text: the upper contingent derivative of V at x in the direction f. over states up to some time t and the regularity information, F { } N In particular, when V is Gateaux differentiable and = v we verify whether the system behaves safely at a given time is a singleton, D V (x) coincides with the gradient T > t . + (cid:0) N (cid:1) ≡ ∇ Corollary 1. Consider differential inclusion (9) with u D+V (x)(v) = ( V (x))T v. 0 and let T > t . If there exist a function B ∈ ≡ C R R ∩ C ∞N R TaCRn1hd(eRol≥reRet0m;TR2)>. aCtn0od.nasIfidptehoresridetiifvfeeexriedsntetﬁiaanliftueinnccfultuinoscnitoionBn(1∈W0)Cw∈1(iRthLn(u;RRn) ×∩0 tLio1(nRsnn×W; [+t)N, ∞∈1)(;[t[LNtN(,R, ∞n)×;)) [stuNca,hn∞dth)ta;wt[toNp,o∞si)t)iveadnedﬁniWte−func∈- ≥ ; ≥ ) such that 0 0 − ∈ X B (x(T), T) B (x(t ), t ) > 0, x(T) , (13) − ∈ X N N u B (x(T), T) B (x(t ), t ) > 0, x(T) , (11) (cid:16) (cid:17) 0 0 u (cid:16) (cid:17) ≤ − ∈ D B(t, x) X˙ −, 1 W−(t, x), t [t , T], (14) + N ≤ − ∈ ∈ F D B(t, x)(v, 1) W(t, x), t [t , T], v (t, x), ≤ − ∈ + 0 D B(t, x) X˙ , 1 W (t, x), t [t , T], (15) (12) + + + N ∈ X ∈ X then the solutions of (10) satisfy x(T) / u. then the solutions of (9) satisfy x(T) / u. Proof. The proof is carried out by contradiction. Assume it Proof. Inequality (13) ensures that (11) holds. Multiplying ∈ X ≤ ≤ holds that x(T) u. Then, (11) implies that both sides of inequality (14) with a co≤nstant ≤0 α− 1 and inequality (15) with a constant 0 α 1 such that B (x(T), T) > B (x(t ), t ) . + 0 0 α− + α = 1 and adding them, we obtain + Furthermore, using the comparison theorem for differential inclusions [26, Proposition 8, p. 289(cid:90)] and inequality (12), we α−D+B(t, x)(X˙ −, 1) + α+D+B(t, x)(X˙ +, 1) ≤ − − can infer that α−W−(t, x) α W (t, x). + + B(x(s), s) − B(x(t0), t0) ≤ − s W(x, τ) dτ ≤ 0. Since D+ is a linear operator, we have That is, t0 D+B(t, x)(α−X˙ − + α+X˙ +, 1) ≤ ≤ B(x(s), s) B(x(tN), tN). D+B(t, x)(α−≤X˙ −−) + D+B(t, x−)(α−X˙ +) Since s was chosen arbitrary, this is a contradiction. Thus, the α−W−(t, x) α+W+(t, x), ∈ X solutions of (10) satisfy x(T) / . where, in the last line above, we applied inequalities (14) and u { } Theorem 2 presents conditions to check the safety of the (15). Let W(t, x) = min W−(t, x), W+(t, x) . We obtain solutions of a differential inclusion. The above formulation D B(t, x)(α−X˙ − + α X˙ , 1) only necessitates the barrier certiﬁcate to have an upper + ≤ − + + − contingent derivative. Hence, our formulation here includes α−W−(t, x) α+W+(t, x) ≤ − − non-smooth barrier certiﬁcates. The closet work on this class (α− + α )W(t, x) = W(t, x). + of barrier certiﬁcates is the recent research [39], wherein the That is, authors considered min/max barrier functions with application ≤ − ∈ { } to control of multi-robot systems. Nonetheless, the formulation D B(t, x)(v, 1) W(t, x), v co X˙ −, X˙ . + + in [39] is based on ﬁxing the the barrier functions, i.e., the 0- Thus inequality (12) is also satisﬁed. This completes the proof. super level sets of the barrier functions deﬁne the unsafe sets. This assumption can be restrictive in many cases, in particular, general semi-algebraic sets. In this study, we do not make such We show in Section that if we parametrize B and W by X assumptions on the structure of the barrier functions and we polynomials of ﬁxed degree and let be a semi-algebraic u allow them to be variables in the computational formulation set, then we can check inequalities (13) through (15) using in Section . sum of squares programming. We next address Problem 2 by proposing a method to design B. Safety Controller Synthesis for Data-Driven Differential controllers such that solutions to (7) are safe with respect to a X Inclusions given unsafe set for all t > t . In other words, we address u N Problem 2. Having established a safety analysis theorem for differential (cid:0) (cid:1) inclusions, we proceed by proposing conditions for safety anal- Corollary 2. Consider differential inclusion (9) ysis and safe controller synthesis for data-driven differential and let T > t . If there exist a function ∈ C R R ∩ C N ∞ R inclusion (9). We begin with the safety analysis result, which B 1 n; ) 1([t , ); and a positive deﬁnite ∈ L R × N ∞ ∞ follows from Theorem 2. In other words, given the limited data function W ( n [t , ); [t , )) such that N N ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: the

text: upper

text: contingent

text: derivative

text: of

text: V

text: at

text: x

text: in

text: the

text: direction

text: f.

text: over

text: states

text: up

text: to

text: some

text: time

text: t

text: and

text: the

text: regularity

text: information,

text: F

text: N

text: In

text: particular,

text: when

text: V

text: is

text: Gateaux

text: differentiable

text: and

text: v

text: we

text: verify

text: whether

text: the

text: system

text: behaves

text: safely

text: at

text: a

text: given

text: time

text: is

text: a

text: singleton,

text: D

text: V

text: (x)

text: coincides

text: with

text: the

text: gradient

text: T

text: t

text: N

text: Corollary

text: Consider

text: differential

text: inclusion

text: with

text: u

text: D+V

text: (x)(v)

text: V

text: (x))T

text: v.

text: and

text: let

text: T

text: t

text: If

text: there

text: exist

text: a

text: function

text: B

text: C

text: R

text: R

text: C

text: ∞N

text: R

text: ×∩0

text: [+t)N,

text: ∞n)×;))

text: such

text: that

text: X

text: B

text: (x(T),

text: T)

text: B

text: (x(t

text: t

text: x(T)

text: X

text: N

text: N

text: u

text: B

text: (x(T),

text: T)

text: B

text: (x(t

text: t

text: x(T)

text: u

text: D

text: B(t,

text: x)

text: X˙

text: W−(t,

text: x),

text: t

text: [t

text: T],

text: N

text: F

text: D

text: B(t,

text: x)(v,

text: W(t,

text: x),

text: t

text: [t

text: T],

text: v

text: (t,

text: x),

text: D

text: B(t,

text: x)

text: X˙

text: W

text: (t,

text: x),

text: t

text: [t

text: T],

text: N

text: X

text: X

text: then

text: the

text: solutions

text: of

text: satisfy

text: x(T)

text: u.

text: then

text: the

text: solutions

text: of

text: satisfy

text: x(T)

text: u.

text: Proof.

text: The

text: proof

text: is

text: carried

text: out

text: by

text: contradiction.

text: Assume

text: it

text: Proof.

text: Inequality

text: ensures

text: that

text: holds.

text: Multiplying

text: X

text: holds

text: that

text: x(T)

text: u.

text: Then,

text: implies

text: that

text: both

text: sides

text: of

text: inequality

text: with

text: a

text: co≤nstant

text: ≤0

text: α−

text: and

text: inequality

text: with

text: a

text: constant

text: α

text: such

text: that

text: B

text: (x(T),

text: T)

text: B

text: (x(t

text: t

text: α−

text: α

text: and

text: adding

text: them,

text: we

text: obtain

text: Furthermore,

text: using

text: the

text: comparison

text: theorem

text: for

text: differential

text: inclusions

text: Proposition

text: p.

text: and

text: inequality

text: we

text: α−D+B(t,

text: x)(X˙

text: α+D+B(t,

text: x)(X˙

text: can

text: infer

text: that

text: α−W−(t,

text: x)

text: α

text: W

text: (t,

text: x).

text: B(x(s),

text: s)

text: s

text: W(x,

text: τ)

text: dτ

text: Since

text: D+

text: is

text: a

text: linear

text: operator,

text: we

text: have

text: That

text: is,

text: D+B(t,

text: x)(α−X˙

text: α+X˙

text: B(x(s),

text: s)

text: B(x(tN),

text: tN).

text: D+B(t,

text: x)(α−≤X˙

text: D+B(t,

text: x−)(α−X˙

text: Since

text: s

text: was

text: chosen

text: arbitrary,

text: this

text: is

text: a

text: contradiction.

text: Thus,

text: the

text: α−W−(t,

text: x)

text: α+W+(t,

text: x),

text: X

text: solutions

text: of

text: satisfy

text: x(T)

text: where,

text: in

text: the

text: last

text: line

text: above,

text: we

text: applied

text: inequalities

text: and

text: u

text: Theorem

text: presents

text: conditions

text: to

text: check

text: the

text: safety

text: of

text: the

text: Let

text: W(t,

text: x)

text: min

text: W−(t,

text: x),

text: W+(t,

text: x)

text: We

text: obtain

text: solutions

text: of

text: a

text: differential

text: inclusion.

text: The

text: above

text: formulation

text: D

text: B(t,

text: x)(α−X˙

text: α

text: X˙

text: only

text: necessitates

text: the

text: barrier

text: certiﬁcate

text: to

text: have

text: an

text: upper

text: contingent

text: derivative.

text: Hence,

text: our

text: formulation

text: here

text: includes

text: α−W−(t,

text: x)

text: α+W+(t,

text: x)

text: non-smooth

text: barrier

text: certiﬁcates.

text: The

text: closet

text: work

text: on

text: this

text: class

text: (α−

text: α

text: )W(t,

text: x)

text: W(t,

text: x).

text: of

text: barrier

text: certiﬁcates

text: is

text: the

text: recent

text: research

text: wherein

text: the

text: That

text: is,

text: authors

text: considered

text: min/max

text: barrier

text: functions

text: with

text: application

text: to

text: control

text: of

text: multi-robot

text: systems.

text: Nonetheless,

text: the

text: formulation

text: D

text: B(t,

text: x)(v,

text: W(t,

text: x),

text: v

text: co

text: X˙

text: X˙

text: in

text: is

text: based

text: on

text: ﬁxing

text: the

text: the

text: barrier

text: functions,

text: i.e.,

text: the

text: Thus

text: inequality

text: is

text: also

text: satisﬁed.

text: This

text: completes

text: the

text: proof.

text: super

text: level

text: sets

text: of

text: the

text: barrier

text: functions

text: deﬁne

text: the

text: unsafe

text: sets.

text: This

text: assumption

text: can

text: be

text: restrictive

text: in

text: many

text: cases,

text: in

text: particular,

text: general

text: semi-algebraic

text: sets.

text: In

text: this

text: study,

text: we

text: do

text: not

text: make

text: such

text: We

text: show

text: in

text: Section

text: that

text: if

text: we

text: parametrize

text: B

text: and

text: W

text: by

text: X

text: assumptions

text: on

text: the

text: structure

text: of

text: the

text: barrier

text: functions

text: and

text: we

text: polynomials

text: of

text: ﬁxed

text: degree

text: and

text: let

text: be

text: a

text: semi-algebraic

text: u

text: allow

text: them

text: to

text: be

text: variables

text: in

text: the

text: computational

text: formulation

text: set,

text: then

text: we

text: can

text: check

text: inequalities

text: through

text: using

text: in

text: Section

text: sum

text: of

text: squares

text: programming.

text: We

text: next

text: address

text: Problem

text: by

text: proposing

text: a

text: method

text: to

text: design

text: B.

text: Safety

text: Controller

text: Synthesis

text: for

text: Data-Driven

text: Differential

text: controllers

text: such

text: that

text: solutions

text: to

text: are

text: safe

text: with

text: respect

text: to

text: a

text: X

text: Inclusions

text: given

text: unsafe

text: set

text: for

text: all

text: t

text: t

text: In

text: other

text: words,

text: we

text: address

text: u

text: N

text: Problem

text: Having

text: established

text: a

text: safety

text: analysis

text: theorem

text: for

text: differential

text: inclusions,

text: we

text: proceed

text: by

text: proposing

text: conditions

text: for

text: safety

text: anal-

text: Corollary

text: Consider

text: differential

text: inclusion

text: ysis

text: and

text: safe

text: controller

text: synthesis

text: for

text: data-driven

text: differential

text: and

text: let

text: T

text: t

text: If

text: there

text: exist

text: a

text: function

text: C

text: R

text: R

text: C

text: N

text: R

text: inclusion

text: We

text: begin

text: with

text: the

text: safety

text: analysis

text: result,

text: which

text: B

text: n;

text: and

text: a

text: positive

text: deﬁnite

text: L

text: R

text: N

text: follows

text: from

text: Theorem

text: In

text: other

text: words,

text: given

text: the

text: limited

text: data

text: function

text: W

text: n

text: [t

text: [t

text: such

text: that

text: N

text: N

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: landingsafe2222-eps-converted-to.pdf 12 { } 10 Fig. 6: The data points x(ti) 2i=51 (black circles), the piecewise-polynomial approximation of the state X(t) (black line), the solution set of the data-driven differential inclusion 8 (the meshed cone) and the actual solution of the system (green). 6 − 4 g = 9.8ms 2, thrust T , lift L and drag D (see Figure 5). The equations of motion are then described by 2 1 − − V˙ = [T cos(α) D(α, V ) mg sin(γ)] , 0 m 0 2 4 6 8 10 1 − γ˙ = [T sin(α) + L(α, V ) mg cos(γ)] , mV Fig. 4: The boundary of the unsafe set x = 10 (red line), z˙ = V sin(γ), (41) { } the data points x(t ) 20 (black circles), and the solution of i i=1 the system (solid blue). The safe controller starts operating at where V the velocity, γ the ﬂight path angle and z the t = 3. altitude are the states of the system. The angle of attack α and thrust T are the inputs of the system. Typically, landing is operated at T = 0.2T where idle max T is the maximal thrust. This value enables the aircraft max to counteract the drag due to ﬂaps, slats and landing gear. Most of the parameters for the DC9-30 can be found in the literature. The values of the numerical parameters used for the DC9-30 in this ﬂight conﬁguration are m = 60, 000 (kg), and T = 160, 000 (N). The lift and drag forces are thus max (cid:2) (cid:3) L(α, V ) = 68.6(1.25 + 4.2α)V 2, D(α, V ) = 2.7 + 3.08(1.25 + 4.2α)2 V 2, (42) Fig. 5: Point mass force diagram for the longitudinal dynamics of the aircraft [45, p. 108]. both in Newtons. For safety analysis and controller synthesis, we consider two separate aircraft failure scenarios in which the aircraft At this point, we consider 20 samples of the solution to ∈ model (41) is not valid anymore and we use data-driven the differential equation (40) from t [0, 2] and we allow differential inclusion models instead. In both cases, we are one seconds for the computations. Thus, the safe controller would kick in at around t = 3. The values of u for learning interested in the safe landing speed of the aircraft Vsafe as it phase are drawn from 10 sin(t). The unsafe set given by reaches the ground z = 0. Therefore, we deﬁne the unsafe set X { ∈ R | − ≥ } as = x x 10 0 and we are interested in making X { | } u sure that the system solutions remain safe from [3, 10], i.e., u = (V, γ, z) V > Vsafe, z = 0 . T = 10. Figure 4 shows the application of the designed safe We consider the following regularity side information controller based on Corollary 2. As it can be observed as (cid:107) (cid:107) ≤ | − | ≤ the safe controller starts acting on the system, the solutions x C1 10, X˙ (t) x˙(t) 10. ≥ become safe for t 3. The computed certiﬁcates B and W In the ﬁrst scenario, we model wing failure by sudden drop of degree 2 in the synthesis problem are given below in the lift force L to 0.2L at t = 0.5(s). We collect 25 data − − samples until t = 2(s) and we are interested in checking B(t, x) = 0.3845t2 0.0002269tx 1.594t whether the aircraft satisﬁes the safe landing speed at the time + 0.0006x2 + 0.0001038x, of landing. Using Lemma 1, we found a barrier certiﬁcate of degree 5 proving that the system is safe. Figure 6 illustrates − W(t, x) = 0.1664t2 0.6558t + 1.5543x2 + 0.6664. the collected data, the actual system state evolution and the solutions of the data-driven differential inclusion given the side B. Example II: Aircraft Landing information. We consider a point mass longitudinal model of an aircraft In the second scenario, we consider an engine control failure subject to the gravity force mg with m being the mass and which is modeled as a sudden surge in thrust (to 0.8T from max ##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##

label: paragraph

font: LPBABI+NimbusMonL-Regu

font: default

label: caption

text: Fig.

text: The

text: data

text: points

text: x(ti)

text: (black

text: circles),

text: the

text: piecewise-polynomial

text: approximation

text: of

text: the

text: state

text: X(t)

text: (black

text: line),

text: the

text: solution

text: set

text: of

text: the

text: data-driven

text: differential

text: inclusion

text: (the

text: meshed

text: cone)

text: and

text: the

text: actual

text: solution

text: of

text: the

text: system

text: (green).

text: g

text: thrust

text: T

text: lift

text: L

text: and

text: drag

text: D

text: (see

text: Figure

text: The

text: equations

text: of

text: motion

text: are

text: then

text: described

text: by

label: equation

text: V˙

text: [T

text: cos(α)

text: D(α,

text: V

text: mg

text: sin(γ)]

text: m

text: γ˙

text: [T

text: sin(α)

text: L(α,

text: V

text: mg

text: cos(γ)]

text: mV

text: Fig.

text: The

text: boundary

text: of

text: the

text: unsafe

text: set

text: x

text: (red

text: line),

text: z˙

text: V

text: sin(γ),

text: the

text: data

text: points

text: x(t

text: (black

text: circles),

text: and

text: the

text: solution

text: of

text: i

text: the

text: system

text: (solid

text: blue).

text: The

text: safe

text: controller

text: starts

text: operating

text: at

text: where

text: V

text: the

text: velocity,

text: γ

text: the

text: ﬂight

text: path

text: angle

text: and

text: z

text: the

text: t

text: altitude

text: are

text: the

text: states

text: of

text: the

text: system.

text: The

text: angle

text: of

text: attack

text: α

text: and

text: thrust

text: T

text: are

text: the

text: inputs

text: of

text: the

text: system.

text: Typically,

text: landing

text: is

text: operated

text: at

text: T

text: where

text: idle

text: max

text: T

text: is

text: the

text: maximal

text: thrust.

text: This

text: value

text: enables

text: the

text: aircraft

text: max

text: to

text: counteract

text: the

text: drag

text: due

text: to

text: ﬂaps,

text: slats

text: and

text: landing

text: gear.

text: Most

text: of

text: the

text: parameters

text: for

text: the

text: can

text: be

text: found

text: in

text: the

text: literature.

text: The

text: values

text: of

text: the

text: numerical

text: parameters

text: used

text: for

text: the

text: in

text: this

text: ﬂight

text: conﬁguration

text: are

text: m

text: (kg),

text: and

text: T

text: (N).

text: The

text: lift

text: and

text: drag

text: forces

text: are

text: thus

text: max

text: L(α,

text: V

text: D(α,

text: V

text: V

text: Fig.

text: Point

text: mass

text: force

text: diagram

text: for

text: the

text: longitudinal

text: dynamics

text: of

text: the

text: aircraft

text: p.

text: both

text: in

text: Newtons.

text: For

text: safety

text: analysis

text: and

text: controller

text: synthesis,

text: we

text: consider

text: two

text: separate

text: aircraft

text: failure

text: scenarios

text: in

text: which

text: the

text: aircraft

text: At

text: this

text: point,

text: we

text: consider

text: samples

text: of

text: the

text: solution

text: to

text: model

text: is

text: not

text: valid

text: anymore

text: and

text: we

text: use

text: data-driven

text: the

text: differential

text: equation

text: from

text: t

text: and

text: we

text: allow

text: differential

text: inclusion

text: models

text: instead.

text: In

text: both

text: cases,

text: we

text: are

text: one

text: seconds

text: for

text: the

text: computations.

text: Thus,

text: the

text: safe

text: controller

text: would

text: kick

text: in

text: at

text: around

text: t

text: The

text: values

text: of

text: u

text: for

text: learning

text: interested

text: in

text: the

text: safe

text: landing

text: speed

text: of

text: the

text: aircraft

text: Vsafe

text: as

text: it

text: phase

text: are

text: drawn

text: from

text: sin(t).

text: The

text: unsafe

text: set

text: given

text: by

text: reaches

text: the

text: ground

text: z

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text: we

text: deﬁne

text: the

text: unsafe

text: set

text: X

text: R

text: as

text: x

text: x

text: and

text: we

text: are

text: interested

text: in

text: making

text: X

text: u

text: sure

text: that

text: the

text: system

text: solutions

text: remain

text: safe

text: from

text: i.e.,

text: u

text: (V,

text: γ,

text: z)

text: V

text: Vsafe,

text: z

text: T

text: Figure

text: shows

text: the

text: application

text: of

text: the

text: designed

text: safe

text: We

text: consider

text: the

text: following

text: regularity

text: side

text: information

text: controller

text: based

text: on

text: Corollary

text: As

text: it

text: can

text: be

text: observed

text: as

text: the

text: safe

text: controller

text: starts

text: acting

text: on

text: the

text: system,

text: the

text: solutions

text: x

text: X˙

text: (t)

text: x˙(t)

text: become

text: safe

text: for

text: t

text: The

text: computed

text: certiﬁcates

text: B

text: and

text: W

text: In

text: the

text: ﬁrst

text: scenario,

text: we

text: model

text: wing

text: failure

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text: sudden

text: drop

text: of

text: degree

text: in

text: the

text: synthesis

text: problem

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text: given

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text: lift

text: force

text: L

text: to

text: at

text: t

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text: collect

text: data

text: samples

text: until

text: t

text: and

text: we

text: are

text: interested

text: in

text: checking

text: B(t,

text: x)

text: whether

text: the

text: aircraft

text: satisﬁes

text: the

text: safe

text: landing

text: speed

text: at

text: the

text: time

text: of

text: landing.

text: Using

text: Lemma

text: we

text: found

text: a

text: barrier

text: certiﬁcate

text: of

text: degree

text: proving

text: that

text: the

text: system

text: is

text: safe.

text: Figure

text: illustrates

text: W(t,

text: x)

text: the

text: collected

text: data,

text: the

text: actual

text: system

text: state

text: evolution

text: and

text: the

text: solutions

text: of

text: the

text: data-driven

text: differential

text: inclusion

text: given

text: the

text: side

text: B.

text: Example

text: II:

text: Aircraft

text: Landing

text: information.

text: We

text: consider

text: a

text: point

text: mass

text: longitudinal

text: model

text: of

text: an

text: aircraft

text: In

text: the

text: second

text: scenario,

text: we

text: consider

text: an

text: engine

text: control

text: failure

text: subject

text: to

text: the

text: gravity

text: force

text: mg

text: with

text: m

text: being

text: the

text: mass

text: and

text: which

text: is

text: modeled

text: as

text: a

text: sudden

text: surge

text: in

text: thrust

text: (to

text: from

text: max

text: ##LTFigure##

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text: 17 They can be derived from the f(1) states by adding one more e-h pair (cid:55)→ (cid:55)→ E A + (e˜ , h˜ ), F B + (e˜ , h˜ ), 2 4 1 3 (cid:55)→ (cid:55)→ (39) G C + (e˜ , h˜ ), H D + (e˜ , h˜ ) 1 3 1 3 where h˜ , h˜ , e˜ , and e˜ are t orbitals close to the 3 4 1 2 2g Fermi level. As is shown in Fig. 6 (d), orbital e˜ has 1 ./xps_spin-eps-converted-to.pdf signiﬁcant spatial overlap with h˜ (sharing the d char- 3 xz acter at the Cr atom next to the excited O), and so does orbital e˜ with h˜ (near the oxygens at the corners of the 2 4 plot), albeit weaker. This overlap makes E, F, G, and H also bright transitions. There are alternative pathways to access these states with two e-h pairs. For instant, F can also be mapped to A + (e˜ , h˜ ), i.e., A coupled with 3 3− an e-h pair (e˜ , h˜ ) (a shake-up d d transition). 3 3 The shake-up excitations can also be found in the satel- lite features of XPS, as shown in Fig. 6 (b). Recently these excitations were investigated with a cumulant ex- pansion technique [61, 62]. Here, we show that these FIG. 7. Spin-wise spectral function σ ↑ and σ ↓. The x- P P satellite features can also be included naturally within axis is the binding energy (E ). E = 0 is aligned with the B B the determinant formalism of the non-interacting MND threshold. theory (albeit poorly approximating their energies due to missing additional interactions between these extra e- h pairs). The strongest transition (labelled as state I) ∗ ∗ originates from the overlap of the N-electron states, de- σ = σ ↑ σ ↓ + σ ↓ σ ↑ (41) A A P A P scribing the initial ground state valence system and the ﬁnal core-excited valence system (assuming the excited ≡ ≡ ∗ where σ σ , σ σ , represents the convo- electron has escaped, approximated using the full-core- A XAS P XPS (cid:104) | (cid:105) lution integral in Eq. (30), σ and σ are spectra of hole approach): ΨN ΨN . This corresponds to the Aµ Pµ f,FCH i,GS one-spin channel before convolution. Then the spectral charge-transfer state in Fig. 6 (d). We may deﬁne I as functions σ can be considered as weighting factors of the only zero-order conﬁguration (f(0)) of XPS. f(1) con- Pµ the two absorption channels σ . If the weighting factors ﬁgurations emerge at larger binding energies and appear Aµ are not considered, the hypothetical spectrum as satellite features in the XPS proﬁle. Two representa- tive states are J and K (cid:48) σ = σ ↑ + σ ↓ (42) A A A (cid:55)→ (cid:55)→ J (e˜1, h˜2), K (e˜1, h˜1) (40) ∼ has a peak-intensity ratio of 1.3 that still deviates signiﬁcantly from experiment (Fig. 6 (c)). This im- which are shake-up excitations from h˜ (a Cr 3d - O plies that the modulation eﬀects of σ ↑ and σ ↓ on their 2 P P 2p hybrid with mixed bonding and anti-bonding char- counter-spin channel are quite diﬀerent. acter) and h˜ (a deep O 2p orbital) to the e˜ orbital, The spectral functions, σ ↑ and σ ↓, are shown in Fig. 1 1 P P respectively. The charge transfer associated with K is 7. In both cases, most spectral weight is concentrated at ↑ particularly strong. zero binding energy, E = 0. But for the metallic B E. Many-body wavefunction overlap eﬀects in channel, more spectral weight is transferred to shake-up CrO2 satellites at higher energies because its lack of a band gap makes e-h pair production easier. As a result, σ ↓ P As shown in Sec. , the projection onto empty initial- is more intense than σ ↑ near E = 0. The integrated ∼ P B state orbitals alone cannot account for the XAS line- intensity of σ ↑ is 2/3 of σ ↓ for E < 1.7 eV (shaded P P B shape for CrO , and one must employ the determinant areas). The more intense σ ↓ enhances the contribution 2 P ↑ formalism. This suggests that there are important many- of σ ↑, especially the lowest-energy peak deﬁned by t A ∼ 2g electron eﬀects in the determinantal amplitude that lead orbitals, leading to a peak-intensity ratio of 1.7 as ∼ to the ultimate peak-intensity ratio of 1.7 between the measured. ﬁrst and second absorption features. To explain this, we To conclude, the three contributing factors leading to rewrite the spectrum as the convolution deﬁned in Eq. the near-edge lineshape of CrO are: (a) the core-level ex- 2 (30) citonic eﬀect in the metallic screening environment lead ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: They

text: can

text: be

text: derived

text: from

text: the

text: states

text: by

text: adding

text: one

text: more

text: e-h

text: pair

text: E

text: A

text: (e˜

text: h˜

text: F

text: B

text: (e˜

text: h˜

text: G

text: C

text: (e˜

text: h˜

text: H

text: D

text: (e˜

text: h˜

text: where

text: h˜

text: h˜

text: e˜

text: and

text: e˜

text: are

text: t

text: orbitals

text: close

text: to

text: the

text: Fermi

text: level.

text: As

text: is

text: shown

text: in

text: Fig.

text: (d),

text: orbital

text: e˜

text: has

text: signiﬁcant

text: spatial

text: overlap

text: with

text: h˜

text: (sharing

text: the

text: d

text: char-

text: xz

text: acter

text: at

text: the

text: Cr

text: atom

text: next

text: to

text: the

text: excited

text: O),

text: and

text: so

text: does

text: orbital

text: e˜

text: with

text: h˜

text: (near

text: the

text: oxygens

text: at

text: the

text: corners

text: of

text: the

text: plot),

text: albeit

text: weaker.

text: This

text: overlap

text: makes

text: E,

text: F,

text: G,

text: and

text: H

text: also

text: bright

text: transitions.

text: There

text: are

text: alternative

text: pathways

text: to

text: access

text: these

text: states

text: with

text: two

text: e-h

text: pairs.

text: For

text: instant,

text: F

text: can

text: also

text: be

text: mapped

text: to

text: A

text: (e˜

text: h˜

text: i.e.,

text: A

text: coupled

text: with

text: 3−

text: an

text: e-h

text: pair

text: (e˜

text: h˜

text: (a

text: shake-up

text: d

text: d

text: transition).

text: The

text: shake-up

text: excitations

text: can

text: also

text: be

text: found

text: in

text: the

text: satel-

text: lite

text: features

text: of

text: XPS,

text: as

text: shown

text: in

text: Fig.

text: (b).

text: Recently

text: these

text: excitations

text: were

text: investigated

text: with

text: a

text: cumulant

text: ex-

text: pansion

text: technique

text: Here,

text: we

text: show

text: that

text: these

text: FIG.

text: Spin-wise

text: spectral

text: function

text: σ

text: and

text: σ

text: The

text: x-

text: P

text: P

text: satellite

text: features

text: can

text: also

text: be

text: included

text: naturally

text: within

text: axis

text: is

text: the

text: binding

text: energy

text: (E

text: E

text: is

text: aligned

text: with

text: the

text: B

text: B

text: the

text: determinant

text: formalism

text: of

text: the

text: non-interacting

text: MND

text: threshold.

text: theory

text: (albeit

text: poorly

text: approximating

text: their

text: energies

text: due

text: to

text: missing

text: additional

text: interactions

text: between

text: these

text: extra

text: e-

text: h

text: pairs).

text: The

text: strongest

text: transition

text: (labelled

text: as

text: state

text: I)

text: originates

text: from

text: the

text: overlap

text: of

text: the

text: N-electron

text: states,

text: de-

text: σ

text: σ

text: σ

text: σ

text: σ

text: A

text: A

text: P

text: A

text: P

text: scribing

text: the

text: initial

text: ground

text: state

text: valence

text: system

text: and

text: the

text: ﬁnal

text: core-excited

text: valence

text: system

text: (assuming

text: the

text: excited

text: where

text: σ

text: σ

text: σ

text: σ

text: represents

text: the

text: convo-

text: electron

text: has

text: escaped,

text: approximated

text: using

text: the

text: full-core-

text: A

text: XAS

text: P

text: XPS

text: lution

text: integral

text: in

text: Eq.

text: σ

text: and

text: σ

text: are

text: spectra

text: of

text: hole

text: approach):

text: ΨN

text: ΨN

text: This

text: corresponds

text: to

text: the

text: Aµ

text: Pµ

text: f,FCH

text: i,GS

text: one-spin

text: channel

text: before

text: convolution.

text: Then

text: the

text: spectral

text: charge-transfer

text: state

text: in

text: Fig.

text: (d).

text: We

text: may

text: deﬁne

text: I

text: as

text: functions

text: σ

text: can

text: be

text: considered

text: as

text: weighting

text: factors

text: of

text: the

text: only

text: zero-order

text: conﬁguration

text: of

text: XPS.

text: con-

text: Pµ

text: the

text: two

text: absorption

text: channels

text: σ

text: If

text: the

text: weighting

text: factors

text: ﬁgurations

text: emerge

text: at

text: larger

text: binding

text: energies

text: and

text: appear

text: Aµ

text: are

text: not

text: considered,

text: the

text: hypothetical

text: spectrum

text: as

text: satellite

text: features

text: in

text: the

text: XPS

text: proﬁle.

text: Two

text: representa-

text: tive

text: states

text: are

text: J

text: and

text: K

text: σ

text: σ

text: σ

text: A

text: A

text: A

text: J

text: K

text: has

text: a

text: peak-intensity

text: ratio

text: of

text: that

text: still

text: deviates

text: signiﬁcantly

text: from

text: experiment

text: (Fig.

text: (c)).

text: This

text: im-

text: which

text: are

text: shake-up

text: excitations

text: from

text: h˜

text: (a

text: Cr

text: O

text: plies

text: that

text: the

text: modulation

text: eﬀects

text: of

text: σ

text: and

text: σ

text: on

text: their

text: P

text: P

text: hybrid

text: with

text: mixed

text: bonding

text: and

text: anti-bonding

text: char-

text: counter-spin

text: channel

text: are

text: quite

text: diﬀerent.

text: acter)

text: and

text: h˜

text: (a

text: deep

text: O

text: orbital)

text: to

text: the

text: e˜

text: orbital,

text: The

text: spectral

text: functions,

text: σ

text: and

text: σ

text: are

text: shown

text: in

text: Fig.

text: P

text: P

text: respectively.

text: The

text: charge

text: transfer

text: associated

text: with

text: K

text: is

text: In

text: both

text: cases,

text: most

text: spectral

text: weight

text: is

text: concentrated

text: at

text: particularly

text: strong.

text: zero

text: binding

text: energy,

text: E

text: But

text: for

text: the

text: metallic

text: B

text: E.

text: Many-body

text: wavefunction

text: overlap

text: eﬀects

text: in

text: channel,

text: more

text: spectral

text: weight

text: is

text: transferred

text: to

text: shake-up

text: satellites

text: at

text: higher

text: energies

text: because

text: its

text: lack

text: of

text: a

text: band

text: gap

text: makes

text: e-h

text: pair

text: production

text: easier.

text: As

text: a

text: result,

text: σ

text: P

text: As

text: shown

text: in

text: Sec.

text: the

text: projection

text: onto

text: empty

text: initial-

text: is

text: more

text: intense

text: than

text: σ

text: near

text: E

text: The

text: integrated

text: P

text: B

text: state

text: orbitals

text: alone

text: cannot

text: account

text: for

text: the

text: XAS

text: line-

text: intensity

text: of

text: σ

text: is

text: of

text: σ

text: for

text: E

text: eV

text: (shaded

text: P

text: P

text: B

text: shape

text: for

text: CrO

text: and

text: one

text: must

text: employ

text: the

text: determinant

text: areas).

text: The

text: more

text: intense

text: σ

text: enhances

text: the

text: contribution

text: P

text: formalism.

text: This

text: suggests

text: that

text: there

text: are

text: important

text: many-

text: of

text: σ

text: especially

text: the

text: lowest-energy

text: peak

text: deﬁned

text: by

text: t

text: A

text: electron

text: eﬀects

text: in

text: the

text: determinantal

text: amplitude

text: that

text: lead

text: orbitals,

text: leading

text: to

text: a

text: peak-intensity

text: ratio

text: of

text: as

text: to

text: the

text: ultimate

text: peak-intensity

text: ratio

text: of

text: between

text: the

text: measured.

text: ﬁrst

text: and

text: second

text: absorption

text: features.

text: To

text: explain

text: this,

text: we

text: To

text: conclude,

text: the

text: three

text: contributing

text: factors

text: leading

text: to

text: rewrite

text: the

text: spectrum

text: as

text: the

text: convolution

text: deﬁned

text: in

text: Eq.

text: the

text: near-edge

text: lineshape

text: of

text: CrO

text: are:

text: (a)

text: the

text: core-level

text: ex-

text: citonic

text: eﬀect

text: in

text: the

text: metallic

text: screening

text: environment

text: lead

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 3 studied ﬁrst in Ref. [21]. The dark sector Lagrangian with the renormalizable interactions is then given by L − 1 − − = ψ¯(i∂/ M ) ψ + (∂ S) (∂µS) g ψ¯ψS V (S, H), (2) DM ψ0 2 µ S S where the Higgs portal potential is 1 † † λ λ V (S, H) = M2S2 + λ H HS + λ H HS2 + 3 S3 + 4 S4. (3) S 2 S 1 2 3! 4! Note that the singlet fermionic DM ﬁeld ψ couples only to the singlet scalar S, and the interactions of the singlet † √ sector to the SM sector arise only through the Higgs portal H H. After electroweak symmetry breaking, the neutral (cid:104) (cid:105) (cid:104) (cid:105) component of the SM Higgs and the singlet scalar develop nonzero VEVs, H0 = v / 2 and S = v , respectively. h s Minimizing the full scalar potentials V + V , where S SM − † † V = µ2H H + λ (H H)2, (4) SM 0 (cid:18) (cid:19) (cid:18) (cid:19) the scalar mass parameters M2 and µ2 are expressed in terms of the scalar VEVs as follows [12] S − λ − λ λ M2 = 1 + λ v2 3 + 4 v2, S 2v 2 h 2v 6 s s s µ2 = λ v2 + (λ + λ v )v . (5) 0 h 1 2 s√ s The neutral scalar ﬁelds h and s deﬁned by H0 = (v + h)/ 2 and S = v + s are mixed to yield the mass matrix h s given by µ2 = 2λ v2, h 0 h −λ v2 λ λ µ2 = 1 h + 3 v + 4 v2, s 2v 2 s 3 s s µ2 = (λ + 2λ v )v . (6) hs 1 2 s h (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) The corresponding scalar mass eigenstates h and h are admixtures of h and s, 1 2 h cos θ sin θ h 1 = − , (7) h sin θ cos θ s 2 where the mixing angle θ is given by (cid:112) y tan θ = , (8) 1 + 1 + y2 ≡ − with y 2µ2 /(µ2 µ2). After the mass matrix is diagonalized, we obtain the physical masses of the two scalar hs h s bosons h as follows: 1,2 (cid:112) − µ2 + µ2 ± µ2 µ2 M2 = h s h s 1 + y2, (9) 1,2 2 2 where the upper (lower) sign corresponds to M (M ). We assume that M corresponds to the observed SM-like 1 2 1 Higgs boson mass in what follows. The singlet fermion ψ has mass M = M + g v as an independent parameter of the model since M is just ψ ψ0 S s ψ0 a free model parameter, and the Yukawa coupling g measures the interaction of ψ with singlet component of the S scalar particles. In total, we have eight independent model parameters relevant for DM phenomenology. The six model parameters λ , λ , λ , λ , λ and v determine the masses M , the mixing angle θ, and self-couplings of the 0 1 2 3 4 s 1,2 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: studied

text: ﬁrst

text: in

text: Ref.

text: The

text: dark

text: sector

text: Lagrangian

text: with

text: the

text: renormalizable

text: interactions

text: is

text: then

text: given

text: by

text: L

text: ψ¯(i∂/

text: M

text: ψ

text: S)

text: (∂µS)

text: g

text: ψ¯ψS

text: V

text: (S,

text: H),

text: DM

text: ψ0

text: µ

text: S

text: S

text: where

text: the

text: Higgs

text: portal

text: potential

text: is

text: λ

text: λ

text: V

text: (S,

text: H)

text: λ

text: H

text: HS

text: λ

text: H

text: S

text: S

text: Note

text: that

text: the

text: singlet

text: fermionic

text: DM

text: ﬁeld

text: ψ

text: couples

text: only

text: to

text: the

text: singlet

text: scalar

text: S,

text: and

text: the

text: interactions

text: of

text: the

text: singlet

text: sector

text: to

text: the

text: SM

text: sector

text: arise

text: only

text: through

text: the

text: Higgs

text: portal

text: H

text: H.

text: After

text: electroweak

text: symmetry

text: breaking,

text: the

text: neutral

text: component

text: of

text: the

text: SM

text: Higgs

text: and

text: the

text: singlet

text: scalar

text: develop

text: nonzero

text: VEVs,

text: v

text: and

text: S

text: v

text: respectively.

text: h

text: s

text: Minimizing

text: the

text: full

text: scalar

text: potentials

text: V

text: V

text: where

text: S

text: SM

text: V

text: H

text: λ

text: (H

text: SM

text: the

text: scalar

text: mass

text: parameters

text: and

text: µ2

text: are

text: expressed

text: in

text: terms

text: of

text: the

text: scalar

text: VEVs

text: as

text: follows

text: S

text: λ

text: λ

text: λ

text: λ

text: S

text: h

text: s

text: s

text: s

text: µ2

text: λ

text: (λ

text: λ

text: v

text: )v

text: h

text: s√

text: s

text: The

text: neutral

text: scalar

text: ﬁelds

text: h

text: and

text: s

text: deﬁned

text: by

text: (v

text: h)/

text: and

text: S

text: v

text: s

text: are

text: mixed

text: to

text: yield

text: the

text: mass

text: matrix

text: h

text: s

text: given

text: by

text: µ2

text: 2λ

text: h

text: h

text: −λ

text: λ

text: λ

text: µ2

text: h

text: v

text: s

text: s

text: s

text: s

text: µ2

text: (λ

text: 2λ

text: v

text: )v

text: hs

text: s

text: h

text: The

text: corresponding

text: scalar

text: mass

text: eigenstates

text: h

text: and

text: h

text: are

text: admixtures

text: of

text: h

text: and

text: s,

text: h

text: cos

text: θ

text: sin

text: θ

text: h

text: h

text: sin

text: θ

text: cos

text: θ

text: s

text: where

text: the

text: mixing

text: angle

text: θ

text: is

text: given

text: by

text: y

text: tan

text: θ

text: with

text: y

text: After

text: the

text: mass

text: matrix

text: is

text: diagonalized,

text: we

text: obtain

text: the

text: physical

text: masses

text: of

text: the

text: two

text: scalar

text: hs

text: h

text: s

text: bosons

text: h

text: as

text: follows:

text: µ2

text: µ2

text: µ2

text: µ2

text: h

text: s

text: h

text: s

text: where

text: the

text: upper

text: (lower)

text: sign

text: corresponds

text: to

text: M

text: (M

text: We

text: assume

text: that

text: M

text: corresponds

text: to

text: the

text: observed

text: SM-like

text: Higgs

text: boson

text: mass

text: in

text: what

text: follows.

text: The

text: singlet

text: fermion

text: ψ

text: has

text: mass

text: M

text: M

text: g

text: v

text: as

text: an

text: independent

text: parameter

text: of

text: the

text: model

text: since

text: M

text: is

text: just

text: ψ

text: ψ0

text: S

text: s

text: ψ0

text: a

text: free

text: model

text: parameter,

text: and

text: the

text: Yukawa

text: coupling

text: g

text: measures

text: the

text: interaction

text: of

text: ψ

text: with

text: singlet

text: component

text: of

text: the

text: S

text: scalar

text: particles.

text: In

text: total,

text: we

text: have

text: eight

text: independent

text: model

text: parameters

text: relevant

text: for

text: DM

text: phenomenology.

text: The

text: six

text: model

text: parameters

text: λ

text: λ

text: λ

text: λ

text: λ

text: and

text: v

text: determine

text: the

text: masses

text: M

text: the

text: mixing

text: angle

text: θ,

text: and

text: self-couplings

text: of

text: the

text: s

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 4 H,S ψ S S H,S H,S (a) (b) FIG. 1. One-loop quadratic divergence contributions of the (a) scalars and (b) fermionic DM to the Higgs and the new scalar masses from the hidden sector. two physical scalars h . Given the ﬁxed Higgs mass M , we constrain seven independent NP parameters taking into 1,2 1 account various theoretical consideration and experimental measurements in the next section. III. NATURALNESS AND UNITARITY We apply the VC to this model as done similarly in Refs.[8, 9, 11], and constrain the above model parameters from the ﬁne-tuning problem originating from the radiat(cid:18)ive corrections to the Higgs mass.(cid:19)Cutting oﬀ the loop integral momenta at a scale Λ, and keeping only the dominant contributions in this scale, we obtain 3Λ2 − λ M2 = (M0)2 + M2 + 2M2 + M2 4m2 + 2 v2 , (10) h h 16π2v2 h W Z t 3 h h where M0 is the bare mass constrained in the unrenormalized Lagrangian, and small mixing eﬀect proportional to h θ is neglected. The last term proportional to λ in Eq. (10) is obtained from the one-loop diagram in Fig. 1(a). 2 Following Veltman [4], note that by choosing the parameter λ to be 2 − − − ∼ λ = 3(4m2 M2 2M2 M2)/v2 4.17, (11) 2 t h W Z h the quadratic divergences can be canceled and this could be a prediction for the parameter λ . This result agrees well 2 with the prior result of Ref. [9] where the additional scalar mass is protected by a new hidden vector. Instead, in our model, a singlet fermionic DM stabilizes the mass of the scalar mediator. Similarly to the Higgs mass case, as for the (cid:0) (cid:1) scalar singlet mass, we obtain from Fig. 1 that Λ2 − M2 = (M0)2 + λ + 2λ 8g2 , (12) s s 32π2 4 2 S and the VC requires − λ = 8g2 2λ . (13) 4 S 2 The conditions obtained above were given only at the one-loop level, and cannot be the legitimate solution to the ﬁne-tuning problem especially if the scale of NP is extremely large. For scales not much larger than the electroweak scale, however, one does not need very large cancellations. If the Veltman solution is by chance satisﬁed, the scale Λ can be pushed at the two-loop level to a much higher value proportional to Λ2, and we have almost no or only a ∼ small amount of ﬁne-tuning up to about Λ 15 TeV. Of course, one can set up the model parameters to apply the VC at the two-loop or higher level. The couplings λ can grow signiﬁcantly with increasing renormalization scale Λ, and one can constraint those 2,4 by applying the tree-level perturbative unitarity to scalar elastic scattering processes for the zeroth partial wave amplitude [3]. For a singlet scalar Higgs portal, the bounds on the couplings λ are given by [23] 2,4 | | ≤ ≤ λ 4π, λ 8π. (14) 2 4 The value of λ obtained from the VC in Eq.(11) lies obviously in the range of the above bound. From the fact that 2 the quartic coupling λ should be positive, and from Eqs. (13) and (14) we obtain 4 1.02 < g < 2.05. (15) S ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine##

text: H,S

text: ψ

text: S

text: S

text: H,S

text: H,S

text: (a)

text: (b)

text: FIG.

text: One-loop

text: quadratic

text: divergence

text: contributions

text: of

text: the

text: (a)

text: scalars

text: and

text: (b)

text: fermionic

text: DM

text: to

text: the

text: Higgs

text: and

text: the

text: new

text: scalar

text: masses

text: from

text: the

text: hidden

text: sector.

text: two

text: physical

text: scalars

text: h

text: Given

text: the

text: ﬁxed

text: Higgs

text: mass

text: M

text: we

text: constrain

text: seven

text: independent

text: NP

text: parameters

text: taking

text: into

text: account

text: various

text: theoretical

text: consideration

text: and

text: experimental

text: measurements

text: in

text: the

text: next

text: section.

text: III.

label: section

text: NATURALNESS

text: AND

text: UNITARITY

text: We

text: apply

text: the

text: VC

text: to

text: this

text: model

text: as

text: done

text: similarly

text: in

text: and

text: constrain

text: the

text: above

text: model

text: parameters

text: from

text: the

text: ﬁne-tuning

text: problem

text: originating

text: from

text: the

text: corrections

text: to

text: the

text: Higgs

text: oﬀ

text: the

text: loop

text: integral

text: momenta

text: at

text: a

text: scale

text: Λ,

text: and

text: keeping

text: only

text: the

text: dominant

text: contributions

text: in

text: this

text: scale,

text: we

text: obtain

text: λ

text: h

text: h

text: h

text: W

text: Z

text: t

text: h

text: h

text: where

text: is

text: the

text: bare

text: mass

text: constrained

text: in

text: the

text: unrenormalized

text: Lagrangian,

text: and

text: small

text: mixing

text: eﬀect

text: proportional

text: to

text: h

text: θ

text: is

text: neglected.

text: The

text: last

text: term

text: proportional

text: to

text: λ

text: in

text: Eq.

text: is

text: obtained

text: from

text: the

text: one-loop

text: diagram

text: in

text: Fig.

text: Following

text: Veltman

text: note

text: that

text: by

text: choosing

text: the

text: parameter

text: λ

text: to

text: be

text: λ

text: t

text: h

text: W

text: Z

text: h

text: the

text: quadratic

text: divergences

text: can

text: be

text: canceled

text: and

text: this

text: could

text: be

text: a

text: prediction

text: for

text: the

text: parameter

text: λ

text: This

text: result

text: agrees

text: well

text: with

text: the

text: prior

text: result

text: of

text: Ref.

text: where

text: the

text: additional

text: scalar

text: mass

text: is

text: protected

text: by

text: a

text: new

text: hidden

text: vector.

text: Instead,

text: in

text: our

text: model,

text: a

text: singlet

text: fermionic

text: DM

text: stabilizes

text: the

text: mass

text: of

text: the

text: scalar

text: mediator.

text: Similarly

text: to

text: the

text: Higgs

text: mass

text: case,

text: as

text: for

text: the

text: scalar

text: singlet

text: mass,

text: we

text: obtain

text: from

text: Fig.

text: that

text: Λ2

text: λ

text: 2λ

text: s

text: s

text: S

text: and

text: the

text: VC

text: requires

text: λ

text: 2λ

text: S

text: The

text: conditions

text: obtained

text: above

text: were

text: given

text: only

text: at

text: the

text: one-loop

text: level,

text: and

text: cannot

text: be

text: the

text: legitimate

text: solution

text: to

text: the

text: ﬁne-tuning

text: problem

text: especially

text: if

text: the

text: scale

text: of

text: NP

text: is

text: extremely

text: large.

text: For

text: scales

text: not

text: much

text: larger

text: than

text: the

text: electroweak

text: scale,

text: however,

text: one

text: does

text: not

text: need

text: very

text: large

text: cancellations.

text: If

text: the

text: Veltman

text: solution

text: is

text: by

text: chance

text: satisﬁed,

text: the

text: scale

text: Λ

text: can

text: be

text: pushed

text: at

text: the

text: two-loop

text: level

text: to

text: a

text: much

text: higher

text: value

text: proportional

text: to

text: and

text: we

text: have

text: almost

text: no

text: or

text: only

text: a

text: small

text: amount

text: of

text: ﬁne-tuning

text: up

text: to

text: about

text: Λ

text: TeV.

text: Of

text: course,

text: one

text: can

text: set

text: up

text: the

text: model

text: parameters

text: to

text: apply

text: the

text: VC

text: at

text: the

text: two-loop

text: or

text: higher

text: level.

text: The

text: couplings

text: λ

text: can

text: grow

text: signiﬁcantly

text: with

text: increasing

text: renormalization

text: scale

text: Λ,

text: and

text: one

text: can

text: constraint

text: those

text: by

text: applying

text: the

text: tree-level

text: perturbative

text: unitarity

text: to

text: scalar

text: elastic

text: scattering

text: processes

text: for

text: the

text: zeroth

text: partial

text: wave

text: amplitude

text: For

text: a

text: singlet

text: scalar

text: Higgs

text: portal,

text: the

text: bounds

text: on

text: the

text: couplings

text: λ

text: are

text: given

text: by

text: λ

text: λ

text: The

text: value

text: of

text: λ

text: obtained

text: from

text: the

text: VC

text: in

text: lies

text: obviously

text: in

text: the

text: range

text: of

text: the

text: above

text: bound.

text: From

text: the

text: fact

text: that

text: the

text: quartic

text: coupling

text: λ

text: should

text: be

text: positive,

text: and

text: from

text: Eqs.

text: and

text: we

text: obtain

text: g

text: S

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:90) Furthermore, using Hölder’s and Young’s inequality gives t (cid:104) − − (cid:105) ≤ 1 (cid:107) − (cid:107) σ (cid:107) − (cid:107) gn gm, z˙n z˙m V ∗,V 2σ gn gm 2L2(I;V ∗) + 2 z˙n z˙m 2L2(I,V ). 0 Applying the previous estimates in equation (4.2), yields σ (cid:107) − (cid:107) 1(cid:107) − (cid:107) ≤ | | · | − | 1 (cid:107) − (cid:107) 2 z˙n z˙m 2L2(0,t;V ) + 2 zn(t) zm(t) 2V 2T Ω ρm ρn + 2σ gn gm 2L2(I;V ∗). ∈ f.a.a. t I, which is the asserted inequality. We are now ready to prove existence and uniqueness of solutions for the non-smooth state equation. Proof of Theorem 2.1. Let g ∈ H1(I, V ∗) be given. Let us take a sequence (ρ ) ∈N ∈ R be a (cid:38) (cid:63) S n n sequence with ρ 0 and deﬁne z := g. Using Theorem 3.10 we obtain that the sequence (z ) is boundend in H1(I, V ) ∩ Hn2(I, Vρ)n. Due to reﬂexivity, we have a weakly convergent n n (cid:63) ∈ ∩ subsequence (which we denote again by z ) and a function z H2(I, V ) H1(I, V ) such that n (cid:63) z (cid:42) z in these spaces. n (cid:107) (cid:107) ≤ | | Moreover, we have z˙ (0) ρn Ω , see (3.17), and hence z˙(0) = 0 is satisﬁed. n V σ → H1(LIe,mVm).aD4u(cid:90).e1 tsohothwes,cotnhva(cid:90)etxzitny iosf a| ·C| auwcehyhasveeqfuoernace.a.int ∈H(cid:63)I1(,Ia,lVl n),∈wNhicahndimalpllives∈zVn z in (cid:63) ρ | | ≥ | | (cid:104) − (cid:105) v dx z˙ (t) dx + σ∆z˙ (t) + ∆z (t) + g(t), v z˙ ∗ . (4.3) ρn n ρn n n n V ,V Ω Ω It is easy to show that we can pass to the limit in this inequality and obtain (cid:107) (cid:107) ≥ (cid:107) (cid:107) (cid:104) − (cid:105) v z˙(t) + σ∆z˙(t) + ∆z(t) + g(t), v z˙(t) ∗ , L1(Ω) L1(Ω) V ,V ∈ | | i.e. σ∆z˙(t) + ∆z(t) + g(t) ∂ z˙(t) . ∈ It remains to prove uniqueness of solutions. Let two solutions z , z H1(I, V ) of the ∈ ∈1 2 (cid:63) non-smooth state equation be given. Then for all v, w V and a.a. t I we have (cid:107) (cid:107) ≥ (cid:107) (cid:107) (cid:104) − (cid:105) v z˙ (t) + σ∆z˙ (t) + ∆z (t) + g(t), v z˙ (t) ∗ , L1(Ω) 1 L1(Ω) 1 2 1 V ,V (cid:107) (cid:107) ≥ (cid:107) (cid:107) (cid:104) − (cid:105) w z˙ (t) + σ∆z˙ (t) + ∆z (t) + g(t), w z˙ (t) ∗ . L1(Ω) 2 L1(Ω) 2 2 2 V ,V (cid:0) (cid:1) Choosing v := z˙ (t), w := z˙ (t), adding the resulting inequalities, and canceling out some 2 1 summands gives ≥ (cid:107) − (cid:107) − − 0 z˙ (t) z˙ (t) 2 + z (t) z (t), z˙ (t) z˙ (t) . 1 2 V 2 1 2 1 V Integrating this inequality from 0 to t yields (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) ≥ (cid:107) − (cid:107) 1(cid:107) − (cid:107) − 1(cid:107) − (cid:107) 0 z˙ (t) z˙ (t) 2 + z (t) z (t) 2 z (0) z (0) 2 , 1 2 L2(0,t;V 2 2 1 V 2 2 1 V =0 =0 hence z = z on I. 1 2 In particular, this proof yields the following corollary. ∈ ∗ S → S (cid:38) Corollary 4.2. For every g H1(I, V ) we have (g) (g) in H1(I, V ) for ρ 0. (cid:63) ρ (cid:63) (cid:38) In the next theorem we show stronger convergence for ρ 0. Theorem 4.3. Let sequences (ρ ) ∈N ∈ R and (g ) ∈ ∈ H1(I, V ∗) be given with ρ (cid:38) 0 and → ∗ n∈n ∗ n n N (cid:63)S S n g g in L2(I, V ) for some g H1(I, V ). Deﬁne z := (g ) and z := (g). Then it hnolds z → z in H1(I, V ) and C(I¯, V )(cid:63). n ρn n n (cid:63) Proof. We have the estimate (cid:107) − (cid:107) ≤ (cid:107) − S (cid:107) (cid:107)S − (cid:107) z z z (g) + (g) z . n H(cid:63)1(I,V ) n ρn H(cid:63)1(I,V ) ρn H(cid:63)1(I,V ) The right-hand side converges to zero due to Lemma 3.5 and Corollary 4.2. The claim follows → ∞ by passing to the limit m in inequality (4.1). 13 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Furthermore,

text: using

text: Hölder’s

text: and

text: Young’s

text: inequality

text: gives

text: t

text: σ

text: gn

text: gm,

text: z˙n

text: z˙m

text: V

text: ∗,V

text: 2σ

text: gn

text: gm

text: z˙n

text: z˙m

text: Applying

text: the

text: previous

text: estimates

text: in

text: equation

text: yields

text: σ

text: z˙n

text: z˙m

text: zn(t)

text: zm(t)

text: Ω

text: ρm

text: ρn

text: 2σ

text: gn

text: gm

text: f.a.a.

text: t

text: I,

text: which

text: is

text: the

text: asserted

text: inequality.

text: We

text: are

text: now

text: ready

text: to

text: prove

text: existence

text: and

text: uniqueness

text: of

text: solutions

text: for

text: the

text: non-smooth

text: state

text: equation.

text: Proof

text: of

text: Theorem

text: Let

text: g

text: V

text: be

text: given.

text: Let

text: us

text: take

text: a

text: sequence

text: (ρ

text: ∈N

text: R

text: be

text: a

text: S

text: n

text: n

text: sequence

text: with

text: ρ

text: and

text: deﬁne

text: z

text: g.

text: Using

text: Theorem

text: we

text: obtain

text: that

text: the

text: sequence

text: (z

text: is

text: boundend

text: in

text: V

text: Vρ)n.

text: Due

text: to

text: reﬂexivity,

text: we

text: have

text: a

text: weakly

text: convergent

text: n

text: n

text: subsequence

text: (which

text: we

text: denote

text: again

text: by

text: z

text: and

text: a

text: function

text: z

text: V

text: V

text: such

text: that

text: n

text: z

text: z

text: in

text: these

text: spaces.

text: n

text: Moreover,

text: we

text: have

text: z˙

text: ρn

text: Ω

text: see

text: and

text: hence

text: is

text: satisﬁed.

text: n

text: V

text: σ

text: iosf

text: a|

text: ·C|

text: auwcehyhasveeqfuoernace.a.int

text: n),∈wNhicahndimalpllives∈zVn

text: z

text: in

text: ρ

text: v

text: dx

text: z˙

text: (t)

text: dx

text: σ∆z˙

text: (t)

text: ∆z

text: (t)

text: g(t),

text: v

text: z˙

text: ρn

text: n

text: ρn

text: n

text: n

text: n

text: V

text: ,V

text: Ω

text: Ω

text: It

text: is

text: easy

text: to

text: show

text: that

text: we

text: can

text: pass

text: to

text: the

text: limit

text: in

text: this

text: inequality

text: and

text: obtain

text: v

text: z˙(t)

text: σ∆z˙(t)

text: ∆z(t)

text: g(t),

text: v

text: z˙(t)

text: V

text: ,V

text: i.e.

text: σ∆z˙(t)

text: ∆z(t)

text: g(t)

text: z˙(t)

text: It

text: remains

text: to

text: prove

text: uniqueness

text: of

text: solutions.

text: Let

text: two

text: solutions

text: z

text: z

text: V

text: of

text: the

text: ∈1

text: non-smooth

text: state

text: equation

text: be

text: given.

text: Then

text: for

text: all

text: v,

text: w

text: V

text: and

text: a.a.

text: t

text: I

text: we

text: have

text: v

text: z˙

text: (t)

text: σ∆z˙

text: (t)

text: ∆z

text: (t)

text: g(t),

text: v

text: z˙

text: (t)

text: V

text: ,V

text: w

text: z˙

text: (t)

text: σ∆z˙

text: (t)

text: ∆z

text: (t)

text: g(t),

text: w

text: z˙

text: (t)

text: V

text: ,V

text: Choosing

text: v

text: z˙

text: (t),

text: w

text: z˙

text: (t),

text: adding

text: the

text: resulting

text: inequalities,

text: and

text: canceling

text: out

text: some

text: summands

text: gives

text: z˙

text: (t)

text: z˙

text: (t)

text: z

text: (t)

text: z

text: (t),

text: z˙

text: (t)

text: z˙

text: (t)

text: V

text: V

text: Integrating

text: this

text: inequality

text: from

text: to

text: t

text: yields

text: z˙

text: (t)

text: z˙

text: (t)

text: z

text: (t)

text: z

text: (t)

text: z

text: z

text: V

text: V

text: hence

text: z

text: z

text: on

text: I.

text: In

text: particular,

text: this

text: proof

text: yields

text: the

text: following

text: corollary.

text: S

text: S

text: Corollary

text: For

text: every

text: g

text: V

text: we

text: have

text: (g)

text: (g)

text: in

text: V

text: for

text: ρ

text: ρ

text: In

text: the

text: next

text: theorem

text: we

text: show

text: stronger

text: convergence

text: for

text: ρ

text: Theorem

text: Let

text: sequences

text: (ρ

text: ∈N

text: R

text: and

text: (g

text: V

text: be

text: given

text: with

text: ρ

text: and

text: n∈n

text: n

text: n

text: N

text: S

text: n

text: g

text: g

text: in

text: V

text: for

text: some

text: g

text: V

text: Deﬁne

text: z

text: (g

text: and

text: z

text: (g).

text: Then

text: it

text: hnolds

text: z

text: z

text: in

text: V

text: and

text: C(I¯,

text: V

text: n

text: ρn

text: n

text: n

text: Proof.

text: We

text: have

text: the

text: estimate

text: S

text: z

text: z

text: z

text: (g)

text: (g)

text: z

text: n

text: n

text: ρn

text: ρn

text: The

text: right-hand

text: side

text: converges

text: to

text: zero

text: due

text: to

text: Lemma

text: and

text: Corollary

text: The

text: claim

text: follows

text: by

text: passing

text: to

text: the

text: limit

text: m

text: in

text: inequality

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: The condition (5.1e) involving the Fréchet normal cone of M can be written as the following system of pointwise properties: ⇒ z˙(t, x) > 0, g(t, x) + ∆z(t, x) + σ∆z˙(t, x) = 1 = q(t, x) = 0, (5.2a) ⇒ ≥ ≥ z˙(t, x) = 0, g(t, x) + ∆z(t, x) + σ∆z˙(t, x) = 1 = q(t, x) 0, ξ(t, x) 0, (5.2b) | | ⇒ z˙(t, x) = 0, g(t, x) + ∆z(t, x) + σ∆z˙(t, x) < 1 = ξ(t, x) = 0, (5.2c) − ⇒ ≤ ≤ z˙(t, x) = 0, g(t, x) + ∆z(t, x) + σ∆z˙(t, x) = 1 = q(t, x) 0, ξ(t, x) 0, (5.2d) − ⇒ z˙(t, x) < 0, g(t, x) + ∆z(t, x) + σ∆z˙(t, x) = 1 = q(t, x) = 0. (5.2e) Our aim in the next sections is to prove that some of the these formally derived conditions are optimality conditions for (P). 5.2 Smooth optimal control problem Now, we are going to consider an optimal control problem with regularized state equation depending on the parameter ρ. We show existence of solutions for this problem and investigate (cid:38) what happens with them for ρ 0. This section is based on [16, Section 4.4]. Let ∈ × (z¯, g¯) H1(I, V ) H1(I, H) (cid:63) (cid:63) ≤ S be a local solution of (P). Hence, there exists δ > 0 such that J(z¯, g¯) J( (g), g) for all ∈ (cid:107) − (cid:107) g H1(I, H) with g g¯ < δ holds. We deﬁne (cid:63) H1(I,H) × → R J : H1(I, V ) H1(I, H) ρ (cid:63) (cid:55)→ 1(cid:107) − (cid:107) (z, g) J(z, g) + g g¯ 2 . (5.3) 2 H1(I,H) We will now consider the optimal control problem  1(cid:107) − (cid:107) ∈ × minJ(z, g) + g g¯ 2 for (z, g) H1(I, V ) H1(I, H), 2 H1(I,H) (cid:107) − (cid:107) ≤  g g¯ H1(I,H) δ, s.t. g(0) = 0, z(0) = 0, (P ) ρ − − − | |(cid:48)(cid:48) ∗ ∈ σ∆z˙ ∆z g + z˙ = 0 in V for a.a. t I. ρ Augmenting the original problem with additional penalty terms and constraints is a well- (cid:107) − (cid:107) ≤ known technique for nonsmooth optimal control problem. Here, the constraint g g¯ H1(I,H) δ will give us the existence of global solutions of such a smoothed problem. The additional term (cid:107) − (cid:107) (cid:38) 1 g g¯ 2 will be used to force strong convergence of solutions of (P ) to g¯ for ρ 0, 2 H1(I,H) ρ see the proof of Theorem 5.2, which is from [16, Proof of Theorem 4.9]. Lemma 5.1. For all ρ > 0 the optimal control problem (P ) has global solutions (z , g ). In ∈ ρ ρ ρ addition it holds z H2(I, V ). ρ (cid:63) Proof. The proof of existence is similar to the proof of Lemma 2.6. The optimal state z is in ρ the asserted spaces due to Theorem 3.10. For convergence of global solutions, we have the following theorem. Its proof is identical to the proof of [16, Theorem 4.9]. { } (cid:38) Theorem 5.2. Let (z , g ) a family of global solutions of (P ). Then for ρ 0 it holds ρ ρ ρ>0 ρ → g g¯ in H1(I, H), ρ → (cid:63) z z¯ in H1(I, V ). ρ (cid:63) 15 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: The

text: condition

text: involving

text: the

text: Fréchet

text: normal

text: cone

text: of

text: M

text: can

text: be

text: written

text: as

text: the

text: following

text: system

text: of

text: pointwise

text: properties:

text: z˙(t,

text: x)

text: g(t,

text: x)

text: ∆z(t,

text: x)

text: σ∆z˙(t,

text: x)

text: q(t,

text: x)

text: z˙(t,

text: x)

text: g(t,

text: x)

text: ∆z(t,

text: x)

text: σ∆z˙(t,

text: x)

text: q(t,

text: x)

text: ξ(t,

text: x)

text: z˙(t,

text: x)

text: g(t,

text: x)

text: ∆z(t,

text: x)

text: σ∆z˙(t,

text: x)

text: ξ(t,

text: x)

text: z˙(t,

text: x)

text: g(t,

text: x)

text: ∆z(t,

text: x)

text: σ∆z˙(t,

text: x)

text: q(t,

text: x)

text: ξ(t,

text: x)

text: z˙(t,

text: x)

text: g(t,

text: x)

text: ∆z(t,

text: x)

text: σ∆z˙(t,

text: x)

text: q(t,

text: x)

text: Our

text: aim

text: in

text: the

text: next

text: sections

text: is

text: to

text: prove

text: that

text: some

text: of

text: the

text: these

text: formally

text: derived

text: conditions

text: are

text: optimality

text: conditions

text: for

text: (P).

text: Smooth

text: optimal

text: control

text: problem

text: Now,

text: we

text: are

text: going

text: to

text: consider

text: an

text: optimal

text: control

text: problem

text: with

text: regularized

text: state

text: equation

text: depending

text: on

text: the

text: parameter

text: ρ.

text: We

text: show

text: existence

text: of

text: solutions

text: for

text: this

text: problem

text: and

text: investigate

text: what

text: happens

text: with

text: them

text: for

text: ρ

text: This

text: section

text: is

text: based

text: on

text: Section

text: Let

text: (z¯,

text: g¯)

text: V

text: H)

text: S

text: be

text: a

text: local

text: solution

text: of

text: (P).

text: Hence,

text: there

text: exists

text: δ

text: such

text: that

text: J(z¯,

text: g¯)

text: J(

text: (g),

text: g)

text: for

text: all

text: g

text: H)

text: with

text: g

text: g¯

text: δ

text: holds.

text: We

text: deﬁne

text: R

text: J

text: V

text: H)

text: ρ

text: (z,

text: g)

text: J(z,

text: g)

text: g

text: g¯

text: We

text: will

text: now

text: consider

text: the

text: optimal

text: control

text: problem

text: minJ(z,

text: g)

text: g

text: g¯

text: for

text: (z,

text: g)

text: V

text: H),

text: g

text: g¯

text: δ,

text: s.t.

text: (P

text: ρ

text: σ∆z˙

text: ∆z

text: g

text: z˙

text: in

text: V

text: for

text: a.a.

text: t

text: I.

text: ρ

text: Augmenting

text: the

text: original

text: problem

text: with

text: additional

text: penalty

text: terms

text: and

text: constraints

text: is

text: a

text: well-

text: known

text: technique

text: for

text: nonsmooth

text: optimal

text: control

text: problem.

text: Here,

text: the

text: constraint

text: g

text: g¯

text: δ

text: will

text: give

text: us

text: the

text: existence

text: of

text: global

text: solutions

text: of

text: such

text: a

text: smoothed

text: problem.

text: The

text: additional

text: term

text: g

text: g¯

text: will

text: be

text: used

text: to

text: force

text: strong

text: convergence

text: of

text: solutions

text: of

text: (P

text: to

text: g¯

text: for

text: ρ

text: ρ

text: see

text: the

text: proof

text: of

text: Theorem

text: which

text: is

text: from

text: Proof

text: of

text: Theorem

text: Lemma

text: For

text: all

text: ρ

text: the

text: optimal

text: control

text: problem

text: (P

text: has

text: global

text: solutions

text: (z

text: g

text: In

text: ρ

text: ρ

text: ρ

text: addition

text: it

text: holds

text: z

text: V

text: ρ

text: Proof.

text: The

text: proof

text: of

text: existence

text: is

text: similar

text: to

text: the

text: proof

text: of

text: Lemma

text: The

text: optimal

text: state

text: z

text: is

text: in

text: ρ

text: the

text: asserted

text: spaces

text: due

text: to

text: Theorem

text: For

text: convergence

text: of

text: global

text: solutions,

text: we

text: have

text: the

text: following

text: theorem.

text: Its

text: proof

text: is

text: identical

text: to

text: the

text: proof

text: of

text: Theorem

text: Theorem

text: Let

text: (z

text: g

text: a

text: family

text: of

text: global

text: solutions

text: of

text: (P

text: Then

text: for

text: ρ

text: it

text: holds

text: ρ

text: ρ

text: ρ

text: g

text: g¯

text: in

text: H),

text: ρ

text: z

text: z¯

text: in

text: V

text: ρ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:90) → ∗ (cid:104) (cid:105) − ∇ · ∇ ∆ : V V : ∆u, v ∗ := u(x) v(x) dx V ,V (cid:82) Ω (cid:112) ∇ ·∇ The inner product on V is deﬁned by (u, v) = (u, v) := u(x) v(x) dx and its induced H1(Ω) V 0 (cid:107) (cid:107) Ω norm by u := (u, u) . Due to the zero boundary condition this norm is equivalent to the V V H1(Ω)-norm in the space V . Since we are analyzing an evolution equation, we need Bochner spaces like Lp(I, X) and H1(I, X), where X is a real Banach space. The state equation of interest is equipped with a zero initial condition. Therefore we deﬁne the spaces { ∈ | } H1(I, X) := u H1(I, X) u(0) = 0 , (cid:63) { ∈ | } H2(I, X) := u H2(I, X) u(0) = 0, u˙(0) = 0 , (cid:63) where u˙ denotes the weak derivativ→e wit∼h re∗sp→ect t∗o the time variable t. Moreover, we work with the standard Hilbert triple V (cid:44) H = H (cid:44) V induced by the L2-inner product in order ∗ to use L2(Ω)-functions as elements of V . 2 The non-smooth optimal control problem Let us ﬁrst the deﬁne the notion of weak solutions of the diﬀerential inclusion (1.3). ∈ Deﬁnition 2.1. A function z H1(I, V ) is called weak solution of (1.3) if and only if for ∈ (cid:63) almost all t I it holds ∈ | | − − − ∗ 0 ∂ z˙(t) ∆z(t) σ∆z˙(t) g(t) in V . | | Here, ∂ v denotes the subdiﬀerential of the L1(Ω)-norm with respect to the space V . The state equation is uniquely solvable and we have the following theorem, which will be proven in section . ∈ ∗ ∈ Theorem 2.1. For all g H1(I, V ) there exists a unique solution z H2(I, V ) of the non- (cid:63) (cid:63) smooth state equation (P). ∈ ∗ ∈ Note that g H1(I, V ) and z H1(I, V ) include the conditions g(0) = 0 and z(0) = 0. In (cid:63) (cid:63) view of the results derived below, these conditions can be viewed as compatibility conditions at t = 0. As a conclusion of Theorem 2.1 we can deﬁne a solution operator, which maps a control to the corresponding state. S → (cid:55)→ : H1(I, L2(Ω)) H1(I, H1(Ω)) g z. (2.1) (cid:63) (cid:63) 0 D We now take a closer look at the subdiﬀerential of the non-smooth part of , which is D (cid:107) (cid:107) ˆ(v) := v and give characterizations of the state equation via cones. To this end we L1(Ω) deﬁne { ∈ ∗ | ∈ − ≤ ≤ } K := v V v H, 1 v 1 a.e. in Ω , { ∈ | (cid:104) − (cid:105) ≤ ∀ ∈ } N (v) := w V w, v˜ v ∗ 0 v˜ K , K V,V { ∈ | ∈ − ≤ ≤ } − K˜ := w V ∆w H, 1 ∆w 1 a.e. in Ω = ∆ 1(K), { ∈ | − ≤ ∀ ∈ } NHilbert(v) := w V (w, v˜ v) 0 v˜ K˜ . K˜ V ∈ ∈ ∗ Lemma 2.2. Let v V and f V . Then it holds ∈ D ⇔ ∈ ∈ f ∂ ˆ(v) f K and v N (f). K 3 ##LTLine##

text: V

text: V

text: ∆u,

text: v

text: u(x)

text: v(x)

text: dx

text: V

text: ,V

text: Ω

text: The

text: inner

text: product

text: on

text: V

text: is

text: deﬁned

text: by

text: (u,

text: v)

text: (u,

text: v)

text: u(x)

text: v(x)

text: dx

text: and

text: its

text: induced

text: V

text: Ω

text: norm

text: by

text: u

text: (u,

text: u)

text: Due

text: to

text: the

text: zero

text: boundary

text: condition

text: this

text: norm

text: is

text: equivalent

text: to

text: the

text: V

text: V

text: in

text: the

text: space

text: V

text: Since

text: we

text: are

text: analyzing

text: an

text: evolution

text: equation,

text: we

text: need

text: Bochner

text: spaces

text: like

text: Lp(I,

text: X)

text: and

text: X),

text: where

text: X

text: is

text: a

text: real

text: Banach

text: space.

text: The

text: state

text: equation

text: of

text: interest

text: is

text: equipped

text: with

text: a

text: zero

text: initial

text: condition.

text: Therefore

text: we

text: deﬁne

text: the

text: spaces

text: X)

text: u

text: X)

text: X)

text: u

text: X)

text: where

text: u˙

text: denotes

text: the

text: weak

text: derivativ→e

text: wit∼h

text: re∗sp→ect

text: t∗o

text: the

text: time

text: variable

text: t.

text: Moreover,

text: we

text: work

text: with

text: the

text: standard

text: Hilbert

text: triple

text: V

text: H

text: H

text: V

text: induced

text: by

text: the

text: product

text: in

text: order

text: to

text: use

text: as

text: elements

text: of

text: V

text: The

text: non-smooth

text: optimal

text: control

text: problem

text: Let

text: us

text: ﬁrst

text: the

text: deﬁne

text: the

text: notion

text: of

text: weak

text: solutions

text: of

text: the

text: diﬀerential

text: inclusion

text: Deﬁnition

text: A

text: function

text: z

text: V

text: is

text: called

text: weak

text: solution

text: of

text: if

text: and

text: only

text: if

text: for

text: almost

text: all

text: t

text: I

text: it

text: holds

text: z˙(t)

text: ∆z(t)

text: σ∆z˙(t)

text: g(t)

text: in

text: V

text: Here,

text: v

text: denotes

text: the

text: subdiﬀerential

text: of

text: the

text: with

text: respect

text: to

text: the

text: space

text: V

text: The

text: state

text: equation

text: is

text: uniquely

text: solvable

text: and

text: we

text: have

text: the

text: following

text: theorem,

text: which

text: will

text: be

text: proven

text: in

text: section

text: Theorem

text: For

text: all

text: g

text: V

text: there

text: exists

text: a

text: unique

text: solution

text: z

text: V

text: of

text: the

text: non-

text: smooth

text: state

text: equation

text: (P).

text: Note

text: that

text: g

text: V

text: and

text: z

text: V

text: include

text: the

text: conditions

text: and

text: In

text: view

text: of

text: the

text: results

text: derived

text: below,

text: these

text: conditions

text: can

text: be

text: viewed

text: as

text: compatibility

text: conditions

text: at

text: t

text: As

text: a

text: conclusion

text: of

text: Theorem

text: we

text: can

text: deﬁne

text: a

text: solution

text: operator,

text: which

text: maps

text: a

text: control

text: to

text: the

text: corresponding

text: state.

text: S

text: g

text: z.

text: D

text: We

text: now

text: take

text: a

text: closer

text: look

text: at

text: the

text: subdiﬀerential

text: of

text: the

text: non-smooth

text: part

text: of

text: which

text: is

text: D

text: ˆ(v)

text: v

text: and

text: give

text: characterizations

text: of

text: the

text: state

text: equation

text: via

text: cones.

text: To

text: this

text: end

text: we

text: deﬁne

text: K

text: v

text: V

text: v

text: H,

text: v

text: a.e.

text: in

text: Ω

text: N

text: (v)

text: w

text: V

text: w,

text: v˜

text: v

text: v˜

text: K

text: K

text: V,V

text: K˜

text: w

text: V

text: ∆w

text: H,

text: ∆w

text: a.e.

text: in

text: Ω

text: NHilbert(v)

text: w

text: V

text: (w,

text: v˜

text: v)

text: v˜

text: K˜

text: K˜

text: V

text: Lemma

text: Let

text: v

text: V

text: and

text: f

text: V

text: Then

text: it

text: holds

text: D

text: f

text: ˆ(v)

text: f

text: K

text: and

text: v

text: N

text: (f).

text: K

text: ##LTLine##


text: (cid:48) (cid:48) In that reference, α, α (β, β ) run over all occupied spin orbitals of A (B). Equation (13) applies in the open-shell case as well (for the diagonal term) but we have to break up the summation over e.g. the occupied spin orbitals of A into the inactive orbitals with spins α and β and active orbitals with spin α. When we do that and perform all spin integrations, the resulting expression for the diagonal term is (cid:104) P(cid:105) − (cid:104) (cid:105)(cid:104)P(cid:105) − − − E(10) = V V = v˜ja v˜bi 2v˜ji exch,diag aj ib ij (cid:48) − (cid:48) (cid:48) − (cid:48) − − +v˜a aSj(cid:48) v˜aa Sj(cid:48) + v˜jj Sa(cid:48) 2v˜j jSa(cid:48) + v˜aiSj 2v˜iaSj v˜jbSa aj a aj a aj j aj j ij a ij a ab j (cid:48) − (cid:48) (cid:48) − (cid:48) − − +v˜bb Si(cid:48) v˜b bSi(cid:48) + v˜i iSb(cid:48) 2v˜ii Sb(cid:48) + v˜jbSi 2v˜bjSi v˜aiSb ib b ib b ib i ib i ij b ij b ab i − − +v˜iaSj + v˜bjSi 2v˜aiSj 2v˜jbSi aj i ib j aj i ib j (cid:48) (cid:48) − (cid:48) − (cid:48) +2v˜i iSj(cid:48) + 2v˜jj Si(cid:48) 4v˜ii Sj(cid:48) 4v˜j jSi(cid:48) ij i ij j ij i ij j (cid:48) (cid:48) +v˜a bSj(cid:48)Sa + v˜ab SbSi(cid:48) + 2v˜abSjSi + 2v˜abSjSi ab a j ab i b ib a j aj i b (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) − (cid:48) (cid:48) +2v˜a jSj(cid:48)Sa(cid:48) + 2v˜aj SjSi(cid:48) + v˜aj Sj(cid:48)Sa(cid:48) v˜a j Sj(cid:48)Sa(cid:48) aj a j aj i j aj a j aj a j (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) − (cid:48) (cid:48) +2v˜ib Sb(cid:48)Si(cid:48) + 2v˜i bSj(cid:48)Si + v˜i bSb(cid:48) Si(cid:48) v˜i b Sb(cid:48)Si(cid:48) ib i b ib i j ib i b ib i b (cid:48) − (cid:48) (cid:48) − (cid:48) +4v˜ajSj Si(cid:48) 2v˜aj SjSi(cid:48) + 4v˜ibSj(cid:48)Si 2v˜i bSj(cid:48)Si ij a j ij a j ij i b ij i b (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) +2v˜i jSb(cid:48)Si + 4v˜i jSj(cid:48) Si(cid:48) + 2v˜ij SjSa(cid:48) + 4v˜ij Sj(cid:48)Si(cid:48) ij i b ij i j ij a j ij i j − (cid:48) (cid:48) 2v˜i j Sj(cid:48)Si(cid:48) (14) ij i j where the implicit summations now run over orbitals: i–(inactive A), j–(inactive B), a– (active A), b–(active B). It is worth noting that an active-active exchange term v˜ba is absent ab from Eq. (14) due to the fact that all active orbitals on A are paired with α spins and all active orbitals on B are paired with β spins. For the ﬁrst oﬀ-diagonal term, corresponding to a single spin ﬂip between the in- (cid:104) | P| ↓ ↑ (cid:105) teracting monomers, Eq. (12) involves two matrix elements, Ψ Ψ V Ψ Ψ and A B A B (cid:104) |P| ↓ ↑ (cid:105) Ψ Ψ Ψ Ψ . Analogously to the conventional closed-shell formalism, these elements A B A B can be expressed through the “spin-ﬂip interac(cid:90)tion density matrix”: (cid:104) | P| ↓ ↑ (cid:105) ↓↑ Ψ Ψ V Ψ Ψ = v˜(12)ρ (12) dτ (15) A B A B int 12 11

text: In

text: that

text: reference,

text: α,

text: α

text: (β,

text: β

text: run

text: over

text: all

text: occupied

text: spin

text: orbitals

text: of

text: A

text: (B).

text: Equation

text: applies

text: in

text: the

text: open-shell

text: case

text: as

text: well

text: (for

text: the

text: diagonal

text: term)

text: but

text: we

text: have

text: to

text: break

text: up

text: the

text: summation

text: over

text: e.g.

text: the

text: occupied

text: spin

text: orbitals

text: of

text: A

text: into

text: the

text: inactive

text: orbitals

text: with

text: spins

text: α

text: and

text: β

text: and

text: active

text: orbitals

text: with

text: spin

text: α.

text: When

text: we

text: do

text: that

text: and

text: perform

text: all

text: spin

text: integrations,

text: the

text: resulting

text: expression

text: for

text: the

text: diagonal

text: term

text: is

text: V

text: V

text: v˜ja

text: v˜bi

text: exch,diag

text: aj

text: ib

text: ij

text: +v˜a

text: v˜aa

text: v˜jj

text: v˜aiSj

text: v˜jbSa

text: aj

text: a

text: aj

text: a

text: aj

text: j

text: aj

text: j

text: ij

text: a

text: ij

text: a

text: ab

text: j

text: +v˜bb

text: v˜b

text: v˜i

text: v˜jbSi

text: v˜aiSb

text: ib

text: b

text: ib

text: b

text: ib

text: i

text: ib

text: i

text: ij

text: b

text: ij

text: b

text: ab

text: i

text: +v˜iaSj

text: v˜bjSi

text: aj

text: i

text: ib

text: j

text: aj

text: i

text: ib

text: j

text: ij

text: i

text: ij

text: j

text: ij

text: i

text: ij

text: j

text: +v˜a

text: v˜ab

text: ab

text: a

text: j

text: ab

text: i

text: b

text: ib

text: a

text: j

text: aj

text: i

text: b

text: v˜aj

text: v˜a

text: j

text: aj

text: a

text: j

text: aj

text: i

text: j

text: aj

text: a

text: j

text: aj

text: a

text: j

text: v˜i

text: v˜i

text: b

text: ib

text: i

text: b

text: ib

text: i

text: j

text: ib

text: i

text: b

text: ib

text: i

text: b

text: ij

text: a

text: j

text: ij

text: a

text: j

text: ij

text: i

text: b

text: ij

text: i

text: b

text: ij

text: i

text: b

text: ij

text: i

text: j

text: ij

text: a

text: j

text: ij

text: i

text: j

text: j

text: ij

text: i

text: j

text: where

text: the

text: implicit

text: summations

text: now

text: run

text: over

text: orbitals:

text: i–(inactive

text: A),

text: j–(inactive

text: B),

text: a–

text: (active

text: A),

text: b–(active

text: B).

text: It

text: is

text: worth

text: noting

text: that

text: an

text: active-active

text: exchange

text: term

text: v˜ba

text: is

text: absent

text: ab

text: from

text: Eq.

text: due

text: to

text: the

text: fact

text: that

text: all

text: active

text: orbitals

text: on

text: A

text: are

text: paired

text: with

text: α

text: spins

text: and

text: all

text: active

text: orbitals

text: on

text: B

text: are

text: paired

text: with

text: β

text: spins.

text: For

text: the

text: ﬁrst

text: oﬀ-diagonal

text: term,

text: corresponding

text: to

text: a

text: single

text: spin

text: ﬂip

text: between

text: the

text: in-

text: P|

text: teracting

text: monomers,

text: Eq.

text: involves

text: two

text: matrix

text: elements,

text: Ψ

text: Ψ

text: V

text: Ψ

text: Ψ

text: and

text: A

text: B

text: A

text: B

text: |P|

text: Ψ

text: Ψ

text: Ψ

text: Ψ

text: Analogously

text: to

text: the

text: conventional

text: closed-shell

text: formalism,

text: these

text: elements

text: A

text: B

text: A

text: B

text: can

text: be

text: expressed

text: through

text: the

text: “spin-ﬂip

text: density

text: matrix”:

text: P|

text: Ψ

text: Ψ

text: V

text: Ψ

text: Ψ

text: dτ

text: A

text: B

text: A

text: B

text: int


text: ↑↑ Ψ are proportional to Sˆ−Sˆ−Ψ and Sˆ Sˆ Ψ (cid:90), respectively. Similar to the previous term, B A + + B (cid:104) | P| ↓↓ ↑↑(cid:105) ↓↓↑↑ Ψ Ψ V Ψ Ψ = v˜(12)ρ (12) dτ (27) A B A B int 12 ↓↓↑↑ where the double spin-ﬂip interaction density matrix ρ (12) is given by a formula anal- int ogous to Eq. (18). However, as a double spin ﬂip can be treated as a special case of a double excitation, the one- and two-electron reduced spin-ﬂip density matrices in the modiﬁed Eq. (18) are now given by Eqs. (93) and (95), respectively, of Ref. [48]. Consequently, ↓↓ | (cid:48) ρ (1 1 ) = 0 (28) A and the two-electron density matrix is proportional to (cid:88) ↓↓ | (cid:48) (cid:48) ∼ 2SA (cid:48) (cid:48) Γ (12 1 2 ) [χ α(1)χ α(2)χ β(1 )χ β(2 ) A an am an am m,n=1 − (cid:48) (cid:48) χ α(1)χ α(2)χ β(1 )χ β(2 ) am an an am − (cid:48) (cid:48) χ α(1)χ α(2)χ β(1 )χ β(2 ) an am am an (cid:48) (cid:48) +χ α(1)χ α(2)χ β(1 )χ β(2 )] (29) am an am an ↓↓↑↑ According to Eq. (28), only the las(cid:90)t term in the expression for ρ survives: int ↓↓↑↑ − ↓↓ | ↑↑ | ρ (12) = Γ (13 14)Γ (24 23) dτ dτ (30) int A B 3 4 Such an expression vanishes upon the integration over dτ and dτ , either alone or with a 1 2 spin-independent operator such as v˜(12) (like in Eq. (27)). In particular, in the integration ↓↓ | over dτ , the two spinorbitals in Γ (13 14) that depend on the coordinates of electron 1 1 A always occur with opposite spins (cf. Eq. (29)), leading to a zero spin integral. This concludes the proof that the double spin-ﬂip contribution to Eq. (12) vanishes. To conclude, the ﬁrst-order SAPT exchange correction for an interaction of two high-spin | − | ≤ ≤ open-shell molecules with spins S and S , in an arbitrary dimer spin state S S S A B A B 15

text: Ψ

text: are

text: proportional

text: to

text: Sˆ−Sˆ−Ψ

text: and

text: Sˆ

text: Sˆ

text: Ψ

text: respectively.

text: Similar

text: to

text: the

text: previous

text: term,

text: B

text: A

text: B

text: P|

text: Ψ

text: Ψ

text: V

text: Ψ

text: Ψ

text: dτ

text: A

text: B

text: A

text: B

text: int

text: where

text: the

text: double

text: spin-ﬂip

text: interaction

text: density

text: matrix

text: ρ

text: is

text: given

text: by

text: a

text: formula

text: anal-

text: int

text: ogous

text: to

text: Eq.

text: However,

text: as

text: a

text: double

text: spin

text: ﬂip

text: can

text: be

text: treated

text: as

text: a

text: special

text: case

text: of

text: a

text: double

text: excitation,

text: the

text: one-

text: and

text: two-electron

text: reduced

text: spin-ﬂip

text: density

text: matrices

text: in

text: the

text: modiﬁed

text: Eq.

text: are

text: now

text: given

text: by

text: Eqs.

text: and

text: respectively,

text: of

text: Ref.

text: Consequently,

text: ρ

text: A

text: and

text: the

text: two-electron

text: density

text: matrix

text: is

text: proportional

text: to

text: Γ

text: [χ

text: )χ

text: A

text: an

text: am

text: an

text: am

text: χ

text: )χ

text: am

text: an

text: an

text: am

text: χ

text: )χ

text: an

text: am

text: am

text: an

text: +χ

text: )χ

text: am

text: an

text: am

text: an

text: According

text: to

text: Eq.

text: only

text: the

text: term

text: in

text: the

text: expression

text: for

text: ρ

text: survives:

text: int

text: ρ

text: Γ

text: dτ

text: dτ

text: int

text: A

text: B

text: Such

text: an

text: expression

text: vanishes

text: upon

text: the

text: integration

text: over

text: dτ

text: and

text: dτ

text: either

text: alone

text: or

text: with

text: a

text: spin-independent

text: operator

text: such

text: as

text: (like

text: in

text: Eq.

text: In

text: particular,

text: in

text: the

text: integration

text: over

text: dτ

text: the

text: two

text: spinorbitals

text: in

text: Γ

text: that

text: depend

text: on

text: the

text: coordinates

text: of

text: electron

text: A

text: always

text: occur

text: with

text: opposite

text: spins

text: (cf.

text: Eq.

text: leading

text: to

text: a

text: zero

text: spin

text: integral.

text: This

text: concludes

text: the

text: proof

text: that

text: the

text: double

text: spin-ﬂip

text: contribution

text: to

text: Eq.

text: vanishes.

text: To

text: conclude,

text: the

text: ﬁrst-order

text: SAPT

text: exchange

text: correction

text: for

text: an

text: interaction

text: of

text: two

text: high-spin

text: open-shell

text: molecules

text: with

text: spins

text: S

text: and

text: S

text: in

text: an

text: arbitrary

text: dimer

text: spin

text: state

text: S

text: S

text: S

text: A

text: B

text: A

text: B


text: E(10) (Eq. (25)) up to terms that vanish as the fourth or higher powers of intermolecular elst overlap integrals [59]. The apparent Coulomb energy does not have this property and the diﬀerence between the two (that is, E(10) ) does not vanish in the S2 approximation. The exch,diag · · · only exception are the interactions involving one- and two-electron systems, such as H H · · · · · · [37], He (3S) H [60], and He (3S) He (3S) [61], when the mathematical Coulomb energy coincides with the mean energy of physical states. In these cases, E(10) (Eq. (14)) is exch,diag identically zero as there are no inactive orbitals (the range of summation over i and j is · · · empty). For the Li H complex, as already observed in Ref. [38], the two Coulomb energies are particularly close and the E(10) term is much smaller than E(10) . As a result, exch,diag exch,1ﬂip the singlet and triplet exchange energies are almost exactly the negatives of each other. Assuming that this is precisely the case, one could deﬁne the exchange energy at the FCI level as one half of the diﬀerence between the FCI interaction energies of triplet and singlet. The FCI exchange energies deﬁned in this way are also displayed in Fig. 1. Interestingly, E(10) represents about two thirds of the total exchange energy, showing that the ﬁrst-order exch description of exchange is qualitatively correct. This result should be contrasted with the ﬁrst-order electrostatic correction E(10) that is nearly negligible for this dispersion-bound elst complex. As a result, the sum E(10) + E(10) recovers only a small fraction (8% at the elst exch,diag triplet van der Waals minimum distance of 11.5a ) of the FCI apparent Coulomb energy 0 while E(10) recovers 71% of the FCI exchange energy. exch,1ﬂip · · · · · · · · · B. Li Li, Li N, and N N · · · · · · The total spin for the Li Li complex can be either singlet or triplet, for Li N - triplet or quintet, and for two interacting nitrogen atoms it can take multiplicities from singlet to · · · septet. For these systems, unlike for the Li H interaction, the benchmark SAPT correc- tions with exact monomer wavefunctions are not available. In Fig. 2 we show the comparison of exchange splittings from SF-SAPT, CASSCF, and MRCI (Davidson corrected). As ex- pected, the ﬁrst-order result fails to recover the exchange splitting around the chemical · · · · · · minima, which are at 5a for Li Li, 3.5a for Li N, and 2.1a for the nitrogen dimer. 0 0 0 Nonetheless, in all these cases SF-SAPT exchange splitting is very close to the CASSCF one for the separations corresponding to the van der Waals minima of the highest spin states: · · · 7.9a for the lithium dimer [62], 10.2a for Li N (this value was obtained in present work 0 0 using the spin-restricted CCSD(T) method with the aug-cc-pVQZ basis set), and 7.2a for 0 21

text: (Eq.

text: up

text: to

text: terms

text: that

text: vanish

text: as

text: the

text: fourth

text: or

text: higher

text: powers

text: of

text: intermolecular

text: elst

text: overlap

text: integrals

text: The

text: apparent

text: Coulomb

text: energy

text: does

text: not

text: have

text: this

text: property

text: and

text: the

text: diﬀerence

text: between

text: the

text: two

text: (that

text: is,

text: does

text: not

text: vanish

text: in

text: the

text: approximation.

text: The

text: exch,diag

text: only

text: exception

text: are

text: the

text: interactions

text: involving

text: one-

text: and

text: two-electron

text: systems,

text: such

text: as

text: H

text: H

text: He

text: H

text: and

text: He

text: He

text: when

text: the

text: mathematical

text: Coulomb

text: energy

text: coincides

text: with

text: the

text: mean

text: energy

text: of

text: physical

text: states.

text: In

text: these

text: cases,

text: (Eq.

text: is

text: exch,diag

text: identically

text: zero

text: as

text: there

text: are

text: no

text: inactive

text: orbitals

text: (the

text: range

text: of

text: summation

text: over

text: i

text: and

text: j

text: is

text: empty).

text: For

text: the

text: Li

text: H

text: complex,

text: as

text: already

text: observed

text: in

text: Ref.

text: the

text: two

text: Coulomb

text: energies

text: are

text: particularly

text: close

text: and

text: the

text: term

text: is

text: much

text: smaller

text: than

text: As

text: a

text: result,

text: exch,diag

text: the

text: singlet

text: and

text: triplet

text: exchange

text: energies

text: are

text: almost

text: exactly

text: the

text: negatives

text: of

text: each

text: other.

text: Assuming

text: that

text: this

text: is

text: precisely

text: the

text: case,

text: one

text: could

text: deﬁne

text: the

text: exchange

text: energy

text: at

text: the

text: FCI

text: level

text: as

text: one

text: half

text: of

text: the

text: diﬀerence

text: between

text: the

text: FCI

text: interaction

text: energies

text: of

text: triplet

text: and

text: singlet.

text: The

text: FCI

text: exchange

text: energies

text: deﬁned

text: in

text: this

text: way

text: are

text: also

text: displayed

text: in

text: Fig.

text: Interestingly,

text: represents

text: about

text: two

text: thirds

text: of

text: the

text: total

text: exchange

text: energy,

text: showing

text: that

text: the

text: ﬁrst-order

text: exch

text: description

text: of

text: exchange

text: is

text: qualitatively

text: correct.

text: This

text: result

text: should

text: be

text: contrasted

text: with

text: the

text: ﬁrst-order

text: electrostatic

text: correction

text: that

text: is

text: nearly

text: negligible

text: for

text: this

text: dispersion-bound

text: elst

text: complex.

text: As

text: a

text: result,

text: the

text: sum

text: recovers

text: only

text: a

text: small

text: fraction

text: at

text: the

text: elst

text: exch,diag

text: triplet

text: van

text: der

text: Waals

text: minimum

text: distance

text: of

text: of

text: the

text: FCI

text: apparent

text: Coulomb

text: energy

text: while

text: recovers

text: of

text: the

text: FCI

text: exchange

text: energy.

text: B.

text: Li

text: Li,

text: Li

text: N,

text: and

text: N

text: N

text: The

text: total

text: spin

text: for

text: the

text: Li

text: Li

text: complex

text: can

text: be

text: either

text: singlet

text: or

text: triplet,

text: for

text: Li

text: N

text: triplet

text: or

text: quintet,

text: and

text: for

text: two

text: interacting

text: nitrogen

text: atoms

text: it

text: can

text: take

text: multiplicities

text: from

text: singlet

text: to

text: septet.

text: For

text: these

text: systems,

text: unlike

text: for

text: the

text: Li

text: H

text: interaction,

text: the

text: benchmark

text: SAPT

text: correc-

text: tions

text: with

text: exact

text: monomer

text: wavefunctions

text: are

text: not

text: available.

text: In

text: Fig.

text: we

text: show

text: the

text: comparison

text: of

text: exchange

text: splittings

text: from

text: SF-SAPT,

text: CASSCF,

text: and

text: MRCI

text: (Davidson

text: corrected).

text: As

text: ex-

text: pected,

text: the

text: ﬁrst-order

text: result

text: fails

text: to

text: recover

text: the

text: exchange

text: splitting

text: around

text: the

text: chemical

text: minima,

text: which

text: are

text: at

text: for

text: Li

text: Li,

text: for

text: Li

text: N,

text: and

text: for

text: the

text: nitrogen

text: dimer.

text: Nonetheless,

text: in

text: all

text: these

text: cases

text: SF-SAPT

text: exchange

text: splitting

text: is

text: very

text: close

text: to

text: the

text: CASSCF

text: one

text: for

text: the

text: separations

text: corresponding

text: to

text: the

text: van

text: der

text: Waals

text: minima

text: of

text: the

text: highest

text: spin

text: states:

text: for

text: the

text: lithium

text: dimer

text: for

text: Li

text: N

text: (this

text: value

text: was

text: obtained

text: in

text: present

text: work

text: using

text: the

text: spin-restricted

text: CCSD(T)

text: method

text: with

text: the

text: aug-cc-pVQZ

text: basis

text: set),

text: and

text: for


text: · · · FIG. 1: First-order exchange energy for the singlet and triplet states of the Li H complex, computed within the ROHF-based approach of this work (points) and the FCI-based SAPT of Ref. [38] (lines). The singlet-triplet splitting from the FCI calculations is shown for comparison. 300 E(1)exch(singlet) E(1)exch(triplet) E(10)exch(singlet) 200 E(10)exch(triplet) FCI 100 0 -100 -200 -300 8 10 12 14 16 18 R [bohr] 34 ]1-mc[ ygrene egnahcxE ##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##

text: FIG.

text: First-order

text: exchange

text: energy

text: for

text: the

text: singlet

text: and

text: triplet

text: states

text: of

text: the

text: Li

text: H

text: complex,

text: computed

text: within

text: the

text: ROHF-based

text: approach

text: of

text: this

text: work

text: (points)

text: and

text: the

text: FCI-based

text: SAPT

text: of

text: Ref.

text: (lines).

text: The

text: singlet-triplet

text: splitting

text: from

text: the

text: FCI

text: calculations

text: is

text: shown

text: for

text: comparison.

text: FCI

text: R

text: [bohr]

text: ygrene

text: egnahcxE

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