Long range behavior of the van der Waals forces between a molecule - - PowerPoint PPT Presentation

Long range behavior of the van der Waals forces between a molecule and a perfectly conducting metallic plate

based on joint work with Mariam Badalyan and Dirk Hundertmark Ioannis Anapolitanos special thanks to: Kurt Busch, Francesco Intravaia, Semjon Wugalter 21.10.2019, CIRM

Ioannis Anapolitanos special thanks to: Kurt Busch, Francesco Intravaia, Semjon van der Waals force between a molecule and a metallic plate

Van der Waals forces and heuristics

Water-Water Water-Atom Atom-Atom dipole-dipole dipole-induced dipole induced dipole- interaction interaction induced dipole interaction

Ioannis Anapolitanos special thanks to: Kurt Busch, Francesco Intravaia, Semjon van der Waals force between a molecule and a metallic plate

Van der Waals forces in life and science

◮ Stabilize DNA, Influence boiling points ◮ Material sciences, Chemistry ◮ Molecule-Wall Interactions

(i) Geckos climb vertical surfaces (ii) Can change the direction of atomic beams

Ioannis Anapolitanos special thanks to: Kurt Busch, Francesco Intravaia, Semjon van der Waals force between a molecule and a metallic plate

Modeling of the problem (for a hydrogen atom)

◮ nucleus at (0, 0, 0) ◮ position of the electron x = (x1, x2, x3) ◮ infinite plate at the plane x1 = r

Ioannis Anapolitanos special thanks to: Kurt Busch, Francesco Intravaia, Semjon van der Waals force between a molecule and a metallic plate

Modeling of the problem (with perfectly conducting plate)

◮ ”anti-nucleus” at 2re1 ◮ ”anti-electron” with distance x∗ = (−x1, x2, x3) from

anti-nucleus.

Ioannis Anapolitanos special thanks to: Kurt Busch, Francesco Intravaia, Semjon van der Waals force between a molecule and a metallic plate

Hamiltonian of the system

◮ Hamiltonian

H(r) = Hh + I

2

Hh = −∆x − 1 |x| I = − 1 2r

• Attraction

K +/K −

− 1 | − x + 2re1 + x∗|

• Attraction

e−/e+

+ 1 |2re1 − x|

• Repulsion

e−/K −

+ 1 |2re1 + x∗|

• Repulsion

e+/K +

• =

2 |2re1−x| Ioannis Anapolitanos special thanks to: Kurt Busch, Francesco Intravaia, Semjon van der Waals force between a molecule and a metallic plate

Hamiltonian of the system

◮ Hamiltonian

H(r) = Hh + I

2

Hh = −∆x − 1 |x| I = − 1 2r

• Attraction

K +/K −

− 1 | − x + 2re1 + x∗|

• Attraction

e−/e+

+ 1 |2re1 − x|

• Repulsion

e−/K −

+ 1 |2re1 + x∗|

• Repulsion

e+/K +

• =

2 |2re1−x|

◮ Always attractive: I < 0.

Ioannis Anapolitanos special thanks to: Kurt Busch, Francesco Intravaia, Semjon van der Waals force between a molecule and a metallic plate

Form core of the Hamiltonian

◮ Electron ’lives’ in halfspace A = {(x1, x2, x3) | x1 < r}. ◮ In H1(A) (no boundary conditions): Hamiltonian not bounded

from below, attraction of electron from the mirror charge is too singular.

◮ Form core C∞ c (A): Electron can not touch the plane, or pass

through it (Dirichlet b.c. rides to the rescue).

Ioannis Anapolitanos special thanks to: Kurt Busch, Francesco Intravaia, Semjon van der Waals force between a molecule and a metallic plate

Form domain and boundedness from below

◮ We use the ”Hardy inequality”

∞

|u(x)|2 4x2 dx ≤

∞

|u′(x)|2dx, in C∞

c (R+).

∞

|u(x)|2 4x2 dx = −

∞

u(x)u(x) 4

1

x

′

dx = Re

∞

u(x) 2x u′(x)dx

CS

≤

∞

• u(x)

4x

• 2

dx

1

2 ∞

|u′(x)|2dx

1

2 Ioannis Anapolitanos special thanks to: Kurt Busch, Francesco Intravaia, Semjon van der Waals force between a molecule and a metallic plate

Form domain and boundedness from below

◮ We use the ”Hardy inequality”

∞

|u(x)|2 4x2 dx ≤

∞

|u′(x)|2dx, in C∞

c (R+).

∞

|u(x)|2 4x2 dx = −

∞

u(x)u(x) 4

1

x

′

dx = Re

∞

u(x) 2x u′(x)dx

CS

≤

∞

• u(x)

4x

• 2

dx

1

2 ∞

|u′(x)|2dx

1

2

◮ Extension of the form on H1 0(A): Form closed and bounded

from below.

◮ Thus H(r) can be realized as a self-adjoint operator.

Ioannis Anapolitanos special thanks to: Kurt Busch, Francesco Intravaia, Semjon van der Waals force between a molecule and a metallic plate

Plate and Electron

◮ z position of the electron with respect to a point on the plate. ◮ He− = −∆z − 1 2|z−z∗| = −∆z − 1 4z1

Ee− = inf

ψ∈H1

0(A),ψL2(A)=1 ψ | He−ψ

rescaling

<

Ioannis Anapolitanos special thanks to: Kurt Busch, Francesco Intravaia, Semjon van der Waals force between a molecule and a metallic plate

Plate and Electron

◮ z position of the electron with respect to a point on the plate. ◮ He− = −∆z − 1 2|z−z∗| = −∆z − 1 4z1

Ee− = inf

ψ∈H1

0(A),ψL2(A)=1 ψ | He−ψ

rescaling

<

◮ Lower bound: Because 2ab ≤ a2 + b2:

1 4z1 = 2 1 4(2z1) ≤ 1 16 + 1 4z2

1 ”Hardy”

≤ 1 16 − ∂2 ∂z2

1

Thus −∆z −

1 4z1 ≥ − 1 16 = Eh 4 > Eh (hydrogen gs energy) ◮ Thus Ee− > Eh. The electron prefers to be close to the

nucleus and not close to the plate.

Ioannis Anapolitanos special thanks to: Kurt Busch, Francesco Intravaia, Semjon van der Waals force between a molecule and a metallic plate

Existence of a ground state of the system

◮ HVZ type Theorem:

inf σess(H(r)) = Ee−

• minimum energy of

electron at ’infinity’

− 1 4r

nucleus-plate attraction ◮ I < 0 r not too small

= ⇒ inf σ(H(r)) ≤ Eh.

◮ Recall Ee− ≥ Eh 4 > Eh. ◮ Thus inf σ(H(r)) in discrete spectrum.

Ioannis Anapolitanos special thanks to: Kurt Busch, Francesco Intravaia, Semjon van der Waals force between a molecule and a metallic plate

Interaction energy and main result

◮ Ground state energy E(r) = infψ∈H1

0(A),ψL2(A)=1 ψ | H(r)ψ .

◮ Interaction energy W (r) = E(r) − lims→∞ E(s) = E(r) − Eh.

Ioannis Anapolitanos special thanks to: Kurt Busch, Francesco Intravaia, Semjon van der Waals force between a molecule and a metallic plate

Interaction energy and main result

◮ Ground state energy E(r) = infψ∈H1

0(A),ψL2(A)=1 ψ | H(r)ψ .

◮ Interaction energy W (r) = E(r) − lims→∞ E(s) = E(r) − Eh.

Theorem (A., Badalyan, 51,32 e)

There exist a r0 > 0 and D > 0, so that for all r > r0:

• W (r) + 1

r 3

• ≤ D

r 5 holds.

Ioannis Anapolitanos special thanks to: Kurt Busch, Francesco Intravaia, Semjon van der Waals force between a molecule and a metallic plate

Main idea of the proof

Write H = H(r) and E the ground state energy of H.

◮ P orthogonal projection P⊥ = 1 − P, H⊥ = P⊥HP⊥.

Feshbach map FP(λ) = (PHP − PHP⊥(H⊥ − λ)−1P⊥HP)|RanP.

◮ Thm: (H⊥ − E) ≥ γ > 0 =

⇒ E is eigenvalue of FP(E).

Ioannis Anapolitanos special thanks to: Kurt Busch, Francesco Intravaia, Semjon van der Waals force between a molecule and a metallic plate

Main idea of the proof

Write H = H(r) and E the ground state energy of H.

◮ P orthogonal projection P⊥ = 1 − P, H⊥ = P⊥HP⊥.

Feshbach map FP(λ) = (PHP − PHP⊥(H⊥ − λ)−1P⊥HP)|RanP.

◮ Thm: (H⊥ − E) ≥ γ > 0 =

⇒ E is eigenvalue of FP(E).

◮ Choose P = |ψψ| cut off ground state of hydrogen atom. ◮ Step 1: (H⊥ − E) ≥ γ, we need Ee− > Eh. ◮ Step 2: PHP|RanP ≃ Eh + ψ, I 2ψ = Eh − 1 r3 + O( 1 r5 ).

I = −(x · e1)2 − |x|2 8r 3 + fodd(x) 8r 4 + O

1

r 5

• ◮ Step 3: −PHP⊥(H⊥ − E)−1P⊥HP = O( 1

r6 ).

Ioannis Anapolitanos special thanks to: Kurt Busch, Francesco Intravaia, Semjon van der Waals force between a molecule and a metallic plate

Feshbach condition H⊥ − E ≥ c > 0

◮ IMS Localization: H = J1HJ1 + J2HJ2 − |∇J1|2 − |∇J2|2 for

J1, J2 : R3 → R zwei C∞-Functions so that J2

1 + J2 2 = 1 ◮ For the term J1HJ1 we need Ee− > Eh

Ioannis Anapolitanos special thanks to: Kurt Busch, Francesco Intravaia, Semjon van der Waals force between a molecule and a metallic plate

The case of a general molecule

◮ Hamiltonian HN = HN(r, v), N number of electrons, r

distance of molecule and plate, v relative orientation.

Ioannis Anapolitanos special thanks to: Kurt Busch, Francesco Intravaia, Semjon van der Waals force between a molecule and a metallic plate

The case of a general molecule

◮ Hamiltonian HN = HN(r, v), N number of electrons, r

distance of molecule and plate, v relative orientation.

◮ Hk: Hamiltonian with all nuclei but only k electrons present. ◮ Hamiltonian k electrons-plate (no nuclei)

Ak = −

k

• i=1

∆xi +

• 1≤i<j≤k

1 |xi − xj| − 1 2

• 1≤i≤j≤k

1 | − xi + 2rv + x∗

j |

Ioannis Anapolitanos special thanks to: Kurt Busch, Francesco Intravaia, Semjon van der Waals force between a molecule and a metallic plate

The case of a general molecule

◮ Hamiltonian HN = HN(r, v), N number of electrons, r

distance of molecule and plate, v relative orientation.

◮ Hk: Hamiltonian with all nuclei but only k electrons present. ◮ Hamiltonian k electrons-plate (no nuclei)

Ak = −

k

• i=1

∆xi +

• 1≤i<j≤k

1 |xi − xj| − 1 2

• 1≤i≤j≤k

1 | − xi + 2rv + x∗

j | ◮ Condition for existence of ground state

inf σ(HN) < inf σ(HN−k) + inf σ(Ak), ∀k ∈ {1, . . . , N}. For helium atom it is proven.

Ioannis Anapolitanos special thanks to: Kurt Busch, Francesco Intravaia, Semjon van der Waals force between a molecule and a metallic plate

Theorem in the case of a general molecule

◮ Recall the Condition ∀k ∈ {1, . . . , N}

inf σ(HN) < inf σ(HN−k) + inf σ(Ak). (0.1)

◮ Let B be the ground state eigenspace of HN

C(v) = 1 16 sup

ψ∈B,ψ=1

• ψ,

  N

• i=1

xi · v

2

+

• N
• i=1

xi

• 2

 ψ

• .

Ioannis Anapolitanos special thanks to: Kurt Busch, Francesco Intravaia, Semjon van der Waals force between a molecule and a metallic plate

Theorem in the case of a general molecule

◮ Recall the Condition ∀k ∈ {1, . . . , N}

inf σ(HN) < inf σ(HN−k) + inf σ(Ak). (0.1)

◮ Let B be the ground state eigenspace of HN

C(v) = 1 16 sup

ψ∈B,ψ=1

• ψ,

  N

• i=1

xi · v

2

+

• N
• i=1

xi

• 2

 ψ

• .

◮ Theorem: (independent of statistics) If (0.1) holds then

W (r, v) = −C(v) r 3 + O( 1 r 4 ).

Ioannis Anapolitanos special thanks to: Kurt Busch, Francesco Intravaia, Semjon van der Waals force between a molecule and a metallic plate

Theorem in the case of a general molecule

◮ Recall the Condition ∀k ∈ {1, . . . , N}

inf σ(HN) < inf σ(HN−k) + inf σ(Ak). (0.1)

◮ Let B be the ground state eigenspace of HN

C(v) = 1 16 sup

ψ∈B,ψ=1

• ψ,

  N

• i=1

xi · v

2

+

• N
• i=1

xi

• 2

 ψ

• .

◮ Theorem: (independent of statistics) If (0.1) holds then

W (r, v) = −C(v) r 3 + O( 1 r 4 ).

◮ New ingredient in the proof: E is lowest eigenvalue of FP(E)

The eigenfunction minimizes FP(E) Thus multiplicity of ground state energy does not matter.

Ioannis Anapolitanos special thanks to: Kurt Busch, Francesco Intravaia, Semjon van der Waals force between a molecule and a metallic plate

History of the mathematical research

◮ Morgan, Simon (1980) ◮ Lieb,Thirring (1986) ◮ Anapolitanos, Sigal/Anapolitanos (2016/2017) ◮ Anapolitanos, Lewin, Roth (2019) ◮ Barbaroux, Hartig, Hundertmark, Wugalter (2019) ◮ In context of chemical reactions:

Lewin (2004, 2006), Anapolitanos, Lewin (2019)

◮ numerics with proof of convergence: Cances, Scott (2017)

Ioannis Anapolitanos special thanks to: Kurt Busch, Francesco Intravaia, Semjon van der Waals force between a molecule and a metallic plate