text: −76.2 RHF h-RHF RHF h-RHF −76.4 (π∗)2 −76.6 A A 3 3 A A 2 2 −76.8 −77.0 (π)2 0 30 60 90 120 150 180 0 30 60 90 120 150 180 Torsion Angle, φ / deg Torsion Angle, φ / deg T T (a) QX = 6.0 a.u. (symmetric ethene) (b) QX = 6.1 a.u. (asymmetric ethene) ∗ Figure 7: The four h-RHF states for two electrons in the space of π and π orbitals of CH XH 2 2 (STO-3G) freezing the remaining core electrons and vitual orbitals at a C−X bond length of ◦ R = 1.256 ˚A. X is a carbon-like atom with nuclear charge Q . At φ = 90 , every h-RHF CC X T ◦ ◦ solution corresponds to a real RHF state (red/solid). As φ moves towards 0 or 180 , real RHF T states coalesce whilst h-RHF solutions continue with complex orbital coeﬃcients (magenta/dashed). Breaking the molecular symmetry splits the degeneracy of the high energy ionic states, changing the coalescence points from triply degenerate A cusp catastrophes to doubly degenerate A fold 3 2 catastrophes. ∗ ing the emergence of h-RHF solutions, we are ing at dissociation to the (π)2 and (π )2 conﬁg- not restricted to molecular systems containing urations and the degenerate symmetry broken only two electrons. The properties and reac- H C+−C– H and H C– −C+H ionic states. As 2 2 2 2 tivity of many molecules are dominated by a the carbon-carbon bond length R is short- CC ∗ subset of only two electrons which, by freez- ened, the ionic states coalesce with the (π )2 ing the remaining core electrons, can also be state at around R = 1.29 ˚A in a triply degen- CC considered as two-electron problems. The elec- erate A cusp catastrophe analogous to Figure 3 tronic energy levels in the rotation of ethene, for 4(a). example, depend strongly on the two-electron, The evolution of the h-RHF states as the two-centre π bond. torsion angle φ varies for R = 1.256 ˚A is T CC Starting with an orthogonal basis set com- shown in Figure 7(a). As φ increases or de- ◦ T posed of the STO-3G ground state RHF molec- creases from the 90 perpendicular structure ular orbitals at the optimised planar D ge- (D ) towards the planar geometry, the ionic 2∗h 2d ometry, we select the b (π) and b (π ) or- states simultaneously coalesce with the anti- 3u 2g ∗ bitals as an active pair and freeze the remaining bonding (π )2 state at two A cusp catastrophes 3◦ ◦ core electrons and virtual orbitals. An h-RHF located at around φ = 45 and 135 . This T calculation using the π electrons in this ac- mirrors a previous analysis by Fukutome.49 Be- tive space reduces the system to a two-electron yond these singlet instability points, the related problem in two basis functions, yielding 4 so- h-RHF states continue to exist with complex lutions through Equation 15. Due to the sym- orbital coeﬃcients. metry equivalence of the carbon centres, the h- Similarly to HZ, breaking of the molecular RHF states resemble those of H , correspond- symmetry can be modelled by replacing one 2 12 E / E h ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: The

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text: levels

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text: ethene,

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text: example,

text: depend

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text: two-electron,

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text: evolution

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text: h-RHF

text: states

text: as

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text: two-centre

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text: bond.

text: torsion

text: angle

text: φ

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text: R

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text: CC

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text: Figure

text: As

text: φ

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text: or

text: de-

text: T

text: posed

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text: the

text: ground

text: state

text: RHF

text: molec-

text: creases

text: from

text: the

text: perpendicular

text: structure

text: ular

text: orbitals

text: at

text: the

text: optimised

text: planar

text: D

text: ge-

text: (D

text: towards

text: the

text: planar

text: geometry,

text: the

text: ionic

text: ometry,

text: we

text: select

text: the

text: b

text: (π)

text: and

text: b

text: (π

text: or-

text: states

text: simultaneously

text: coalesce

text: with

text: the

text: anti-

text: bitals

text: as

text: an

text: active

text: pair

text: and

text: freeze

text: the

text: remaining

text: bonding

text: (π

text: state

text: at

text: two

text: A

text: cusp

text: catastrophes

text: 3◦

text: core

text: electrons

text: and

text: virtual

text: orbitals.

text: An

text: h-RHF

text: located

text: at

text: around

text: φ

text: and

text: This

text: T

text: calculation

text: using

text: the

text: π

text: electrons

text: in

text: this

text: ac-

text: mirrors

text: a

text: previous

text: analysis

text: by

text: Be-

text: tive

text: space

text: reduces

text: the

text: system

text: to

text: a

text: two-electron

text: yond

text: these

text: singlet

text: instability

text: points,

text: the

text: related

text: problem

text: in

text: two

text: basis

text: functions,

text: yielding

text: so-

text: h-RHF

text: states

text: continue

text: to

text: exist

text: with

text: complex

text: lutions

text: through

text: Equation

text: Due

text: to

text: the

text: sym-

text: orbital

text: coeﬃcients.

text: metry

text: equivalence

text: of

text: the

text: carbon

text: centres,

text: the

text: h-

text: Similarly

text: to

text: HZ,

text: breaking

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text: molecular

text: RHF

text: states

text: resemble

text: those

text: of

text: H

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text: symmetry

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text: modelled

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text: 3A 2 −76.4 1B 1 −76.6 RHF h-RHF UHF −76.8 h-UHF NOCI −77.0 −77.2 0 30 60 90 120 150 180 Torsion Angle, φ / deg T Figure 9: Lowest four h-UHF (blue/cyan) and four h-RHF (red/magenta) states located using two ∗ electrons in the π and π orbitals of ethene (STO-3G) with a frozen core and virtual approximation. At this bond length, the h-UHF states have real orbital coeﬃcients (blue) across all torsion angles φ . The C−H and C−C bond lengths and angles are ﬁxed at their optimal values and the virtual T ◦ ◦ orbitals relaxed as φ varies between 0 and 180 . Using these solutions as a basis for NOCI recovers T the 1B , 3A and 1B states (green), predicting the crossing of the lowest energy singlet and triplet 1 2 2 surfaces in agreement with the exact FCIQMC results for the STO-3G basis set. further four h-UHF states exist across all ge- quires correlation with the core electrons to be ometries. Retaining the frozen core and virtual included. To verify our NOCI results, we com- orbital approximations, these additional h-UHF pute the exact, fully correlated energies of the solutions have real orbital coeﬃcients for all φ , 1B and 3A states within the STO-3G basis set T 1 2 corresponding to the diradical states and the using Full Conﬁguration Interaction Quantum ∗ (π)1 (π )1 conﬁgurations. Unfreezing the vir- Monte–Carlo (FCIQMC)59 and obtain energies tual orbitals and relaxing the SCF states then of −76.98253(5) E and −76.98833(5) E re- h h allows the inclusion of hyperconjugation with spectively, conﬁrming the ordering predicted by ∗ the C-H σ orbitals. Using these eight solu- NOCI. Further comparison with the FCIQMC tions, the three lowest NOCI energy levels are results indicates that, despite only including computed as shown in Figure 9. With this ba- 8 out of 1.1 × 107 determinants from the full sis set and carbon-carbon bond length only the Hilbert space, NOCI captures 93% and 92% of ionic h-RHF states become complex, however the 1B and 3A correlation energies. 1 2 including these states is essential to prevent dis- Although these NOCI energy levels suggest continuities in the singlet NOCI energy levels. the 1B and 3A states do cross in the rota- 1 2 The NOCI results presented in Figure 9 indi- tion of ethene, it is important to remember that cate that the 3A state lies below the 1B state this is a minimal basis set calculation ignor- ◦ 2 1 at the 90 transition structure, as predicted by ing any geometrical relaxation for the triplet Hund’s rules.56,57 However, Schmidt et al. note 3A state or the non-planar structures. Re- 2 that such results can arise when only the two gardless, it is reassuring to observe the qualita- π electrons are correlated53 — for example in tive accuracy of NOCI within the STO-3G ba- the two-conﬁguration SCF calculations of Yam- sis set approximation using a minimal number aguchi et al.58 — whilst the correct ordering re- of determinants and a frozen core approxima- 14 E / E h ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: RHF

text: h-RHF

text: UHF

text: h-UHF

text: NOCI

text: Torsion

text: Angle,

text: φ

text: deg

text: T

text: Figure

text: Lowest

label: caption

text: four

text: h-UHF

text: (blue/cyan)

text: and

text: four

text: h-RHF

text: (red/magenta)

text: states

text: located

text: using

text: two

text: electrons

text: in

text: the

text: π

text: and

text: π

text: orbitals

text: of

text: ethene

text: with

text: a

text: frozen

text: core

text: and

text: virtual

text: approximation.

text: At

text: this

text: bond

text: length,

text: the

text: h-UHF

text: states

text: have

text: real

text: orbital

text: coeﬃcients

text: (blue)

text: across

text: all

text: torsion

text: angles

text: φ

text: The

text: C−H

text: and

text: C−C

text: bond

text: lengths

text: and

text: angles

text: are

text: ﬁxed

text: at

text: their

text: optimal

text: values

text: and

text: the

text: virtual

text: T

text: orbitals

text: relaxed

text: as

text: φ

text: varies

text: between

text: and

text: Using

text: these

text: solutions

text: as

text: a

text: basis

text: for

text: NOCI

text: recovers

text: T

text: the

text: and

text: states

text: (green),

text: predicting

text: the

text: crossing

text: of

text: the

text: lowest

text: energy

text: singlet

text: and

text: triplet

text: surfaces

text: in

text: agreement

text: with

text: the

text: exact

text: FCIQMC

text: results

text: for

text: the

text: basis

text: set.

text: further

text: four

text: h-UHF

text: states

text: exist

text: across

text: all

text: ge-

text: quires

text: correlation

text: with

text: the

text: core

text: electrons

text: to

text: be

text: ometries.

text: Retaining

text: the

text: frozen

text: core

text: and

text: virtual

text: included.

text: To

text: verify

text: our

text: NOCI

text: results,

text: we

text: com-

text: orbital

text: approximations,

text: these

text: additional

text: h-UHF

text: pute

text: the

text: exact,

text: fully

text: correlated

text: energies

text: of

text: the

text: solutions

text: have

text: real

text: orbital

text: coeﬃcients

text: for

text: all

text: φ

text: and

text: states

text: within

text: the

text: basis

text: set

text: T

text: corresponding

text: to

text: the

text: diradical

text: states

text: and

text: the

text: using

text: Full

text: Conﬁguration

text: Interaction

text: Quantum

text: (π

text: conﬁgurations.

text: Unfreezing

text: the

text: vir-

text: Monte–Carlo

text: and

text: obtain

text: energies

text: tual

text: orbitals

text: and

text: relaxing

text: the

text: SCF

text: states

text: then

text: of

text: E

text: and

text: E

text: re-

text: h

text: h

text: allows

text: the

text: inclusion

text: of

text: hyperconjugation

text: with

text: spectively,

text: conﬁrming

text: the

text: ordering

text: predicted

text: by

text: the

text: C-H

text: σ

text: orbitals.

text: Using

text: these

text: eight

text: solu-

text: NOCI.

text: Further

text: comparison

text: with

text: the

text: FCIQMC

text: tions,

text: the

text: three

text: lowest

text: NOCI

text: energy

text: levels

text: are

text: results

text: indicates

text: that,

text: despite

text: only

text: including

text: computed

text: as

text: shown

text: in

text: Figure

text: With

text: this

text: ba-

text: out

text: of

text: determinants

text: from

text: the

text: full

text: sis

text: set

text: and

text: carbon-carbon

text: bond

text: length

text: only

text: the

text: Hilbert

text: space,

text: NOCI

text: captures

text: and

text: of

text: ionic

text: h-RHF

text: states

text: become

text: complex,

text: however

text: the

text: and

text: correlation

text: energies.

text: including

text: these

text: states

text: is

text: essential

text: to

text: prevent

text: dis-

text: Although

text: these

text: NOCI

text: energy

text: levels

text: suggest

text: continuities

text: in

text: the

text: singlet

text: NOCI

text: energy

text: levels.

text: the

text: and

text: states

text: do

text: cross

text: in

text: the

text: rota-

text: The

text: NOCI

text: results

text: presented

text: in

text: Figure

text: indi-

text: tion

text: of

text: ethene,

text: it

text: is

text: important

text: to

text: remember

text: that

text: cate

text: that

text: the

text: state

text: lies

text: below

text: the

text: state

text: this

text: is

text: a

text: minimal

text: basis

text: set

text: calculation

text: ignor-

text: at

text: the

text: transition

text: structure,

text: as

text: predicted

text: by

text: ing

text: any

text: geometrical

text: relaxation

text: for

text: the

text: triplet

text: Hund’s

text: However,

text: Schmidt

text: et

text: al.

text: note

text: state

text: or

text: the

text: non-planar

text: structures.

text: Re-

text: that

text: such

text: results

text: can

text: arise

text: when

text: only

text: the

text: two

text: gardless,

text: it

text: is

text: reassuring

text: to

text: observe

text: the

text: qualita-

text: π

text: electrons

text: are

text: for

text: example

text: in

text: tive

text: accuracy

text: of

text: NOCI

text: within

text: the

text: ba-

text: the

text: two-conﬁguration

text: SCF

text: calculations

text: of

text: Yam-

text: sis

text: set

text: approximation

text: using

text: a

text: minimal

text: number

text: aguchi

text: et

text: whilst

text: the

text: correct

text: ordering

text: re-

text: of

text: determinants

text: and

text: a

text: frozen

text: core

text: approxima-

text: E

text: E

text: h

text: ##LTFigure##

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text: (cid:16) (cid:17) In contrast, employing the algebraic geome- (cid:16) (cid:17) try approach presented above yields an entirely π generalised geometric proof. Furthermore, our φ (r) = sin θ − χ1s,H (r) 4 approach proves that Equation 15 provides not π + cos θ − χ (r) . (23) only an upper bound on the number of real 4 1s,Z RHF solutions, but also the exact number of h-RHF states for two–electron systems. We be- With two basis functions, Equation 15 dic- lieve that using algebraic geometry will subse- tates that four h-RHF states exist for all bond quently enable the number of holomorphic so- lengths R and values of Q . lutions to be computed for both unrestricted or HZ Z We begin by considering the case where Q = many electron systems, however there may be Z 1.00 a.u., corresponding to H , and plot the con- challenges in obtaining a general closed formula 2 ventional Hartree–Fock energy of each h-RHF for these cases. solution from R = 0.5 ˚A to R = 4.0 ˚A Finally, we note that the number of h-RHF HZ HZ in Figure 3 (left panel). In the dissociation states predicted by Equation 15 is much larger limit, each solution has real orbital coeﬃcients than the dimension of the full conﬁguration in- and corresponds to a real Hartree–Fock state teraction (FCI) space for two-electrons, given (red/solid), representing the σ2, σ2 and degen- by n2. However, the non-orthogonality of diﬀer- g u erate ionic H+−Z– and H– −Z+ states in or- ent SCF solutions allows each state to span mul- der of ascending energy. As the internuclear tiple FCI determinants, enabling a more com- distance is reduced, the ionic states coalesce pact description of the Hilbert space through a with the σ2 state at a Hartree–Fock instabil- small number of relevant h-RHF states. u ity threshold and disappear at shorter bond 4 Closed-shell states of HZ lengths. In contrast, the corresponding h-RHF solutions continue to exist with complex or- in STO-3G bital coeﬃcients (magenta/dashed), forming a degenerate pair related by complex conjuga- Since the number of h-RHF states for two- tion. Signiﬁcantly, although the conventional electron systems depends on only the number of Hartree–Fock energy of these states appears basis functions, any pair of distinct two-electron kinked at the coalescence point, their path systems with the same number of basis func- through orbital coeﬃcient space is both smooth tions can be smoothly interconverted by either and continuous, and it is this property that moving the basis function centres (eg. changing is essential for NOCI. Using the classiﬁcation structure) or adjusting the atomic charges (eg. of Hartree–Fock singlet instability thresholds changing atoms). This concept can be demon- developed by Mestechkin,22–24 the coalescence strated by considering the h-RHF solutions of point for Q = 1.00 a.u. can be identiﬁed as the HZ molecule using the STO-3G basis set. z a “conﬂuence” point, where two maxima con- Varying the nuclear charge of the hydrogenic Z verge onto a minimum, as shown in Figure 4(a). atom, Q , between 0 and 2, enables the smooth Z In contrast, the molecular symmetry is bro- interconversion along the isoelectronic sequence ken by moving to Q = 1.15 a.u. (middle panel H– → H → HHe+.45 This simple system is of Z 2 of Figure 3), lifting the degeneracy of the ionic particular interest as an archetypal model for states and leading to the coalescence of only the qualitative nature of the h-RHF states in the σ2 and H+ − Z– states at a “pair annihila- symmetric and asymmetric diatomics. u tion” point. Beyond this point, both real RHF The sole occupied spatial orbital is expressed solutions disappear whilst, again, their h-RHF in terms of the RHF rotation angle θ describ- counterparts continue as a complex degenerate ing the degree of mixing between the 1s atomic pair. The existence of complex h-RHF states orbitals on H and Z, arising at this pair annhilation point indicates the applicability of holomorphic Hartree–Fock 8 ##LTLine## ##LTLine##

label: equation

text: In

text: contrast,

text: employing

text: the

text: algebraic

text: geome-

text: try

text: approach

text: presented

text: above

text: yields

text: an

text: entirely

text: π

text: generalised

text: geometric

text: proof.

text: Furthermore,

text: our

text: φ

text: (r)

text: sin

text: θ

text: (r)

text: approach

text: proves

text: that

text: Equation

text: provides

text: not

text: π

text: cos

text: θ

text: χ

text: (r)

text: only

text: an

text: upper

text: bound

text: on

text: the

text: number

text: of

text: real

text: RHF

text: solutions,

text: but

text: also

text: the

text: exact

text: number

text: of

text: h-RHF

text: states

text: for

text: two–electron

text: systems.

text: We

text: be-

text: With

text: two

text: basis

text: functions,

text: Equation

text: dic-

text: lieve

text: that

text: using

text: algebraic

text: geometry

text: will

text: subse-

text: tates

text: that

text: four

text: h-RHF

text: states

text: exist

text: for

text: all

text: bond

text: quently

text: enable

text: the

text: number

text: of

text: holomorphic

text: so-

text: lengths

text: R

text: and

text: values

text: of

text: Q

text: lutions

text: to

text: be

text: computed

text: for

text: both

text: unrestricted

text: or

text: HZ

text: Z

text: We

text: begin

text: by

text: considering

text: the

text: case

text: where

text: Q

text: many

text: electron

text: systems,

text: however

text: there

text: may

text: be

text: Z

text: a.u.,

text: corresponding

text: to

text: H

text: and

text: plot

text: the

text: con-

text: challenges

text: in

text: obtaining

text: a

text: general

text: closed

text: formula

text: ventional

text: Hartree–Fock

text: energy

text: of

text: each

text: h-RHF

text: for

text: these

text: cases.

text: solution

text: from

text: R

text: ˚A

text: to

text: R

text: ˚A

text: Finally,

text: we

text: note

text: that

text: the

text: number

text: of

text: h-RHF

text: HZ

text: HZ

text: in

text: Figure

text: (left

text: panel).

text: In

text: the

text: dissociation

text: states

text: predicted

text: by

text: Equation

text: is

text: much

text: larger

text: limit,

text: each

text: solution

text: has

text: real

text: orbital

text: coeﬃcients

text: than

text: the

text: dimension

text: of

text: the

text: full

text: conﬁguration

text: in-

text: and

text: corresponds

text: to

text: a

text: real

text: Hartree–Fock

text: state

text: teraction

text: (FCI)

text: space

text: for

text: two-electrons,

text: given

text: (red/solid),

text: representing

text: the

text: σ2

text: and

text: degen-

text: by

text: However,

text: the

text: non-orthogonality

text: of

text: diﬀer-

text: g

text: u

text: erate

text: ionic

text: H+−Z–

text: and

text: H–

text: −Z+

text: states

text: in

text: or-

text: ent

text: SCF

text: solutions

text: allows

text: each

text: state

text: to

text: span

text: mul-

text: der

text: of

text: ascending

text: energy.

text: As

text: the

text: internuclear

text: tiple

text: FCI

text: determinants,

text: enabling

text: a

text: more

text: com-

text: distance

text: is

text: reduced,

text: the

text: ionic

text: states

text: coalesce

text: pact

text: description

text: of

text: the

text: Hilbert

text: space

text: through

text: a

text: with

text: the

text: σ2

text: state

text: at

text: a

text: Hartree–Fock

text: instabil-

text: small

text: number

text: of

text: relevant

text: h-RHF

text: states.

text: u

text: ity

text: threshold

text: and

text: disappear

text: at

text: shorter

text: bond

text: Closed-shell

text: states

text: of

text: HZ

text: lengths.

text: In

text: contrast,

text: the

text: corresponding

text: h-RHF

text: solutions

text: continue

text: to

text: exist

text: with

text: complex

text: or-

text: in

text: bital

text: coeﬃcients

text: (magenta/dashed),

text: forming

text: a

text: degenerate

text: pair

text: related

text: by

text: complex

text: conjuga-

text: Since

text: the

text: number

text: of

text: h-RHF

text: states

text: for

text: two-

text: tion.

text: Signiﬁcantly,

text: although

text: the

text: conventional

text: electron

text: systems

text: depends

text: on

text: only

text: the

text: number

text: of

text: Hartree–Fock

text: energy

text: of

text: these

text: states

text: appears

text: basis

text: functions,

text: any

text: pair

text: of

text: distinct

text: two-electron

text: kinked

text: at

text: the

text: coalescence

text: point,

text: their

text: path

text: systems

text: with

text: the

text: same

text: number

text: of

text: basis

text: func-

text: through

text: orbital

text: coeﬃcient

text: space

text: is

text: both

text: smooth

text: tions

text: can

text: be

text: smoothly

text: interconverted

text: by

text: either

text: and

text: continuous,

text: and

text: it

text: is

text: this

text: property

text: that

text: moving

text: the

text: basis

text: function

text: centres

text: (eg.

text: changing

text: is

text: essential

text: for

text: NOCI.

text: Using

text: the

text: classiﬁcation

text: structure)

text: or

text: adjusting

text: the

text: atomic

text: charges

text: (eg.

text: of

text: Hartree–Fock

text: singlet

text: instability

text: thresholds

text: changing

text: atoms).

text: This

text: concept

text: can

text: be

text: demon-

text: developed

text: by

text: the

text: coalescence

text: strated

text: by

text: considering

text: the

text: h-RHF

text: solutions

text: of

text: point

text: for

text: Q

text: a.u.

text: can

text: be

text: identiﬁed

text: as

text: the

text: HZ

text: molecule

text: using

text: the

text: basis

text: set.

text: z

text: a

text: “conﬂuence”

text: point,

text: where

text: two

text: maxima

text: con-

text: Varying

text: the

text: nuclear

text: charge

text: of

text: the

text: hydrogenic

text: Z

text: verge

text: onto

text: a

text: minimum,

text: as

text: shown

text: in

text: Figure

text: atom,

text: Q

text: between

text: and

text: enables

text: the

text: smooth

text: Z

text: In

text: contrast,

text: the

text: molecular

text: symmetry

text: is

text: bro-

text: interconversion

text: along

text: the

text: isoelectronic

text: sequence

text: ken

text: by

text: moving

text: to

text: Q

text: a.u.

text: (middle

text: panel

text: H–

text: H

text: This

text: simple

text: system

text: is

text: of

text: Z

text: of

text: Figure

text: lifting

text: the

text: degeneracy

text: of

text: the

text: ionic

text: particular

text: interest

text: as

text: an

text: archetypal

text: model

text: for

text: states

text: and

text: leading

text: to

text: the

text: coalescence

text: of

text: only

text: the

text: qualitative

text: nature

text: of

text: the

text: h-RHF

text: states

text: in

text: the

text: σ2

text: and

text: H+

text: Z–

text: states

text: at

text: a

text: “pair

text: annihila-

text: symmetric

text: and

text: asymmetric

text: diatomics.

text: u

text: tion”

text: point.

text: Beyond

text: this

text: point,

text: both

text: real

text: RHF

text: The

text: sole

text: occupied

text: spatial

text: orbital

text: is

text: expressed

text: solutions

text: disappear

text: whilst,

text: again,

text: their

text: h-RHF

text: in

text: terms

text: of

text: the

text: RHF

text: rotation

text: angle

text: θ

text: describ-

text: counterparts

text: continue

text: as

text: a

text: complex

text: degenerate

text: ing

text: the

text: degree

text: of

text: mixing

text: between

text: the

text: atomic

text: pair.

text: The

text: existence

text: of

text: complex

text: h-RHF

text: states

text: orbitals

text: on

text: H

text: and

text: Z,

text: arising

text: at

text: this

text: pair

text: annhilation

text: point

text: indicates

text: the

text: applicability

text: of

text: holomorphic

text: Hartree–Fock

text: ##LTLine##

text: ##LTLine##


text: SECOND ORDER MODELS FOR OPTIMAL TRANSPORT AND CUBIC SPLINES ON THE WASSERSTEIN SPACE JEAN-DAVID BENAMOU, THOMAS O. GALLOUE¨T, AND FRANC¸OIS-XAVIER VIALARD Abstract. On the space of probability densities, we extend the Wasserstein geodesics to the case of higher-order interpolation such as cubic spline interpolation. After presenting the natural exten- sion of cubic splines to the Wasserstein space, we propose simpler approach, similarly to Brenier’s generalized Euler solutions. Our method is based on the relaxation of the variational problem on the path space. We propose an eﬃcient implementation based on multimarginal optimal transport and entropic regularization in 1D and 2D. Our framework also enables extrapolation of Wasserstein geodesics via a natural convex relaxation. 1. Introduction In this paper, we propose a variational method to generalize cubic splines on the space of densities using multimarginal optimal transport. In short, the proposed method consists in minimizing, on the space of measures on the path space, under marginal constraints, the norm squared of the acceleration. In this setting, entropic regularization can be applied and give an eﬃcient algorithms to approximate the solution. We showcase our methodology on 1D and 2D data. In the past few years, higher-order interpolations methods have been investigated for applications in computer vision or medical imaging, for time-sequence interpolation or regression. The most usual setting is when data are modeled as shapes, which object embedded in the Euclidean space with no preferred parametrization: space of unparametrized curves or surfaces, or images are some of the most important examples. These examples are inﬁnite dimensional but the ﬁnite dimensional case was interesting for camera motion interpolation as ﬁrst introduced in [17] and further developed in [4, 6]. Motivated by sometimes diﬀerent applications, the problem of interpolation between two shapes is usually treated via the use of a Riemannian metric on the space of shapes and computing a geodesic between the two shapes. From a mathematical point of view, shape spaces are often inﬁnite dimensional and thus, non-trivial analytical questions arise such as existence of minimizing geodesics or global well-posedness of the initial value problem associated with geodesics. A ﬁnite dimensional approximation is still possible such as in [23], in which spline interpolation is proposed for a diﬀeomorphic group action on a ﬁnite dimensional manifold. It has been extended for invariant higher-order lagrangians in [9, 10] on a group, still ﬁnite dimensional. A numerical implementation of the variational and shooting splines has been developed in [21] with applications to medical imaging. The question of existence of an extremum is not addressed in these publications. An attempt is given in [22] where the exact relaxation of the problem is computed in the case of the group of diﬀeomorphisms of the unit interval. In a similar direction, in [11] discuss the convergence of the discretization of cubic splines in some particular inﬁnite dimensional Riemannian context on the space of shapes. As a shape space, we are, in this paper, interested in probability densities or more generally densities endowed with the Wasserstein metric. Since the Wasserstein metric shares some similarities with a Riemannian metric on this space of probability densities, it is natural to study further higher- order models in this context. Our motivation is to answer the following practical question, what is a natural extension of cubic splines to the Wasserstein space. To the best of our knowledge, this question has not been yet addressed in the literature on optimal transport. We present in Section the notion of cubic splines on a Riemannian manifold and detail its variational formulation in Hamiltonian coordinates. We then discuss independently in Section a 1

text: SECOND

text: ORDER

text: MODELS

text: FOR

text: OPTIMAL

text: TRANSPORT

text: AND

text: CUBIC

text: SPLINES

text: ON

text: THE

text: WASSERSTEIN

text: SPACE

text: JEAN-DAVID

text: BENAMOU,

text: THOMAS

text: O.

text: GALLOUE¨T,

text: AND

text: FRANC¸OIS-XAVIER

text: VIALARD

text: Abstract.

text: On

label: abstract

text: the

text: space

text: of

text: probability

text: densities,

text: we

text: extend

text: the

text: Wasserstein

text: geodesics

text: to

text: the

text: case

text: of

text: higher-order

text: interpolation

text: such

text: as

text: cubic

text: spline

text: interpolation.

text: After

text: presenting

text: the

text: natural

text: exten-

text: sion

text: of

text: cubic

text: splines

text: to

text: the

text: Wasserstein

text: space,

text: we

text: propose

text: simpler

text: approach,

text: similarly

text: to

text: Brenier’s

text: generalized

text: Euler

text: solutions.

text: Our

text: method

text: is

text: based

text: on

text: the

text: relaxation

text: of

text: the

text: variational

text: problem

text: on

text: the

text: path

text: space.

text: We

text: propose

text: an

text: eﬃcient

text: implementation

text: based

text: on

text: multimarginal

text: optimal

text: transport

text: and

text: entropic

text: regularization

text: in

text: and

text: Our

text: framework

text: also

text: enables

text: extrapolation

text: of

text: Wasserstein

text: geodesics

text: via

text: a

text: natural

text: convex

text: relaxation.

text: Introduction

text: In

text: this

text: paper,

text: we

text: propose

text: a

text: variational

text: method

text: to

text: generalize

text: cubic

text: splines

text: on

text: the

text: space

text: of

text: densities

text: using

text: multimarginal

text: optimal

text: transport.

text: In

text: short,

text: the

text: proposed

text: method

text: consists

text: in

text: minimizing,

text: on

text: the

text: space

text: of

text: measures

text: on

text: the

text: path

text: space,

text: under

text: marginal

text: constraints,

text: the

text: norm

text: squared

text: of

text: the

text: acceleration.

text: In

text: this

text: setting,

text: entropic

text: regularization

text: can

text: be

text: applied

text: and

text: give

text: an

text: eﬃcient

text: algorithms

text: to

text: approximate

text: the

text: solution.

text: We

text: showcase

text: our

text: methodology

text: on

text: and

text: data.

text: In

text: the

text: past

text: few

text: years,

text: higher-order

text: interpolations

text: methods

text: have

text: been

text: investigated

text: for

text: applications

text: in

text: computer

text: vision

text: or

text: medical

text: imaging,

text: for

text: time-sequence

text: interpolation

text: or

text: regression.

text: The

text: most

text: usual

text: setting

text: is

text: when

text: data

text: are

text: modeled

text: as

text: shapes,

text: which

text: object

text: embedded

text: in

text: the

text: Euclidean

text: space

text: with

text: no

text: preferred

text: parametrization:

text: space

text: of

text: unparametrized

text: curves

text: or

text: surfaces,

text: or

text: images

text: are

text: some

text: of

text: the

text: most

text: important

text: examples.

text: These

text: examples

text: are

text: inﬁnite

text: dimensional

text: but

text: the

text: ﬁnite

text: dimensional

text: case

text: was

text: interesting

text: for

text: camera

text: motion

text: interpolation

text: as

text: ﬁrst

text: introduced

text: in

text: and

text: further

text: developed

text: in

text: Motivated

text: by

text: sometimes

text: diﬀerent

text: applications,

text: the

text: problem

text: of

text: interpolation

text: between

text: two

text: shapes

text: is

text: usually

text: treated

text: via

text: the

text: use

text: of

text: a

text: Riemannian

text: metric

text: on

text: the

text: space

text: of

text: shapes

text: and

text: computing

text: a

text: geodesic

text: between

text: the

text: two

text: shapes.

text: From

text: a

text: mathematical

text: point

text: of

text: view,

text: shape

text: spaces

text: are

text: often

text: inﬁnite

text: dimensional

text: and

text: thus,

text: non-trivial

text: analytical

text: questions

text: arise

text: such

text: as

text: existence

text: of

text: minimizing

text: geodesics

text: or

text: global

text: well-posedness

text: of

text: the

text: initial

text: value

text: problem

text: associated

text: with

text: geodesics.

text: A

text: ﬁnite

text: dimensional

text: approximation

text: is

text: still

text: possible

text: such

text: as

text: in

text: in

text: which

text: spline

text: interpolation

text: is

text: proposed

text: for

text: a

text: diﬀeomorphic

text: group

text: action

text: on

text: a

text: ﬁnite

text: dimensional

text: manifold.

text: It

text: has

text: been

text: extended

text: for

text: invariant

text: higher-order

text: lagrangians

text: in

text: on

text: a

text: group,

text: still

text: ﬁnite

text: dimensional.

text: A

text: numerical

text: implementation

text: of

text: the

text: variational

text: and

text: shooting

text: splines

text: has

text: been

text: developed

text: in

text: with

text: applications

text: to

text: medical

text: imaging.

text: The

text: question

text: of

text: existence

text: of

text: an

text: extremum

text: is

text: not

text: addressed

text: in

text: these

text: publications.

text: An

text: attempt

text: is

text: given

text: in

text: where

text: the

text: exact

text: relaxation

text: of

text: the

text: problem

text: is

text: computed

text: in

text: the

text: case

text: of

text: the

text: group

text: of

text: diﬀeomorphisms

text: of

text: the

text: unit

text: interval.

text: In

text: a

text: similar

text: direction,

text: in

text: discuss

text: the

text: convergence

text: of

text: the

text: discretization

text: of

text: cubic

text: splines

text: in

text: some

text: particular

text: inﬁnite

text: dimensional

text: Riemannian

text: context

text: on

text: the

text: space

text: of

text: shapes.

text: As

text: a

text: shape

text: space,

text: we

text: are,

text: in

text: this

text: paper,

text: interested

text: in

text: probability

text: densities

text: or

text: more

text: generally

text: densities

text: endowed

text: with

text: the

text: Wasserstein

text: metric.

text: Since

text: the

text: Wasserstein

text: metric

text: shares

text: some

text: similarities

text: with

text: a

text: Riemannian

text: metric

text: on

text: this

text: space

text: of

text: probability

text: densities,

text: it

text: is

text: natural

text: to

text: study

text: further

text: higher-

text: order

text: models

text: in

text: this

text: context.

text: Our

text: motivation

text: is

text: to

text: answer

text: the

text: following

text: practical

text: question,

text: what

text: is

text: a

text: natural

text: extension

text: of

text: cubic

text: splines

text: to

text: the

text: Wasserstein

text: space.

text: To

text: the

text: best

text: of

text: our

text: knowledge,

text: this

text: question

text: has

text: not

text: been

text: yet

text: addressed

text: in

text: the

text: literature

text: on

text: optimal

text: transport.

text: We

text: present

text: in

text: Section

text: the

text: notion

text: of

text: cubic

text: splines

text: on

text: a

text: Riemannian

text: manifold

text: and

text: detail

text: its

text: variational

text: formulation

text: in

text: Hamiltonian

text: coordinates.

text: We

text: then

text: discuss

text: independently

text: in

text: Section

text: a


text: 4 JEAN-DAVID BENAMOU, THOMAS O. GALLOUE¨T, AND FRANC¸OIS-XAVIER VIALARD diﬀeomorphism group descends to the Wasserstein L2 metric on the space of densities, or in other (cid:55)→ words, the pushforward action ϕ ϕ∗µ is a Riemannian submersion. An important property 0 of Riemannian submersion is that geodesics on Dens(M) are in correspondence with geodesics on the group, given by horizontal lift. This property is actually contained in Brenier’s polar factoriza- tion theorem, which shows that the horizontal lift is the gradient of a convex function. The ﬁrst order variational calculus is now quite well understood in the Wasserstein space, as well as some higher-order geometrical quantities such as curvature. However, for higher-order variational problems, this reduction does not hold in general even if the Riemannian submersion structure is present. It means that a variational problem on the group can lead to a diﬀerent model. More precisely, instead of minimizing the kinetic energy of the map, one can minimize the acceleration of the map, that is ≥ ∈ Deﬁnition 1 (Monge formulation). Let 0 = t < . . . < t = 1, n 2 and ρ , . . . , ρ be n 0 n 1 n probability measures on M. (cid:55)→ (cid:90) (cid:90) Minimize, among time dependent maps ϕ(t) : M M, 1 | | (4.1) ϕ¨ 2 dµ dt , 0 0 M under the marginal constraints ϕ(t )∗µ = ρ . This minimizing problem is denoted by (MP). i 0 i It is a Monge formulation of the variational problem, similar to standard optimal transport. On a Riemannian manifold M, the notation ϕ¨ stands for D ϕ˙. By the change of variable with the map Dt ϕ, the problem can be written in Eulerian coordinates, that is using the vector ﬁeld associated with ◦ (cid:90) (cid:90) the Lagrangian map ϕ, ∂ ϕ = v ϕ, one aims at minimizing t 1 | | (4.2) u 2ρ dµ dt 0 0 M (cid:40) under the constraints ρ˙ + div(ρv) = 0 (4.3) ∇ v˙ + (v, )v = u , and some marginals constraints such as ρ(t ) = ρ . i i Remark 1. It is now very clear that the two models (3.7) and (4.1) do not coincide a priori, since the optimization set of the ﬁrst model (3.7) is a subset of the second (4.1). Indeed, if one takes ∇ v = p, the two formulations coincide. In this direction, another formal argument is that the Wasserstein space has nonnegative curvature if the underlying space M has nonnegative curvature, but the space of maps in the Euclidean space is ﬂat. Therefore, the two Euler-Lagrange equations (2.2) lead to a diﬀerent evolution equations: for instance, if M is the Euclidean space then the Euler-Lagrange equation for the second model is simply ... ϕ = 0, which is a priori diﬀerent from the splines Euler-Lagrange equation in the Wasserstein case. Since, as is well-known in standard optimal transport, the Monge formulation is not well-posed, we propose to solve instead a relaxation of the problem which takes the form: ≥ ∈ Deﬁnition 2 (Kantorovich relaxation). Let 0 = t < . . . < t = 1, n 3 and ρ , . . . , ρ be n 1 n 1 n probability measures on M. P Minimize on the space of probability measures on the path space H2([0, 1], M) denoted by in (cid:90) short, | | (4.4) min x¨ 2 dµ(x) , µ P ##LTLine##

text: JEAN-DAVID

text: BENAMOU,

text: THOMAS

text: O.

text: GALLOUE¨T,

text: AND

text: FRANC¸OIS-XAVIER

text: VIALARD

text: diﬀeomorphism

text: group

text: descends

text: to

text: the

text: Wasserstein

text: metric

text: on

text: the

text: space

text: of

text: densities,

text: or

text: in

text: other

text: words,

text: the

text: pushforward

text: action

text: ϕ

text: ϕ∗µ

text: is

text: a

text: Riemannian

text: submersion.

text: An

text: important

text: property

text: of

text: Riemannian

text: submersion

text: is

text: that

text: geodesics

text: on

text: Dens(M)

text: are

text: in

text: correspondence

text: with

text: geodesics

text: on

text: the

text: group,

text: given

text: by

text: horizontal

text: lift.

text: This

text: property

text: is

text: actually

text: contained

text: in

text: Brenier’s

text: polar

text: factoriza-

text: tion

text: theorem,

text: which

text: shows

text: that

text: the

text: horizontal

text: lift

text: is

text: the

text: gradient

text: of

text: a

text: convex

text: function.

text: The

text: ﬁrst

text: order

text: variational

text: calculus

text: is

text: now

text: quite

text: well

text: understood

text: in

text: the

text: Wasserstein

text: space,

text: as

text: well

text: as

text: some

text: higher-order

text: geometrical

text: quantities

text: such

text: as

text: curvature.

text: However,

text: for

text: higher-order

text: variational

text: problems,

text: this

text: reduction

text: does

text: not

text: hold

text: in

text: general

text: even

text: if

text: the

text: Riemannian

text: submersion

text: structure

text: is

text: present.

text: It

text: means

text: that

text: a

text: variational

text: problem

text: on

text: the

text: group

text: can

text: lead

text: to

text: a

text: diﬀerent

text: model.

text: More

text: precisely,

text: instead

text: of

text: minimizing

text: the

text: kinetic

text: energy

text: of

text: the

text: map,

text: one

text: can

text: minimize

text: the

text: acceleration

text: of

text: the

text: map,

text: that

text: is

text: Deﬁnition

text: (Monge

text: formulation).

text: Let

text: t

text: t

text: n

text: and

text: ρ

text: ρ

text: be

text: n

text: n

text: n

text: probability

text: measures

text: on

text: M.

text: Minimize,

text: among

text: time

text: dependent

text: maps

text: ϕ(t)

text: M

text: M,

text: ϕ¨

text: dµ

text: dt

text: M

text: under

text: the

text: marginal

text: constraints

text: ϕ(t

text: )∗µ

text: ρ

text: This

text: minimizing

text: problem

text: is

text: denoted

text: by

text: (MP).

text: i

text: i

text: It

text: is

text: a

text: Monge

text: formulation

text: of

text: the

text: variational

text: problem,

text: similar

text: to

text: standard

text: optimal

text: transport.

text: On

text: a

text: Riemannian

text: manifold

text: M,

text: the

text: notation

text: ϕ¨

text: stands

text: for

text: D

text: ϕ˙.

text: By

text: the

text: change

text: of

text: variable

text: with

text: the

text: map

text: Dt

text: ϕ,

text: the

text: problem

text: can

text: be

text: written

text: in

text: Eulerian

text: coordinates,

text: that

text: is

text: using

text: the

text: vector

text: ﬁeld

text: associated

text: with

text: the

text: Lagrangian

text: map

text: ϕ,

text: ϕ

text: v

text: ϕ,

text: one

text: aims

text: at

text: minimizing

text: t

text: u

text: 2ρ

text: dµ

text: dt

text: M

text: under

text: the

text: constraints

text: ρ˙

text: div(ρv)

text: v˙

text: (v,

text: )v

text: u

text: and

text: some

text: marginals

text: constraints

text: such

text: as

text: ρ(t

text: ρ

text: i

text: i

text: Remark

text: It

text: is

text: now

text: very

text: clear

text: that

text: the

text: two

text: models

text: and

text: do

text: not

text: coincide

text: a

text: priori,

text: since

text: the

text: optimization

text: set

text: of

text: the

text: ﬁrst

text: model

text: is

text: a

text: subset

text: of

text: the

text: second

text: Indeed,

text: if

text: one

text: takes

text: v

text: p,

text: the

text: two

text: formulations

text: coincide.

text: In

text: this

text: direction,

text: another

text: formal

text: argument

text: is

text: that

text: the

text: Wasserstein

text: space

text: has

text: nonnegative

text: curvature

text: if

text: the

text: underlying

text: space

text: M

text: has

text: nonnegative

text: curvature,

text: but

text: the

text: space

text: of

text: maps

text: in

text: the

text: Euclidean

text: space

text: is

text: ﬂat.

text: Therefore,

text: the

text: two

text: Euler-Lagrange

text: equations

text: lead

text: to

text: a

text: diﬀerent

text: evolution

text: equations:

text: for

text: instance,

text: if

text: M

text: is

text: the

text: Euclidean

text: space

text: then

text: the

text: Euler-Lagrange

text: equation

text: for

text: the

text: second

text: model

text: is

text: simply

text: ϕ

text: which

text: is

text: a

text: priori

text: diﬀerent

text: from

text: the

text: splines

text: Euler-Lagrange

text: equation

text: in

text: the

text: Wasserstein

text: case.

text: Since,

text: as

text: is

text: well-known

text: in

text: standard

text: optimal

text: transport,

text: the

text: Monge

text: formulation

text: is

text: not

text: well-posed,

text: we

text: propose

text: to

text: solve

text: instead

text: a

text: relaxation

text: of

text: the

text: problem

text: which

text: takes

text: the

text: form:

text: Deﬁnition

text: (Kantorovich

text: relaxation).

text: Let

text: t

text: t

text: n

text: and

text: ρ

text: ρ

text: be

text: n

text: n

text: n

text: probability

text: measures

text: on

text: M.

text: P

text: Minimize

text: on

text: the

text: space

text: of

text: probability

text: measures

text: on

text: the

text: path

text: space

text: M)

text: denoted

text: by

text: in

text: short,

text: min

text: x¨

text: dµ(x)

text: µ

text: P

text: ##LTLine##


text: 8 JEAN-DAVID BENAMOU, THOMAS O. GALLOUE¨T, AND FRANC¸OIS-XAVIER VIALARD (cid:88) (cid:107) − (cid:107) x + x − 2 x 2 (5.6) c (x , ..., x ) := i+1 i 1 i dτ 1 N − dτ3 i=1,N 1 (cid:90) For all times, marginals (4.5) are computed as : (cid:55)→ (cid:78) (5.7) τ dµ (x , ..x ) j dτ 1 N i(cid:54)=j Mi In order to simplify the presentation we will assume that the marginal constraints (4.6) are set at times t , ..t which coincide with times steps of the discretization (of course n < N, meaning the 1 n number of constraint is not the same as the number of time steps). ∈ In short, there exist (j , ..j ) [0, N] such that 1 n (t , .., t ) = (τ , ..., τ ). 1 n j1 jn (cid:90) The constraint (4.6) becomes for all k = 1, ..n (cid:78) (5.8) dµ (x , ..x ) = ρ (x ) dτ 1 N jk jk i(cid:54)=jk Mi where ρ is the prescribed density to interpolate at time τ = t . jk jk k The time discretized problem is the multimarginal problem (5.5 -5.8). The simplest space discretization strategy is to use a regular cartesian grid. In dimension 2 and ∈ for M = [0, 1]2 and at time t , the grid will be denoted x = (α h, β h) for (α , β ) [0, N ] and { } { i } αi,βi i i i i x h = 1 , a = α and b = β will be the vectors of indices. ThNex time andi space discretiization of the pr(cid:88)oblem then becomes (5.9) min C T a,b a,b T a,b × × Where T is the N N N tensor of grid values µ (x , .., x ) and x x dτ α1,β1 αN,βN (5.10) C = c (x , .., x ) a,b dτ α1,β1 αN,βN The marginals (5.7) at all times τ are give(cid:88)n by j (5.11) T a,b \{ } \{ } a αj ,b βj The constraints (5.8) therfore becom(cid:88)es for all k (5.12) T = ρ (x ) \{ } \{ } a,b jk αjk,βjk a αjk ,b βjk \ { } a α denotes the set of indices a minus α . jk jk ##LTLine## ##LTLine##

text: JEAN-DAVID

text: BENAMOU,

text: THOMAS

text: O.

text: GALLOUE¨T,

text: AND

text: FRANC¸OIS-XAVIER

text: VIALARD

text: x

text: x

text: x

text: c

text: (x

text: x

text: i

text: i

text: dτ

text: N

text: For

text: all

text: times,

text: marginals

text: are

text: computed

text: as

text: τ

text: dµ

text: (x

text: ..x

text: j

text: dτ

text: N

text: Mi

text: In

text: order

text: to

text: simplify

text: the

text: presentation

text: we

text: will

text: assume

text: that

text: the

text: marginal

text: constraints

text: are

text: set

text: at

text: times

text: t

text: ..t

text: which

text: coincide

text: with

text: times

text: steps

text: of

text: the

text: discretization

text: (of

text: course

text: n

text: N,

text: meaning

text: the

text: n

text: number

text: of

text: constraint

text: is

text: not

text: the

text: same

text: as

text: the

text: number

text: of

text: time

text: steps).

text: In

text: short,

text: there

text: exist

text: (j

text: ..j

text: N]

text: such

text: that

text: n

text: (t

text: t

text: (τ

text: τ

text: n

text: jn

text: The

text: constraint

text: becomes

text: for

text: all

text: k

text: ..n

text: dµ

text: (x

text: ..x

text: ρ

text: (x

text: dτ

text: N

text: jk

text: jk

text: Mi

text: where

text: ρ

text: is

text: the

text: prescribed

text: density

text: to

text: interpolate

text: at

text: time

text: τ

text: t

text: jk

text: jk

text: k

text: The

text: time

text: discretized

text: problem

text: is

text: the

text: multimarginal

text: problem

text: The

text: simplest

text: space

text: discretization

text: strategy

text: is

text: to

text: use

text: a

text: regular

text: cartesian

text: grid.

text: In

text: dimension

text: and

text: for

text: M

text: and

text: at

text: time

text: t

text: the

text: grid

text: will

text: be

text: denoted

text: x

text: (α

text: h,

text: β

text: h)

text: for

text: (α

text: β

text: N

text: and

text: i

text: αi,βi

text: i

text: i

text: i

text: i

text: x

text: h

text: a

text: α

text: and

text: b

text: β

text: will

text: be

text: the

text: vectors

text: of

text: indices.

text: ThNex

text: time

text: andi

text: space

text: discretiization

text: of

text: the

text: then

text: becomes

text: min

text: C

text: T

text: a,b

text: a,b

text: T

text: a,b

text: Where

text: T

text: is

text: the

text: N

text: N

text: N

text: tensor

text: of

text: grid

text: values

text: µ

text: (x

text: x

text: and

text: x

text: x

text: dτ

text: αN,βN

text: C

text: c

text: (x

text: x

text: a,b

text: dτ

text: αN,βN

text: The

text: marginals

text: at

text: all

text: times

text: τ

text: are

text: by

text: j

text: T

text: a,b

text: a

text: αj

text: ,b

text: βj

text: The

text: constraints

text: therfore

text: for

text: all

text: k

text: T

text: ρ

text: (x

text: a,b

text: jk

text: αjk,βjk

text: a

text: αjk

text: ,b

text: βjk

text: a

text: α

text: denotes

text: the

text: set

text: of

text: indices

text: a

text: minus

text: α

text: jk

text: jk

text: ##LTLine##

text: ##LTLine##


text: 10 JEAN-DAVID BENAMOU, THOMAS O. GALLOUE¨T, AND FRANC¸OIS-XAVIER VIALARD which yields the same minimization problem. 1D case: We present a 1D test case to highlight some of the qualitative properties of the cubic splines interpolation on the space of densities. Initial time data and targets Figure 2. Four interpolation timepoints, 1, 6, 11, 16 and representation of the four density conﬁgurations, as well as 6 intermediate times. This experiment underlines that the spline curve has more smoothness in time and can present some con- centration or diﬀusion eﬀects depending on the data. The entropic regularization − parameter is ε = 8.10 5. We consider four interpolation time points and the corresponding data are mixture of Gaussians of diﬀerent standard deviations. We use a discretization of 140 points on the interval [0, 1] with 16 time steps. This experiment shows that the mass can concentrate or diﬀuse in some situation. Another important point here is that the entropic regularization parameter has an important impact − on this concentration/diﬀusion eﬀects: we show the simulations for ε = 0.002 and ε = 8.10 5. In the simulation with a large ε, the concentration eﬀect is not present and it is due to the diﬀusion on the path space ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: JEAN-DAVID

text: BENAMOU,

text: THOMAS

text: O.

text: GALLOUE¨T,

text: AND

text: FRANC¸OIS-XAVIER

text: VIALARD

text: which

text: yields

text: the

text: same

text: minimization

text: problem.

text: case:

text: We

text: present

text: a

text: test

text: case

text: to

text: highlight

text: some

text: of

text: the

text: qualitative

text: properties

text: of

text: the

text: cubic

text: splines

text: interpolation

text: on

text: the

text: space

text: of

text: densities.

text: Initial

text: time

text: data

text: and

text: targets

text: Figure

text: Four

text: interpolation

text: timepoints,

text: and

text: representation

text: of

text: the

text: four

text: density

text: conﬁgurations,

text: as

text: well

text: as

text: intermediate

text: times.

text: This

text: experiment

text: underlines

text: that

text: the

text: spline

text: curve

text: has

text: more

text: smoothness

text: in

text: time

text: and

text: can

text: present

text: some

text: con-

text: centration

text: or

text: diﬀusion

text: eﬀects

text: depending

text: on

text: the

text: data.

text: The

text: entropic

text: regularization

text: parameter

text: is

text: ε

text: We

text: consider

text: four

text: interpolation

text: time

text: points

text: and

text: the

text: corresponding

text: data

text: are

text: mixture

text: of

text: Gaussians

text: of

text: diﬀerent

text: standard

text: deviations.

text: We

text: use

text: a

text: discretization

text: of

text: points

text: on

text: the

text: interval

text: with

text: time

text: steps.

text: This

text: experiment

text: shows

text: that

text: the

text: mass

text: can

text: concentrate

text: or

text: diﬀuse

text: in

text: some

text: situation.

text: Another

text: important

text: point

text: here

text: is

text: that

text: the

text: entropic

text: regularization

text: parameter

text: has

text: an

text: important

text: impact

text: on

text: this

text: concentration/diﬀusion

text: eﬀects:

text: we

text: show

text: the

text: simulations

text: for

text: ε

text: and

text: ε

text: In

text: the

text: simulation

text: with

text: a

text: large

text: ε,

text: the

text: concentration

text: eﬀect

text: is

text: not

text: present

text: and

text: it

text: is

text: due

text: to

text: the

text: diﬀusion

text: on

text: the

text: path

text: space

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: Under consideration for publication in J. Plasma Phys. 1 Turbulent heating in an inhomogeneous, magnetized plasma slab Michael Barnes1,2†, P. Abiuso3 and W. Dorland4 1Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX2 8ES, UK 2Euratom/CCFE Fusion Association, Culham Science Centre, Abingdon OX14 3DB, UK 3Dipartimento di Fisica dell’Universita` di Pisa, Scuole Normale Superiore, I-56126 Pisa, Italy 4Department of Physics, University of Maryland, College Park, Maryland 20740, USA (Received xx; revised xx; accepted xx) Observational evidence in space and astrophysical plasmas with long collisional mean free path suggests that more massive charged particles may be preferentially heated. One possible mechanism for this is the turbulent cascade of energy from injection to dissipation scales, where the energy is converted to heat. Here we consider a simple system consisting of a magnetized plasma slab of electrons and a single ion species with a cross-ﬁeld density gradient. We show that such a system is subject to an electron drift wave instability, known as the universal instability, which is stabilized only when the electron and ion thermal speeds are equal. For unequal thermal speeds, we ﬁnd that the instability gives rise to turbulent energy exchange between ions and electrons that acts to equalize the thermal speeds. Consequently, this turbulent heating tends to equalize the component temperatures of pair plasmas and to heat ions to much higher temperatures than electrons for conventional mass-ratio plasmas. 1. Introduction An interesting and fundamental question in plasma physics is how thermal equilibrium is determined in an essentially collisionless plasma. In such a system, there is no reason a priori to assume that the relative temperatures of the component species will be equal: On the contrary, there are numerous collisionless heating mechanisms that have been identiﬁed - shocks, magnetic reconnection, cyclotron resonance (Cranmer et al. 1999), and various forms of turbulent heating (Quataert & Gruzinov 1999; Howes et al. 2008; Chandran et al. 2010; Barnes et al. 2012), to name a few - and each of these mechanisms may drive temperature separation instead of equilibration. Indeed, evidence from observations of collisionless space and astrophysical plasmas, e.g., the solar wind and accretion ﬂows onto compact objects, suggests that more massive charged particles may be preferentially heated; cf. (Schmidt et al. 1980; Collier et al. 1996; Kohl et al. 1997; Quataert & Gruzinov 1999). A thorough treatment of this problem would require exploration of a parameter space spanning a wide range of plasma β, electron-ion temperature ratio, energy injection mechanisms, etc., while including all potentially relevant physics present from the macro- scopic to microscopic space-time scales. As such a study is infeasible, we choose to focus on a single heating mechanism - turbulent Joule heating - in a simpliﬁed system: a † Email address for correspondence: michael.barnes@physics.ox.ac.uk

text: Under

text: consideration

text: for

text: publication

text: in

text: J.

text: Plasma

text: Phys.

text: Turbulent

label: title

text: heating

text: in

text: an

text: inhomogeneous,

text: magnetized

text: plasma

text: slab

text: Michael

label: author

text: P.

text: and

text: W.

text: Peierls

text: Centre

text: for

text: Theoretical

text: Physics,

text: University

text: of

text: Oxford,

text: Oxford

text: UK

text: Fusion

text: Association,

text: Culham

text: Science

text: Centre,

text: Abingdon

text: UK

text: di

text: Fisica

text: dell’Universita`

text: di

text: Pisa,

text: Scuole

text: Normale

text: Superiore,

text: Pisa,

text: Italy

text: of

text: Physics,

text: University

text: of

text: Maryland,

text: College

text: Park,

text: Maryland

text: USA

text: (Received

text: xx;

text: revised

text: xx;

text: accepted

text: xx)

text: Observational

text: evidence

text: in

text: space

text: and

text: astrophysical

text: plasmas

text: with

text: long

text: collisional

text: mean

text: free

text: path

text: suggests

text: that

text: more

text: massive

text: charged

text: particles

text: may

text: be

text: preferentially

text: heated.

text: One

text: possible

text: mechanism

text: for

text: this

text: is

text: the

text: turbulent

text: cascade

text: of

text: energy

text: from

text: injection

text: to

text: dissipation

text: scales,

text: where

text: the

text: energy

text: is

text: converted

text: to

text: heat.

text: Here

text: we

text: consider

text: a

text: simple

text: system

text: consisting

text: of

text: a

text: magnetized

text: plasma

text: slab

text: of

text: electrons

text: and

text: a

text: single

text: ion

text: species

text: with

text: a

text: cross-ﬁeld

text: density

text: gradient.

text: We

text: show

text: that

text: such

text: a

text: system

text: is

text: subject

text: to

text: an

text: electron

text: drift

text: wave

text: instability,

text: known

text: as

text: the

text: universal

text: instability,

text: which

text: is

text: stabilized

text: only

text: when

text: the

text: electron

text: and

text: ion

text: thermal

text: speeds

text: are

text: equal.

text: For

text: unequal

text: thermal

text: speeds,

text: we

text: ﬁnd

text: that

text: the

text: instability

text: gives

text: rise

text: to

text: turbulent

text: energy

text: exchange

text: between

text: ions

text: and

text: electrons

text: that

text: acts

text: to

text: equalize

text: the

text: thermal

text: speeds.

text: Consequently,

text: this

text: turbulent

text: heating

text: tends

text: to

text: equalize

text: the

text: component

text: temperatures

text: of

text: pair

text: plasmas

text: and

text: to

text: heat

text: ions

text: to

text: much

text: higher

text: temperatures

text: than

text: electrons

text: for

text: conventional

text: mass-ratio

text: plasmas.

text: Introduction

text: An

text: interesting

text: and

text: fundamental

text: question

text: in

text: plasma

text: physics

text: is

text: how

text: thermal

text: equilibrium

text: is

text: determined

text: in

text: an

text: essentially

text: collisionless

text: plasma.

text: In

text: such

text: a

text: system,

text: there

text: is

text: no

text: reason

text: a

text: priori

text: to

text: assume

text: that

text: the

text: relative

text: temperatures

text: of

text: the

text: component

text: species

text: will

text: be

text: equal:

text: On

text: the

text: contrary,

text: there

text: are

text: numerous

text: collisionless

text: heating

text: mechanisms

text: that

text: have

text: been

text: identiﬁed

text: shocks,

text: magnetic

text: reconnection,

text: cyclotron

text: resonance

text: (Cranmer

text: et

text: al.

text: and

text: various

text: forms

text: of

text: turbulent

text: heating

text: (Quataert

text: Gruzinov

text: Howes

text: et

text: al.

text: Chandran

text: et

text: al.

text: Barnes

text: et

text: al.

text: to

text: name

text: a

text: few

text: and

text: each

text: of

text: these

text: mechanisms

text: may

text: drive

text: temperature

text: separation

text: instead

text: of

text: equilibration.

text: Indeed,

text: evidence

text: from

text: observations

text: of

text: collisionless

text: space

text: and

text: astrophysical

text: plasmas,

text: e.g.,

text: the

text: solar

text: wind

text: and

text: accretion

text: ﬂows

text: onto

text: compact

text: objects,

text: suggests

text: that

text: more

text: massive

text: charged

text: particles

text: may

text: be

text: preferentially

text: heated;

text: cf.

text: (Schmidt

text: et

text: al.

text: Collier

text: et

text: al.

text: Kohl

text: et

text: al.

text: Quataert

text: Gruzinov

text: A

text: thorough

text: treatment

text: of

text: this

text: problem

text: would

text: require

text: exploration

text: of

text: a

text: parameter

text: space

text: spanning

text: a

text: wide

text: range

text: of

text: plasma

text: β,

text: electron-ion

text: temperature

text: ratio,

text: energy

text: injection

text: mechanisms,

text: etc.,

text: while

text: including

text: all

text: potentially

text: relevant

text: physics

text: present

text: from

text: the

text: macro-

text: scopic

text: to

text: microscopic

text: space-time

text: scales.

text: As

text: such

text: a

text: study

text: is

text: infeasible,

text: we

text: choose

text: to

text: focus

text: on

text: a

text: single

text: heating

text: mechanism

text: turbulent

text: Joule

text: heating

text: in

text: a

text: simpliﬁed

text: system:

text: a

text: Email

text: address

text: for

text: correspondence:

text: michael.barnes@physics.ox.ac.uk


text: VARIATIONAL HIGHER-ORDER INTERPOLATION ON THE GROUP OF DIFFEOMORPHISMS3 manifold without boundary of dimension d. We deﬁne hereafter a group of diﬀeo- morphisms of M which is a complete metric space. Consider a space V a Hilbert space of vector ﬁelds on M (rapidly decreasing at inﬁnity in the unbounded case), → ∞ R left invariant by their ﬂows, such that the inclusion map V (cid:44) W1, (M, d) is continuous. This hypothesis implies that the ﬂow of a time dependent vector ﬁeld in L2([0, 1], V ) is well deﬁned, see [You08, Appendix C]. Then, the set of ﬂows at G time 1 deﬁnes a group of diﬀeomorphisms denoted by . Denoting V (2.1) Fl (ξ) = ϕ(1) 1 where ϕ solves the ﬂow equation (2.2) ∂ ϕ(t, x) = ξ(t, ϕ(t, x)) t ∀ ∈ (2.3) ϕ(0, x) = x x D , we deﬁne G { ∃ ∈ } (2.4) d=ef. ϕ(1) : ξ L2([0, 1], V ) s.t. Fl (ξ) , V 1 which has been introduced by Trouv´e in [Tro95]. On this group, Trouv´e deﬁnes a metric (cid:26)(cid:90) (cid:27) 1 (cid:107) (cid:107) ∈ ◦ (2.5) dist(ψ , ψ )2 = inf ξ 2 dt : ξ L2([0, 1], V ) s.t. ψ = Fl (ξ) ψ 1 0 V 1 1 0 0 G under which he proves that is complete. In full generality, that is for a V general space of vector ﬁelds V , very few properties are known on this group. For instance, it is a priori not a topological group, or more precisely, there is no known topological structure making it a topological group (the inversion need not be continuous). Moreover, there does not need to be a diﬀerentiable structure on this group. However, for certain choices of spaces V , such structures are available and therefore more properties can be derived in this situation. Indeed, consider D the group s(M), with s > d/2 + 1, which consists of all C1-diﬀeomorphisms of Sobolev regularity Hs. It is known since the work of Ebin and Marsden [EM70] D that s(M) is a smooth Hilbert manifold and a topological group. It only remains G D to prove that = s(M) (the connected component of identity) which is done Hs 0 in [BV17, Section 8] and its main result is R Theorem 2. Let M be d or a closed manifold and s > d/2+1. If Gs is a smooth, D right-invariant Sobolev-metric of order s on s(M), then D (1) ( s(M), Gs) is geodesically complete; D (2) ( s(M) , dists) is a complete metric space; 0 D (3) Any two elements of s(M) can be joined by a minimizing geodesic. 0 D D The statements also hold for the subgroups s(M) and s (M) of diﬀeomorphisms µ ω preserving a volume form µ or a symplectic structure ω.

text: VARIATIONAL

text: HIGHER-ORDER

text: INTERPOLATION

text: ON

text: THE

text: GROUP

text: OF

text: manifold

text: without

text: boundary

text: of

text: dimension

text: d.

text: We

text: deﬁne

text: hereafter

text: a

text: group

text: of

text: diﬀeo-

text: morphisms

text: of

text: M

text: which

text: is

text: a

text: complete

text: metric

text: space.

text: Consider

text: a

text: space

text: V

text: a

text: Hilbert

text: space

text: of

text: vector

text: ﬁelds

text: on

text: M

text: (rapidly

text: decreasing

text: at

text: inﬁnity

text: in

text: the

text: unbounded

text: case),

text: R

text: left

text: invariant

text: by

text: their

text: ﬂows,

text: such

text: that

text: the

text: inclusion

text: map

text: V

text: (M,

text: d)

text: is

text: continuous.

text: This

text: hypothesis

text: implies

text: that

text: the

text: ﬂow

text: of

text: a

text: time

text: dependent

text: vector

text: ﬁeld

text: in

text: V

text: is

text: well

text: deﬁned,

text: see

text: Appendix

text: C].

text: Then,

text: the

text: set

text: of

text: ﬂows

text: at

text: G

text: time

text: deﬁnes

text: a

text: group

text: of

text: diﬀeomorphisms

text: denoted

text: by

text: Denoting

text: V

text: Fl

text: (ξ)

text: where

text: ϕ

text: solves

text: the

text: ﬂow

text: equation

text: ϕ(t,

text: x)

text: ξ(t,

text: ϕ(t,

text: x))

text: t

text: x)

text: x

text: x

text: D

text: we

text: deﬁne

text: G

text: d=ef.

text: ξ

text: V

text: s.t.

text: Fl

text: (ξ)

text: V

text: which

text: has

text: been

text: introduced

text: by

text: Trouv´e

text: in

text: On

text: this

text: group,

text: Trouv´e

text: deﬁnes

text: a

text: metric

text: dist(ψ

text: ψ

text: inf

text: ξ

text: dt

text: ξ

text: V

text: s.t.

text: ψ

text: Fl

text: (ξ)

text: ψ

text: V

text: G

text: under

text: which

text: he

text: proves

text: that

text: is

text: complete.

text: In

text: full

text: generality,

text: that

text: is

text: for

text: a

text: V

text: general

text: space

text: of

text: vector

text: ﬁelds

text: V

text: very

text: few

text: properties

text: are

text: known

text: on

text: this

text: group.

text: For

text: instance,

text: it

text: is

text: a

text: priori

text: not

text: a

text: topological

text: group,

text: or

text: more

text: precisely,

text: there

text: is

text: no

text: known

text: topological

text: structure

text: making

text: it

text: a

text: topological

text: group

text: (the

text: inversion

text: need

text: not

text: be

text: continuous).

text: Moreover,

text: there

text: does

text: not

text: need

text: to

text: be

text: a

text: diﬀerentiable

text: structure

text: on

text: this

text: group.

text: However,

text: for

text: certain

text: choices

text: of

text: spaces

text: V

text: such

text: structures

text: are

text: available

text: and

text: therefore

text: more

text: properties

text: can

text: be

text: derived

text: in

text: this

text: situation.

text: Indeed,

text: consider

text: D

text: the

text: group

text: s(M),

text: with

text: s

text: which

text: consists

text: of

text: all

text: of

text: Sobolev

text: regularity

text: Hs.

text: It

text: is

text: known

text: since

text: the

text: work

text: of

text: Ebin

text: and

text: Marsden

text: D

text: that

text: s(M)

text: is

text: a

text: smooth

text: Hilbert

text: manifold

text: and

text: a

text: topological

text: group.

text: It

text: only

text: remains

text: G

text: D

text: to

text: prove

text: that

text: s(M)

text: (the

text: connected

text: component

text: of

text: identity)

text: which

text: is

text: done

text: Hs

text: in

text: Section

text: and

text: its

text: main

text: result

text: is

text: R

text: Theorem

text: Let

text: M

text: be

text: d

text: or

text: a

text: closed

text: manifold

text: and

text: s

text: If

text: Gs

text: is

text: a

text: smooth,

text: D

text: right-invariant

text: Sobolev-metric

text: of

text: order

text: s

text: on

text: s(M),

text: then

text: D

text: s(M),

text: Gs)

text: is

text: geodesically

text: complete;

text: D

text: s(M)

text: dists)

text: is

text: a

text: complete

text: metric

text: space;

text: D

text: Any

text: two

text: elements

text: of

text: s(M)

text: can

text: be

text: joined

text: by

text: a

text: minimizing

text: geodesic.

text: D

text: D

text: The

text: statements

text: also

text: hold

text: for

text: the

text: subgroups

text: s(M)

text: and

text: s

text: (M)

text: of

text: diﬀeomorphisms

text: µ

text: ω

text: preserving

text: a

text: volume

text: form

text: µ

text: or

text: a

text: symplectic

text: structure

text: ω.


text: 12 VLTI is to combine interferometry with high-dispersion spectroscopy such as performed with single-aperture telescopes (e.g., Snellen et al 2015). This tech- nique is currently being explored with VLTI/GRAVITY and can be further extended to be used with a nulling instrument. Finally, one can also consider an interesting new idea of symmetric beam combination scheme that is insen- sitive to polarization states (i.e., the Cross Cuber Nuller, H´enault and Spang 2014). The application of these techniques to the VLTI will be investigated during the Hi-5 study. 5 Synergies with other instruments Hi-5 will be complementary to several future high-angular resolution in- struments operating in the tIR as described below. – MATISSE (Multi AperTure mid-Infrared SpectroScopic Experiment, Lopez et al 2014; Matter et al 2016b) is the second-generation tIR and mIR spec- trograph and imager for the VLTI. MATISSE will provide a wide wave- length coverage, from 2.8 to 13 µm, associated with a milli-arcsecond scale angular resolution (3 mas in L band; 10 mas in N band), and various ∼ ∼ spectral resolutions from R 30 to R 5000. In terms of performance, the- oretical MATISSE visibility accuracies of 1 to 3 percent in L and M bands, and 8 percent in N band, were estimated for a 20 Jy source; estimates based on SNR calculations including the contribution of the fundamental noises (source photon noise, readout noise, thermal background photon noise) and the transfer function variations (Matter et al 2016a). More recently, in the frame of the MATISSE test phase in lab, many instrumental visibilities were measured over 4 hours in LM band, and 3 days in N band. Those mea- surements were performed with a very bright artiﬁcial IR source in order to estimate the instrumental contribution to the accuracy without being limited by the fundamental noises. Such a source would have an equivalent ﬂux, if observed with the UTs, of 20 to 70 Jy in N-band, 400 Jy in M-band, and 600 Jy in L band. This lead to absolute visibility accuracies lower than 0.5 percent in L band, 0.4 percent in M band, and 2.5 percent in N band, on average over the corresponding spectral band. Those promising results are extensively described in internal ESO documents written by the MA- TISSE consortium (private communication with A. Matter). Eventually, the on-sky tests (commissioning), starting in March 2018, will provide the real on-sky performance (sensitivity, accuracy) of MATISSE, which will notably include the eﬀects of the sky thermal background ﬂuctuations, the atmospheric turbulence, and the on-sky calibration. – ELT/METIS (Brandl et al 2016) is the Mid-infrared E-ELT Imager and Spectro-graph for the European Extremely Large Telescope. METIS will provide diﬀraction limited imaging and medium resolution slit spectroscopy in the 3 to 19 µm range, as well as high resolution (R = 100000) integral ﬁeld spectroscopy from 2.9 to 5.3 µm. Assuming a collecting aperture of 39 m in diameter, METIS will provide an angular resolution in the nIR similar to that of Hi-5 in the tIR (see Figure 1). VLTI/Hi-5 will hence provide complementary high-contrast observations to characterize the ob- ##LTLine##

text: VLTI

text: is

text: to

text: combine

text: interferometry

text: with

text: high-dispersion

text: spectroscopy

text: such

text: as

text: performed

text: with

text: single-aperture

text: telescopes

text: (e.g.,

text: Snellen

text: et

text: al

text: This

text: tech-

text: nique

text: is

text: currently

text: being

text: explored

text: with

text: VLTI/GRAVITY

text: and

text: can

text: be

text: further

text: extended

text: to

text: be

text: used

text: with

text: a

text: nulling

text: instrument.

text: Finally,

text: one

text: can

text: also

text: consider

text: an

text: interesting

text: new

text: idea

text: of

text: symmetric

text: beam

text: combination

text: scheme

text: that

text: is

text: insen-

text: sitive

text: to

text: polarization

text: states

text: (i.e.,

text: the

text: Cross

text: Cuber

text: Nuller,

text: H´enault

text: and

text: Spang

text: The

text: application

text: of

text: these

text: techniques

text: to

text: the

text: VLTI

text: will

text: be

text: investigated

text: during

text: the

text: study.

label: section

text: Synergies

text: with

text: other

text: instruments

text: will

text: be

text: complementary

text: to

text: several

text: future

text: high-angular

text: resolution

text: in-

text: struments

text: operating

text: in

text: the

text: tIR

text: as

text: described

text: below.

text: MATISSE

text: (Multi

text: AperTure

text: mid-Infrared

text: SpectroScopic

text: Experiment,

text: Lopez

text: et

text: al

text: Matter

text: et

text: al

text: is

text: the

text: second-generation

text: tIR

text: and

text: mIR

text: spec-

text: trograph

text: and

text: imager

text: for

text: the

text: VLTI.

text: MATISSE

text: will

text: provide

text: a

text: wide

text: wave-

text: length

text: coverage,

text: from

text: to

text: µm,

text: associated

text: with

text: a

text: milli-arcsecond

text: scale

text: angular

text: resolution

text: mas

text: in

text: L

text: band;

text: mas

text: in

text: N

text: band),

text: and

text: various

text: spectral

text: resolutions

text: from

text: R

text: to

text: R

text: In

text: terms

text: of

text: performance,

text: the-

text: oretical

text: MATISSE

text: visibility

text: accuracies

text: of

text: to

text: percent

text: in

text: L

text: and

text: M

text: bands,

text: and

text: percent

text: in

text: N

text: band,

text: were

text: estimated

text: for

text: a

text: Jy

text: source;

text: estimates

text: based

text: on

text: SNR

text: calculations

text: including

text: the

text: contribution

text: of

text: the

text: fundamental

text: noises

text: (source

text: photon

text: noise,

text: readout

text: noise,

text: thermal

text: background

text: photon

text: noise)

text: and

text: the

text: transfer

text: function

text: variations

text: (Matter

text: et

text: al

text: More

text: recently,

text: in

text: the

text: frame

text: of

text: the

text: MATISSE

text: test

text: phase

text: in

text: lab,

text: many

text: instrumental

text: visibilities

text: were

text: measured

text: over

text: hours

text: in

text: LM

text: band,

text: and

text: days

text: in

text: N

text: band.

text: Those

text: mea-

text: surements

text: were

text: performed

text: with

text: a

text: very

text: bright

text: artiﬁcial

text: IR

text: source

text: in

text: order

text: to

text: estimate

text: the

text: instrumental

text: contribution

text: to

text: the

text: accuracy

text: without

text: being

text: limited

text: by

text: the

text: fundamental

text: noises.

text: Such

text: a

text: source

text: would

text: have

text: an

text: equivalent

text: ﬂux,

text: if

text: observed

text: with

text: the

text: UTs,

text: of

text: to

text: Jy

text: in

text: N-band,

text: Jy

text: in

text: M-band,

text: and

text: Jy

text: in

text: L

text: band.

text: This

text: lead

text: to

text: absolute

text: visibility

text: accuracies

text: lower

text: than

text: percent

text: in

text: L

text: band,

text: percent

text: in

text: M

text: band,

text: and

text: percent

text: in

text: N

text: band,

text: on

text: average

text: over

text: the

text: corresponding

text: spectral

text: band.

text: Those

text: promising

text: results

text: are

text: extensively

text: described

text: in

text: internal

text: ESO

text: documents

text: written

text: by

text: the

text: MA-

text: TISSE

text: consortium

text: (private

text: communication

text: with

text: A.

text: Matter).

text: Eventually,

text: the

text: on-sky

text: tests

text: (commissioning),

text: starting

text: in

text: March

text: will

text: provide

text: the

text: real

text: on-sky

text: performance

text: (sensitivity,

text: accuracy)

text: of

text: MATISSE,

text: which

text: will

text: notably

text: include

text: the

text: eﬀects

text: of

text: the

text: sky

text: thermal

text: background

text: ﬂuctuations,

text: the

text: atmospheric

text: turbulence,

text: and

text: the

text: on-sky

text: calibration.

text: ELT/METIS

text: (Brandl

text: et

text: al

text: is

text: the

text: Mid-infrared

text: E-ELT

text: Imager

text: and

text: Spectro-graph

text: for

text: the

text: European

text: Extremely

text: Large

text: Telescope.

text: METIS

text: will

text: provide

text: diﬀraction

text: limited

text: imaging

text: and

text: medium

text: resolution

text: slit

text: spectroscopy

text: in

text: the

text: to

text: µm

text: range,

text: as

text: well

text: as

text: high

text: resolution

text: (R

text: integral

text: ﬁeld

text: spectroscopy

text: from

text: to

text: µm.

text: Assuming

text: a

text: collecting

text: aperture

text: of

text: m

text: in

text: diameter,

text: METIS

text: will

text: provide

text: an

text: angular

text: resolution

text: in

text: the

text: nIR

text: similar

text: to

text: that

text: of

text: in

text: the

text: tIR

text: (see

text: Figure

text: will

text: hence

text: provide

text: complementary

text: high-contrast

text: observations

text: to

text: characterize

text: the

text: ob-

text: ##LTLine##