0
0
0
−0.02
−1
−3
−1
−0.04
−2
−4 −2
−0.06
−3
−5
−4
−4
−3
−2
−1
0
1
2
3
4
5
−3
−4
−4
x
−0.08
−0.1
−3
−2
−1
0
1
2
3
4
5
log10(ρ) (in plane), B = 0.20
4
curl(j/ρ) proj. on B (in plane)
1
4
0.2
3
0
3
0.15
2
2
0.1
−1
1
0.05
y
1
−2
0
0
0
−1
−0.05
−3
−1
−0.1
−2
−4
−2
−0.15
−3
−5 −3
−4
−4
−3
−2
−1
0
1
x
2
3
4
5
−4
−4
−0.2
−3
−2
−1
0
1
2
3
4
5
log (ρ) (in plane), B = 0.05
10
4
curl(j/ρ) proj. on B (in plane)
1
4
0.08
3
3
0.06
0
2
2
0.04
1
0.02
−1
y
1
−2
0
0
0
−0.02
−1
−3
−1
−0.04
−2
−4 −2
−0.06
−3
−5
−4
−4
−3
−2
−1
0
1
2
3
4
5
−3
−4
−4
x
−0.08
−0.1
−3
−2
−1
0
1
2
3
4
5
log10(ρ) (in plane), B = 0.25
4
curl(j/ρ) proj. on B (in plane)
1
4
0.25
3
0
0.2
3
0.15
2
2
−1
0.1
1
1
0.05
y
−2
0
0
0
−0.05
−3
−1
−1
−2
−0.1
−4
−0.15
−2
−3
−0.2
−5
−3
−0.25
−4
−4
−3
−2
−1
0
1
x
2
3
4
5
−4
−4
−3
−2
−1
0
1
2
3
4
5
log (ρ) (in plane), B = 0.05
10
4
curl(j/ρ) proj. on B (in plane)
1
4
0.08
3
3
0.06
0
2
2
0.04
1
0.02
−1
y
1
−2
0
0
0
−0.02
−1
−3
−1
−0.04
−2
−4 −2
−0.06
−3
−5
−4
−4
−3
−2
−1
0
1
2
3
4
5
−3
−4
−4
x
−0.08
−0.1
−3
−2
−1
log10(ρ) (in plane), B = 0.30
0
1
2
3
4
5
curl(j/ρ) proj. on B (in plane)
4
1
4
3
0
3
0.2
2
2
−1
0.1
1
1
y
−2
0
0
0
−3
−1
−0.1
−1
−4
−2
−2
−0.2
−5
−3
−3
−4
−4
−0.3
−6
−3
−2
−1
0
1
x
2
3
4
5
−4
−4
−3
−2
−1
0
1
2
3
4
5
DFT in strong fields. . .
26/35
Vorticity and approximate CDFT functionals
The VRG functional(s)
VRG functionals
Vignale–Rasolt–Geldart9 suggested the functional form
Z
me kF (ρ) χ
xc
FVRG [ρ, ν] =
− 1 ν 2 dr
48π 2
χ0
(28)
where χ is magnetic susceptibility (and χ0 is the susceptibility
of a noninteracting electron gas).
Taking the susceptibility ratio to be a function of ρ only, one
may write
Z
xc
FVRG [ρ, ν] = C g (ρ)ν 2 dr
(29)
g was fitted to RPA results for a uniform electron gas,
rs = (4πρ/3)−1/3 < 10 bohr.
9
G. Vignale et al. Phys. Rev. B 37:2502 (1988)
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
26/35
Different fits proposed: LCH, OMC, TP10
1.4
1.2
g
1.0
æ æ
æ æ
æ
æ
æ
æ
æ
0.8
æ
1.01
0.6
g
1.00
0.4
0.99
0.98
0.2
0.0
0
0.97
0.0
æ
0.5
10
1.0
rs
1.5
æ
2.0
æ
20
30
rs
10
40
50
æ
A. M. Lee, S. M. Colwell, N. C. Handy Chem. Phys. Lett. 229:225 (1994);
E. Orestes, T. Marcasso, K. Capelle Phys. Rev. A 68:022105 (2003); J. Tao
and J. P. Perdew Phys. Rev. Lett. 95:196403 (2005)
DFT in strong fields. . .
28/35
Vorticity and approximate CDFT functionals
The VRG functional(s)
VRG functionals
All forms of g (ρ) proposed to date have problems.
No fitting data for rs > 10 bohr −→ unreliable low-density
limit.
All forms feature an integrand with the asymptotic behavior
ρ1/3 ν 2
(30)
for small ρ ∼ 0.
Formally, the integral over all space seems well-defined.
The integral is evaluated numerically on a grid. Very sensitive
to noise.
ν is in practice computed as
ν =∇×
jp
ρ∇ × jp − ∇ρ × jp
=
ρ
ρ2
(31)
Numerical noise in the numerator gets amplified when ρ ∼ 0.
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
28/35
DFT in strong fields. . .
29/35
Vorticity and approximate CDFT functionals
The VRG functional(s)
VRG functionals
Let η = ρ2 ν = ρ∇ × jp − ∇ρ × jp .
Example: LCH form
−1/3
xc
FVRG-LCH
[ρ, ν] = C −1 + e −b0 ρ
(1 + b2 ρ−1/3 )
η2
dr
4 + ρ4
(32)
Regularization parameter not in the original; LCH used a
hard cut-off at rs = 10 bohr, claiming it does not affect results
Computed nuclear shielding constants,
σαβ = ∂ 2 E /∂µ∂Bµ=B=0 , do depend cut-offs (hard or soft)
Cut-off dependence is a “barrier to any really useful results”11
11
W. Zhu and S. B. Trickey J. Chem. Phys. 125:094317 (2006)
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
29/35
NMR Isotropic Shielding Constants [ppm]
VRG corrections tend to reduce the shieldings on average –
worsening the results
Mol
HF
CO
N2
H2 O
NH3
CH4
AlF
FCN
H2 S
HFCO
LiF
LiH
ME
MAE
Nuc
H
F
C
O
N
O
H
N
H
C
H
Al
F
F
C
N
S
H
O
C
F
H
Li
F
H
Li
LDA
29.44
416.89
−20.74
−87.05
−90.13
335.08
30.99
267.55
31.57
193.86
31.25
532.13
141.25
344.69
64.91
91.91
727.14
30.44
−138.75
17.35
84.48
22.94
85.41
341.83
25.76
86.40
−21.16
21.19
LDA-VRG
29.46
416.58
−25.05
−94.03
−95.66
335.27
30.99
268.44
31.57
193.95
31.25
528.29
135.41
344.55
62.87
89.99
726.50
30.44
−146.91
15.27
77.99
22.85
82.28
281.82
25.44
81.70
−25.60
25.62
KT3
30.24
412.22
5.60
−54.52
−59.88
327.84
31.65
263.16
32.08
193.49
31.68
566.43
174.71
337.46
83.17
122.15
724.58
31.05
−89.27
37.06
116.34
23.58
89.88
359.32
26.76
92.19
−9.15
10.25
KT3-VRG
30.26
411.86
1.80
−60.92
−64.91
328.02
31.65
264.22
32.08
193.62
31.68
562.82
169.17
337.19
81.27
120.63
723.91
31.05
−96.83
35.17
110.02
23.49
85.67
295.89
26.54
89.58
−13.54
13.91
CCSD(T)
29.18
419.75
3.76
−51.02
−59.07
337.45
31.04
270.63
31.74
200.03
31.48
576.70
225.57
379.12
85.21
118.40
740.04
30.72
−92.87
43.24
178.18
24.14
90.06
386.99
26.54
89.87
Exp
28.51
409.6
0.9
−62.74
−61.6
323.6
30.05
264.54
30.68
195
30.61
Vib
−0.33
−11.8
−2.42
−5.75
−4.33
−14.23
−0.52
−8.71
−0.61
−3.74
−0.63
Emp. Eq.
28.84
421.4
3.32
−56.99
−57.27
337.83
30.57
273.25
31.29
198.74
31.24
30.53
−0.41
30.94
147.7
−12.32
160.02
87.5
374.3
25.7
90.6
0.09
1.14
−0.1
0.13
87.41
373.16
25.8
90.47
Summary
Convex formulation of paramagnetic CDFT available if one
works with u = v + 12 A2 .
Enables the calculation of AC curves through convex
optimization.
FCI/CCSD quality approximations to exact AC curves can be
computed
FCI level calculations within BDFT framework implemented
CCSD and CDFT framework nearly implemented
Comparison of model AC curves from existing DFT and
CDFT functionals will hopefully lead to a better
understanding of why they fail. . . and how to improve them
Existing current-dependent VRG functionals show a poor
low-density limit and degrade the accuracy of magnetic
properties for molecules.
DFT in strong fields. . .
Acknowledgments
32/35
Thanks to. . .
CDFT work:
Trygve Helgaker
Andrew M. Teale & his student James Furness
Ulf Ekström
Simen Kvaal
Espen Sagvolden
Other:
Kai Kaarvann Lange, Mark R. Hoffmann (FCI/CASSSCF)
Stella Stopkowicz (CC), Jon Austad (Laplace MP2)—will
soon be useful for AC studies
CTCC
The audience
Erik Tellgren (and T. Helgaker, A. M. Teale, S. Kvaal, U. Ekström,
DFT
E. Sagvolden)
in strong fields. . .
32/35
More slides. . .
Paramagnetic CDFT: counter-example to presumed HK
theorem
Helium atom in uniform magnetic field B = Bez , and
symmetric gauge A = 21 B × r. . .
For fields not strong enough to induce a level-crossing, the
Hamiltonians
H[v , A] = H[v + 12 A2 , 0]
(33)
have the same ground state (with vanishing canonical angular
momentum).
No HK-type mapping between potentials and ground states
Some limited use of HK ideas still possible: (ρ, jp ) does
determine the unique ground state wave function (alt. the
subspace of deg. wave functions), and therefore all
A-independent properties
Paramagnetic CDFT: counter-example to presumed HK
theorem
Helium atom in uniform magnetic field B = Bez , and
symmetric gauge A = 21 B × r. . .
For fields not strong enough to induce a level-crossing, the
Hamiltonians
H[v , A] = H[v + 12 A2 , 0]
(33)
have the same ground state (with vanishing canonical angular
momentum).
No HK-type mapping between potentials and ground states
Some limited use of HK ideas still possible: (ρ, jp ) does
determine the unique ground state wave function (alt. the
subspace of deg. wave functions), and therefore all
A-independent properties
Paramagnetic CDFT and convexity
Example of Banach spaces (S. Kvaal):
w 2 ρ ∈ L1 (R3 ),
w
−2
∞
3
u ∈ L (R ),
w jp ∈ L1 (R3 )3 ,
w
−1
∞
(34)
3 3
A ∈ L (R )
(35)
where, e.g.,
w (r) = 1 + r n√ (require well-defined moments)
w (r) = 1 + e 1+r (require near-exponential decay)
Pairings (ρ|u) and (jp |A) are now always well-defined.
Only allow gauge transformations jp 7→ jp + ρ∇χ,
A 7→ A + ∇χ satisfying
w −1 ∇χ ∈ L∞ (R3 )3
√
Matching constraint on the wave function: w ψ ∈ H 1/2 ,
with H 1/2 a fractional-order Sobolev space.12
12
Follows if |∇w (r)| ≤ C · w (r), which is true for the two examples.
(36)