Vladimir I Korobov

Jean-Philippe Karr

Mohammad Haidar

Zhen-Xiang Zhong

Hyperfine structure in the H + 2 and HD + molecular ions at mα 6 order

A complete effective Hamiltonian for relativistic corrections at orders mα 6 and mα 6 (m/M ) in a one-electron molecular system is derived from the NRQED Lagrangian. It includes spin-independent corrections to the energy levels and spin-spin scalar interactions contributing to the hyperfine splitting, both of which had been studied previously. In addition, corrections to electron spin-orbit and spin-spin tensor interactions are newly obtained. This allows improving the hyperfine structure theory in the hydrogen molecular ions. Improved values of the spin-orbit hyperfine coefficient are calculated for a few transitions of current experimental interest.

I. INTRODUCTION

. While information on fundamental constants is obtained from comparison of spin-averaged transition frequencies with theoretical predictions, the hyperfine splitting of ro-vibrational lines also allows for precise tests of theory.

 at the level of ∼ 1 ppm. The following step to improve the hyperfine structure theory is to evaluate higher-order corrections to the next largest coefficients, i.e. the electron spin-orbit and spin-spin tensor interaction, starting with relativistic corrections at the mα 6 order.

. Then, we use it to calculate numerically the corrections to the electron spin-orbit interaction for a few transitions studied in ongoing experiments. The paper is organized as follows: in Secs. II and III, we recall the expression of the NRQED Lagrangian and associated interaction vertices. We then systematically derive the effective potentials, which are organized in three categories: tree-level interactions involving the exchange of a Coulomb or transverse photon (Sec. IV), terms due to retardation in the transverse photon exchange (Sec. V), and finally those coming from a seagull diagram with simultaneous exchange of two photons (Sec. VI). In Sec. VII, we collect our results to write the total effective Hamiltonian, separating the different types of interactions: spin-independent, electronic spin-orbit, spin-spin scalar and tensor interactions. Finally, in Sec. VIII we present numerical calculations of the spin-orbit interaction coefficient.

II. NRQED LAGRANGIAN

Natural (Lorentz-Heaviside) units (h = c = 1) are used throughout. We assume that e is the electron's charge and thus is negative, the elementary charge is then denoted by |e|.

, including all the terms involved in 1. Decomposition into irreducible tensor components

bound-state energy corrections up to the mα 6 order:

L main = ψ * e i∂ t -eA 0 + D 2 2m + D 4 8m 3 + D 6 16m 5 + . . . ψ e +ψ * e c F e 2m σ•B + c D e 8m 2 D•E -E•D + c S ie 8m 2 σ• D×E -E×D +c W1 e 8m 3 D 2 , σB + 3ie 16m 4 D 2 , σ• D×E -E×D - 3e 64m 4 D 2 , [∇, E] - 5e 128m 4 D 2 , (D•E + E•D) - e 2 8m 3 E 2 ψ e , (1)
where D = ∇ -ieA. The contact terms required in the NRQED theory [START_REF] Caswell | Effective Lagrangians for bound state problems in QED, QCD, and othe field theories[END_REF][START_REF] Kinoshita | Radiative corrections to the muonium hyperfine structure: The α 2 (Zα) correction[END_REF][START_REF] Hill | NRQED Lagrangian at order 1/M 4[END_REF] are not considered here, because they do not play any role in the spin-orbit and spin-spin tensor interactions which are our main focus in the following.

Here and in what follows we use the notation: {X, Y } = XY + Y * X * , [X, Y ] = XY -Y X where the star denotes a Hermitian conjugate. The coupling constants, c i , are determined by requiring that scattering amplitudes in QED and NRQED agree up to a chosen order in α and in v 2 /c 2 . Performing this matching at tree level, which is enough for the work presented here, one gets c

F = c D = c S = c W1 = 1.
As shown in more detail in [START_REF] Haidar | Nonrelativistic QED approach to the fine-and hyperfine-structure corrections of order mα 6 and mα 6 (m/M ): Application to the hydrogen atom[END_REF], the effective Hamiltonian H eff , which stems from the Lagrangian, is equivalent to the Foldy-Wouthuysen Hamiltonian H FW derived in [START_REF] Patkóš | Higher-order recoil corrections for triplet states of the helium atom[END_REF] (see Eq.( 23)). It may be obtained from H FW through the canonical transformation e iS (H -i∂ t )e -iS , where [START_REF] Patkóš | Higher-order recoil corrections for triplet states of the helium atom[END_REF] S = e 8m 2 σ• A×π -π×A , π = p -eA, where p = -i∇ is the electron's impulse. Heavy particles of masses M a charges Z a , and impulses P a with a = 1, 2, are treated within the leading-order interaction Hamiltonian:

H I = -Z a |e| P a 2M a A + A P a 2M a -µ a •B + Z 2 a e 2 2M a A 2 . (2)
The magnetic moments of particles are expressed as follows: µ e and µ a are dimensionless quantities measured in Bohr and nuclear magnetons, respectively. We will consider corrections to the bound states of a one-electron molecular system such as H + 2 or HD + . The zero-order approximation is the nonrelativistic Schrödinger equation with the Hamiltonian

µ e =
H 0 = P 2 1 2M 1 + P 2 2 2M 2 + p 2 e 2m + V, V = - Z 1 α r 1 - Z 2 α r 2 + Z 1 Z 2 α R . (3)
Here r a = r e -R a with a = (1, 2) is the electron's position with respect to the nucleus a, and R = R 2 -R 1 the internuclear vector. It is assumed that M a m. We also assume that the Hamiltonian is written in the center of mass (center of inertia) frame, which implies: p e +P 1 +P 2 = 0.

The potentials A 0 and A are related to electric and magnetic field strengths as follows

E = -∇A 0 - ∂A ∂t , B = ∇ × A.
We define E = -∇A 0 and E ⊥ = -∂A ∂t , while B is always transverse. It is worth noting that E corresponds to an instantaneous interaction, while A propagates in time with the velocity of light.

In order to determine which terms are needed at a given order, it is useful to know the nominal order of expectation values of various operators for a wavefunction of the nonrelativistic bound system. One gets [START_REF] Kinoshita | Radiative corrections to the muonium hyperfine structure: The α 2 (Zα) correction[END_REF]:

p ∼ m(v/c), ∂ t ∼ m(v/c) 2 , eA 0 ∼ m(v/c) 2 , eA ∼ m(v/c) 3 , eE ∼ m 2 (v/c) 3 , eB ∼ m 2 (v/c) 4 ,
where v is the typical velocity of the bound electron.

The photon propagator in the Coulomb gauge is:

G µν =          G 00 = 1 q 2 , -the Coulomb photon propagator, G ij = δ ij -q i q j /q 2 q 2 0 -q 2 + iε
, -the transverse photon propagator.

(4)

We use Feynman's time-ordered perturbation formalism [START_REF] Feynman | The theory of positrons[END_REF], whereby the change of energy of a bound system due to exchange of one photon is expressed

∆E = d 4 q (2π) 4 i G µν (q) ψ 0 V µ (2) e iqr a 1 E 0 -q 0 -H 0 e -iqr b V ν (1) ψ 0 (5)
where V (1) and V (2) are some NRQED vertices for the electron or nucleus, ψ 0 and E 0 are respectively the nonrelativistic bound state wave function and energy for the Hamiltonian (3), and r a , r b the position operators of the involved particles. This formula dates back from the original work by Feynman [START_REF] Baranger | Relativistic Correction to the Lamb Shift[END_REF] (Sec. II) and appears in a slightly modified form in [START_REF] Pachucki | Long-wavelength quantum electrodynamics[END_REF][START_REF] Pachucki | Higher-order effective Hamiltonian for light atomic systems[END_REF].

III. NRQED VERTICES

It is convenient to translate the NRQED Lagrangian [Eq. ( 1)] in terms of NRQED vertices and "Feynman" rules, as done in Fig. 3 of Ref. [START_REF] Kinoshita | Radiative corrections to the muonium hyperfine structure: The α 2 (Zα) correction[END_REF]. Here, we list the vertices contributing to the mα 6 and mα 6 (m/M ) orders, and give their expressions both in momentum and coordinate space, which are connected to each other by a 3D Fourier transformation. In the momentum space we use p and p as momenta of incident and scattered electron, q = p -p is the transferred momentum.

We first give the tree-level vertices related to the electron line:

1. e 3q 2 (p 2 +p 2 ) 64m 4 + 5(p 2 -p 2 ) 2 128m 4 A 0 - 3e 64m 4 p 2 , [∆A 0 ] + 5e 128m 4 p 2 , [p 2 , A 0 ] 2. -e i 3σ[q × p](p 2 +p 2 ) 32m 4 A 0 3e 32m 4 p 2 , σ•[E ×p] 3. e (p 2 +p 2 )(p +p) 8m 3 A e 8m 3 p 2 , p•A+A•p 4. e i[σ×q](p 2 +p 2 ) 8m 3 A e 8m 3 p 2 , σ•B 5. -e p +p 2m A -e p 2m A + A p 2m 6. -e i 2m [σ×q]•A - e 2m σ•B 7. e iq 0 [σ×(p +p)] 8m 2 A - e 8m 2 σ• p×∂ t A -∂ t A×p (6)
The last one appears only in the retardation contribution, see Sec V B. For nuclei, the following tree-level vertices come into play:

1N. Z a |e|A 0 Z a |e|A 0 2N. -Z a |e| P a +P a 2M a A -Z a |e| P a 2M a A + A P a 2M a 3N. i [µ a ×q]•A -µ a •B (7)
The seagull-type vertices for the electron are:

8.

e 2 iq 1 ×σ 4m 2 A 0 (q 1 )A(q 2 ) - e 2 8m 2 σ A×E -E×A 9. -e 2 q i 1 q i 2 8m 3 A 0 (q 1 )A 0 (q 2 ) e 2 8m 3 E 2 10. e 2 δ ij 2m A(q 1 )A(q 2 ) e 2 2m A 2 (8)
Note that two transverse photon vertex 10. only contributes at the (m/M ) 2 order and thus will not be used in the following. However, the corresponding vertex for nuclei should be included:

4N. Z 2 a e 2 δ ij 2M a A(q 1 )•A(q 2 ) Z 2 a e 2 2M a A 2 . (9)
In the following, we obtain from these vertices the effective potentials at orders mα 6 and mα 6 (m/M ) (both spinindependent and spin-dependent) by systematic application of the nonrelativistic Rayleigh-Schrödinger perturbation theory. For each term, we will mention which vertices are involved by referring to the numbering given above. It is understood that all terms should be summed over the nuclear index a (a = 1, 2, and b = 3 -a).

IV. TREE-LEVEL INTERACTIONS

We first consider the tree-level diagrams involving the exchange of one photon between the electron and a nucleus. The derivation of effective potentials is straightforward in this case (one such example is given in [START_REF] Haidar | Nonrelativistic QED approach to the fine-and hyperfine-structure corrections of order mα 6 and mα 6 (m/M ): Application to the hydrogen atom[END_REF]). For the transformation from momentum to coordinate space, useful integrals can be found in Appendix A. Two terms come from a Coulomb photon exchange (vertices 1-1N and 2-1N):

U 1a = - 3 64m 4 p 2 e , [∆V ] + 5 128m 4 p 2 e , [p 2 e , V ]
U 1b = - 3Z a α 32m 4 p 2 e , 1 r 3 a [r a ×p e ]•σ e . (10)
The square brackets around quantities imply that derivatives act only within the bracket, thus [∆V ] in the first line corresponds to the Laplacian of the Coulomb potential i.e. a sum of delta-function operators. The transverse photon exchange produces four terms (3-2N, 3-3N, 4-2N and 4-3N):

U 2a = - Z a α 8m 2 p 2 e , p i e m δ ij r a + r i a r j a r 3 a P j a M a U 2b = Z a α 8m 3 M a p 2 e , 1 r 3 a r a ×P a •σ e U 2c = - α 4m 3 p 2 e , 1 r 3 a [r a ×p e ] µ a |e| U 2d = 1 4m 2 p 2 e , 8π

V. RETARDATION IN THE SINGLE TRANSVERSE PHOTON EXCHANGE

According to Eq. ( 5), the energy correction due to a single transverse photon exchange between the electron and a nucleus is

∆E = 1 (2π) 4 d 4 q q 2 +i δ ij - q i q j q 2 ψ 0 V i (2) e iqr e 1 E 0 -q 0 -H 0 e -iqR a V j (1) ψ 0 (12)
where V (1) and V (2) are some NRQED vertices from Section III, Eqs. ( 6) and [START_REF] Bakalov | High-Precision Calculation of the Hyperfine Structure of the HD + Ion[END_REF].

A. Dipole and Fermi vertices

Let us consider first the contribution from the leading-order vertices: 5 and 6 for the electron, 2N and 3N for the nucleus,

U (5+) 3 = - e (2π) 3 dq 2q δ ij - q i q j q 2 p e m + i[σ e ×q] 2m i e iqr e 1 E 0 -q-H 0 + 1 q e -iqR a - 1 E 0 -q-H 0 + 1 q -Z a |e| P a M a + i [µ a ×q] j + e iqr e ↔ e -iqr e e -iqR a ↔ e iqR a , (13)
where (5+) means orders mα 5 and higher. The term 1/q in parentheses of (13) corresponds to the subtracted leading mα 4 order contribution to the Breit-Pauli interaction [START_REF] Pachucki | Simple derivation of helium Lamb shift[END_REF], and the mα 5 -order is removed by subtracting the term corresponding to the q = 0 limit. We use the retardation expansion:

1 E 0 -q-H 0 + 1 q = H 0 -E 0 q 2 - (H 0 -E 0 ) 2 q 3 + . . . (14)
for transverse photon momenta q ∼ (v/c) (H 0 -E 0 ) ∼ (v/c) 2 (the contribution from smaller momenta is suppressed after the performed subtractions). Here, the first term corresponds to a contribution of order mα 5 [START_REF] Pachucki | Simple derivation of helium Lamb shift[END_REF], and the second term contributes to order mα 6 . Then

U (6) 3 = e (2π) 3 dq 2q 4 δ ij - q i q j q 2 p e m + i[σ e ×q] 2m i × e iqr e (H 0 -E 0 ) 2 e -iqR a -(H 0 -E 0 ) 2 -Z a |e| P a M a + i [µ a ×q] j + e iqr e ↔ e -iqr e e -iqR a ↔ e iqR a (15)
From the relationship (with a = 1, 2 and b = 3 -a)

R a = - m M r a ∓ M b M R,
where M = M 1 + M 2 + m, and ∓ means a minus sign for a = 1 and plus for a = 2, one gets:

H 0 , e -iqR a = e -iqR a E 0 O m M .
As a result,

e iqr e (H 0 -E 0 ) 2 e -iqR a = e iqr a (H 0 -E 0 ) 2 + e iqr e H 0 , e -iqR a (H 0 -E 0 ) + e iqr e (H 0 -E 0 ) H 0 , e -iqR a ≈ e iqr a (H 0 -E 0 ) 2 = (H 0 -E 0 ) e iqr a (H 0 -E 0 ) + e iqr a , H 0 (H 0 -E 0 ) . (16)
In the second line, we have kept only the leading-order term in (m/M ).

Using this relationship, one immediately sees that the terms of Eq. ( 15) involving the nuclear magnetic moment give a zero contribution when applied to the zero-order state |ψ 0 . These terms thus contribute only at higher orders in m/M (mα 6 (m/M ) 2 and above) and will not be considered here. The remaining terms can be separated into a spin-independent term and a term contributing to the spin-orbit interaction.

For the spin-independent part we have

U 3a = Z a e 2 mM a 1 (2π) 3 dq 2q 4 δ ij - q i q j q 2 p i e (H 0 -E 0 ) e iqr a -1 (H 0 -E 0 ) + e iqr a -1, H 0 (H 0 -E 0 ) + (H 0 -E 0 ) e -iqr a -1 (H 0 -E 0 ) + e -iqr a -1, H 0 (H 0 -E 0 ) P j a ,
and after integration we finally obtain (using the third line of Eq. (A2)):

U 3a = Z a α 16mM a p i e , V r i a r j a -3r 2 a δ ij r a V, P j a + p i e r i a r j a -3r 2 a δ ij r a , p 2 e 2m V, P j a + (h.c.) = Z 2 a α 3 8mM a Z 1 r i 1 r 3 1 + Z 2 r i 2 r 3 2 r i a r j a -3r 2 a δ ij r a - r j a r 3 a ± Z b R j R 3 + Z 2 a α 2 8m 2 M a p 2 e r 2 a -3 r a (r a p e )p e r a ∓ Z 2 a Z b α 2 8m 2 M a (Rr a )p 2 e R 3 r a - (Rr a )r a (r a p e )p e R 3 r 3 a -2 R(r a p e )p e R 3 r a . (17)
Here we have used

[V, P a ] = i - Z a α r a r 3 a ± Z a Z b αR R 3 .
Contributions from the R/R 3 terms can be assumed to be small, since at small R the wave function is exponentially small due to the strong Coulomb barrier. Then the interaction may be simplified to

U 3a = α 3 4m Z 3 1 M 1 1 r 3 1 + Z 3 2 M 2 1 r 3 2 + Z 2 1 Z 2 M 1 (r 1 r 2 ) r 2 1 r 3 2 + Z 1 Z 2 2 M 2 (r 1 r 2 ) r 3 1 r 2 2 + Z 2 1 α 2 8m 2 M 1 1 r 4 1 + p e 1 r 2 1 p e -3 (p e r 1 )(r 1 p e ) r 4 1 + Z 2 2 α 2 8m 2 M 2 1 r 4 2 + p e 1 r 2 2 p e -3 (p e r 2 )(r 2 p e ) r 4 2 . ( 18
)
The electron spin-orbit term is

U 3b = Z a e 2 2mM a 1 (2π) 3 dq 2q 4 i[σ e ×q] e iqr a -1, H 0 (H 0 -E 0 ) + e -iqr a -1, H 0 (H 0 -E 0 ) P a
After Fourier transform:

U 3b = - Z a α 16m 2 M a p 2 e ,
r a r a ×σ e [V, P a ] + (h.c.), and using the commutators

P 2 a , r j a r a = 2r j a r 3 a - 2i r a P j a + 2ir j a r 3 a (r a P a ) , P a P b , r j a r a = - i r a P j b + ir j a r 3 a (r a P b ) ,
one gets:

U 3b = Z 2 a α 2 4m 2 M a 1 r 4 a [r a ×p e ] ∓ Z b r a R 3 [R×p e ] ∓ Z b r 3 a R 3 [r a ×R](r a p e ) •σ e = Z 2 1 α 2 4m 2 M 1 1 r 4 1 [r 1 ×p e ]•σ e + Z 2 2 α 2 4m 2 M 2 1 r 4 2 [r 2 ×p e ]•σ e - Z 2 1 Z 2 α 2 4m 2 M 1 1 r 1 R 3 [R×p e ]•σ e + Z 2 2 Z 1 α 2 4m 2 M 2 1 r 2 R 3 [R×p e ]•σ e + Z 2 1 Z 2 α 2 4m 2 M 1 [r 1 ×r 2 ] r 3 1 R 3 (r 1 p e )•σ e - Z 2 2 Z 1 α 2 4m 2 M 2 [r 1 ×r 2 ] r 3 2 R 3 (r 2 p e )•σ e . (19)
Similarly to the U 3a term, one can neglect the last two lines in the above expression.

B. Time derivative vertex

Now we consider a retardation term where the electron interacts via the time derivative vertex (number 7 in Eq. ( 6)) while the nucleus interacts via the lowest-order vertices (dipole (2N) or Fermi (3N)).

U (5+) 3c = (-i) (2π) 4 d 4 q q 2 0 -q 2 +i δ ij - q i q j q 2 ieq 0 8m 2 (p e +p e )×σ e i × × e iqr e 1 E 0 -q 0 -H 0 +i e -iqR a -Z a e P a M a + i [µ a ×q] j + e iqr e ↔ e -iqr e e -iqR ↔ e iqR , (20)
The first step is integration over q 0 . Using integration in the complex plane one gets

U (5+) 3c = ie 16m 2 d 3 q (2π) 3 δ ij - q i q j q 2 [(p e +p e )×σ e ] i × × e iqr e 1 E 0 -q-H 0 e -iqR a -Z a e P a M a + i [µ a ×q] j + e iqr e ↔ e -iqr e e -iqR a ↔ e iqR a , (21)
The time derivative vertex is of nominal order (v/c) 3 ∼ α 3 . The first term in the expansion of 1/(E 0 -H 0 -q) (i.e.

-1/q, see Eq. ( 14)) would produce a contribution of order mα 5 , but this contribution vanishes due to cancellation between both terms of Eq. ( 21). The mα 6 -order term corresponds to the next term, (H 0 -E 0 )/q 2 . Using the relation e iqr e (H 0 -E 0 ) e -iqR a ≈ e iqr a (H 0 -E 0 ), similar to Eq. ( 16), one gets that the term involving the nuclear magnetic moment only contributes at the (m/M ) 2 order and mat thus be ignored. The remaining term is

U (6) 3c ≈ - iZ a e 2 8M a m 2 d 3 q (2π) 3 1 q 2 δ ij - q i q j q 2 × × [p e ×σ e ] i e iqr a (H 0 -E 0 )P j a + [p e ×σ e ] i e -iqr a (H 0 -E 0 )P j a = - iZ a e 2 8M a m 2 d 3 q (2π) 3 1 q 2 δ ij - q i q j q 2 × × [p e ×σ e ] i e iqr a -1 [H 0 , P j a ] + [p e ×σ e ] i e -iqr a -1 [H 0 , P j a ] . (22)
After Fourier transform:

U 3c = - iZ a α 8M a m 2 [p e ×σ e ] i 1 2r a δ ij + r i a r j a r 2 a V, P j a + (h.c) = - Z 2 1 α 2 4M 1 m 2 1 r 4 1 [r 1 ×p e ]•σ e - Z 2 2 α 2 4M 2 m 2 1 r 4 2 [r 2 ×p e ]•σ e + Z 2 1 Z 2 α 2 8M 1 m 2 1 r 1 R 3 [R×p e ]•σ e - Z 2 2 Z 1 α 2 8M 2 m 2 1 r 2 R 3 [R×p e ]•σ e + Z 2 1 Z 2 α 2 8M 1 m 2 (r 1 R) r 3 1 R 3 [r 1 ×p e ]•σ e - Z 2 2 Z 1 α 2 8M 2 m 2 (r 2 R) r 3 2 R 3 [r 2 ×p e ]•σ e . (23)
Once more, the last two lines in the above expression may be neglected. Comparing Eq. ( 19) with Eq. ( 23), we see that the leading terms cancel out, so that U 3b + U 3c is negligibly small. We will thus ignore these two terms when writing the total effective Hamiltonian in Sec. VII.

VI. SEAGULL-TYPE INTERACTIONS

It remains to consider the contributions arising from seagull-type vertices, Eqs. ( 8) and ( 9). All the interactions in momentum space are the convolution of two functions, which represent either the electric field strength E or magnetic field potential A. In the coordinate space, they are directly given by a product (scalar or vector) of the same functions converted to the coordinate space.

The corresponding expressions for the electric field strength of a point-like Coulomb source, and for the magnetic field potential produced by the moving charge and magnetic moment of a nucleus, are

eE = Z a e 2 iq q 2 ⇒ -Z a α r a r 3 a eA 1 = - e q 2 δ ij - q i q j q 2 -Z a |e| P a M a ⇒ - Z a α 2M a δ ij r a + r i a r j a r 3 a P j a eA 2 = - e q 2 δ ij - q i q j q 2 -i [µ a ×(-q)] = ie q 2 [q×µ a ] ⇒ - e r 3 a [r a ×µ a ]
and the potential produced by the electron at the location of a nucleus is

|e|Z a A a = - |e|Z a q 2 δ ij - q i q j q 2 -e p e m + i e [σ e ×(-q)] 2m ⇒ - Z a α 2m δ ij r a + r i a r j a r 3 a p j e - Z a α 2m 1 r 3 a [r a ×σ e ]
The first seagull-type contribution is a double Coulomb photon exchange diagram (vertices 7-1N-1N):

U 4 = e 2 E 2 8m 3 = α 2 8m 3 Z 1 r 1 r 3 1 + Z 2 r 2 r 3 2 2 . ( 24
)
The next terms stem from the seagull vertex with one Coulomb and one transverse photon (vertices 8-1N-2N and 8-1N-3N):

U 5a = σ e 4m 2 -Z b α r i b r 3 b × - Z a α 2M a δ ij r a + r i a r j a r 3 a P j a U 5b = σ e 4m 2 -Z b α r b r 3 b × - e r 3 a [r a ×µ a ] = - αZ b 2m 1 r 3 b [r b ×µ e ] 1 r 3 a [r a ×µ a ]
or finally

U 5a = Z 2 1 α 2 8m 2 M 1 1 r 4 1 [r 1 ×P 1 ]•σ e + Z 2 2 α 2 8m 2 M 2 1 r 4 2 [r 2 ×P 2 ]•σ e + Z 1 Z 2 α 2 8m 2 M 1 1 r 1 r 3 2 [r 2 ×P 1 ]•σ e + Z 1 Z 2 α 2 8m 2 M 2 1 r 3 1 r 2 [r 1 ×P 2 ]•σ e - Z 1 Z 2 α 2 8m 2 M 1 1 r 3 1 r 3 2 [r 1 ×r 2 ] (r 1 P 1 )•σ e + Z 1 Z 2 α 2 8m 2 M 2 1 r 3 1 r 3 2 [r 1 ×r 2 ] (r 2 P 2 )•σ e U 5b = - α 6m Z 1 r 2 1 µ e µ 1 -3(µ e r 1 )(µ 1 r 1 ) r 6 1 + Z 2 r 2 2 µ e µ 2 -3(µ e r 2 )(µ 2 r 2 ) r 6 2 + 2Z 1 µ e µ 1 r 4 1 + 2Z 2 µ e µ 2 r 4 2 - α 6m Z 2 (r 1 r 2 )µ e µ 1 -3(µ e r 1 )(µ 1 r 2 ) r 3 1 r 3 2 + Z 1 (r 1 r 2 )µ e µ 2 -3(µ e r 2 )(µ 2 r 1 ) r 3 1 r 3 2 + 2Z 2 (r 1 r 2 )µ e µ 1 r 3 1 r 3 2 + 2Z 1 (r 1 r 2 )µ e µ 2 r 3 1 r 3 2 (25)
The last contribution to consider is a double transverse photon exchange with the top on a nucleus and two legs on an electron (vertices 4N-5-5, 4N-5-6, and 4N-6-6):

U 6a = Z 2 1 α 2 8m 2 M 1 p e 1 r 2
1

p e + 3 (p e r 1 )(r 1 p e ) r 4 1 + Z 2 2 α 2 8m 2 M 2 p e 1 r 2
2

p e + 3 (p e r 2 )(r 2 p e ) r 4 2
,

U 6b = - Z 2 1 α 2 4m 2 M 1 1 r 4 1 [r 1 ×p e ]•σ e - Z 2 2 α 2 4m 2 M 2 1 r 4 2 [r 2 ×p e ]•σ e , U 6c = Z 2 1 α 2 4m 2 M 1 1 r 4 1 + Z 2 2 α 2 4m 2 M 2 1 r 4 2 . ( 26
)

VII. EFFECTIVE HAMILTONIAN

In this Section, we collect the results obtained in Secs. IV-VI to build the complete effective Hamiltonian at orders mα 6 and mα 6 (m/M ), including as well the second-order terms contributing to these orders. In doing so, we separate the different types of interactions: spin-independent, spin-orbit, spin-spin scalar and tensor interactions. Before that, we recall the expression of the effective Hamiltonian for leading-order relativistic corrections, i.e. the Breit-Pauli Hamiltonian, which comes into play in the second-order terms.

Spin-orbit interactions require a specific discussion. Formally, the leading electronic spin-orbit interaction H so (see Eq. ( 28) below) contains terms of order mα 4 (electronic spin-orbit) and mα 4 (m/M ) (electronic spin-nuclear orbit). However, assuming one considers a σ electronic state, the electronic spin-orbit coupling gives a zero contribution in the Born-Oppenheimer approach. The nonzero value of this term is due to nonadiabatic effects, so that it is actually smaller by a factor ∼ (m/M ) with respect to its nominal order, and thus of the same order as the electronic spinnuclear orbit coupling. The same thing occurs in the relativistic corrections, i.e. spin-orbit terms that are nominally of order mα 6 are of comparable magnitude to the "recoil" (mα 6 (m/M ) terms. We will thus make no distinction between nonrecoil or recoil contributions whenever the spin-orbit interaction is involved.

For the same reasons, the nuclear spin-orbit interaction H so N (last line in Eq. ( 28)), in which the first term is of nominal order mα 4 (m/M ), has an actual contribution of order mα 4 (m/M ) 2 . Relativistic corrections to this interaction (e.g. the effective potential U 2c , Eq. ( 11)), are of order mα 6 (m/M ) 2 . That is why we will not consider the nuclear spin-orbit interaction in the following.

A. Leading-order (mα 4 ) relativistic corrections

We include here all terms of the Breit-Pauli Hamiltonian at orders mα 4 and mα 4 (m/M ) (the electron's anomalous magnetic moment is not taken into account here).

H (4) = H B + H ret + H so + H ss + H so N , (27)
H B = - p 4 e 8m 3 + [∆ e V ] 8m 2 , H ret = Z 1 2 p i e m δ ij r 1 + r i 1 r j 1 r 3 1 P j 1 M 1 + Z 2 2 p i e m δ ij r 2 + r i 2 r j 2 r 3 2 P j 2 M 2 , H so = Z a (1+2a e ) 2m 2
H so N = 1 m [r a ×p e ] r 3 a µ a - 1 M a 1 - Zm p I a M a µ a [r a ×P a ] r 3 a µ a . (28)
B. Spin-independent interaction: leading term and recoil

The nonrecoil effective Hamiltonian is

H (6) no-spin = p 6 e 16m 5 + U 1a + U 4 = p 6 e 16m 5 - 3 64m 4 p 2 e , ∆V + 5 128m 4 p 4 e , V - 5 64m 4 p 2 e V p 2 e + 1 8m 3 E 2 . ( 29
)
The second-order contribution is

∆E 2 nd -order no-spin = H B Q(E 0 -H 0 ) -1 Q H B . (30)
The above expressions coincide with previous results [START_REF] Pachucki | Effective Hamiltonian approach to the bound state: Positronium hyperfine structure[END_REF][START_REF] Korobov | Relativistic corrections of order mα 6 to the two-centre problem[END_REF]. These corrections have been evaluated numerically for the hydrogen molecular ions in [START_REF] Korobov | Relativistic corrections of mα 6 order to the rovibrational spectrum of H + 2 and HD + molecular ions[END_REF][START_REF] Korobov | Fundamental Transitions and Ionization Energies of the Hydrogen Molecular Ions with Few ppt Uncertainty[END_REF]. It should be noted that both the first-order and second-order contributions contain divergences, which need to be cancelled out [START_REF] Korobov | Relativistic corrections of order mα 6 to the two-centre problem[END_REF].

The recoil effective Hamiltonian is ,

H (6) rec = U 2a + U 3a + U 6a + U 6c , U 2a = - Z 1 8m 2 p 2 e , p i e m δ ij r 1 + r i 1 r j 1 r 3 1 P j 1 M 1 - Z 2 8m 2 p 2 e , p i e m δ ij r 2 + r i 2 r j 2 r 3 2 P j 2 M 2 , U 3a = 1 4m Z 3 1 M 1 1 r 3 1 + Z 3 2 M 2 1 r 3 2 + Z 2 1 Z 2 M 1 (r 1 r 2 ) r 2 1 r 3 2 + Z 1 Z 2 2 M 2 (r 1 r 2 ) r 3 1 r 2 2 + Z 2 1 8m 2 M 1 1 r 4 1 + p e 1 r 2
U 6c = Z 2 1 4m 2 M 1 1 r 4 1 + Z 2 2 4m 2 M 2 1 r 4 2 . ( 31
)
The second-order contribution is

∆E 2 nd -order rec = ∆E ret + ∆E (0) so-so , ∆E ret = 2 H B Q(E 0 -H 0 ) -1 Q H ret ∆E (0) so-so = H so Q(E 0 -H 0 ) -1 QH so (0) . ( 32
)
where A (0) denotes the scalar part of an operator A (H so being a vector operator, the second-order term has contributions of rank 0, 1, and 2), see Appendix B, Eq. (B3) for details.

It should be noted that ∆E (0) so-so was considered together with nonrecoil terms in Refs. [START_REF] Korobov | Relativistic corrections of order mα 6 to the two-centre problem[END_REF][START_REF] Korobov | Relativistic corrections of mα 6 order to the rovibrational spectrum of H + 2 and HD + molecular ions[END_REF]. For the reasons explained above, we prefer to include it in the recoil part.

The effective Hamiltonian ( 31) is actually incomplete, because it does not include contributions from the contact terms of the NRQED Lagrangian [START_REF] Caswell | Effective Lagrangians for bound state problems in QED, QCD, and othe field theories[END_REF][START_REF] Kinoshita | Radiative corrections to the muonium hyperfine structure: The α 2 (Zα) correction[END_REF][START_REF] Patkóš | Higher-order recoil corrections for triplet states of the helium atom[END_REF]. A complete consideration of the recoil effective Hamiltonian for the hydrogen molecular ions, including contact terms and explicit cancellation of divergences, can be found in [START_REF] Zhong | Spin-averaged effective Hamiltonian of orders mα 6 and mα 6 (m/M ) for hydrogen molecular ions[END_REF]. Our results coincide with those of that reference: the potentials U 2a , U 3a , and U 6a respectively appear in the terms denoted δH 4 (Eq. ( 42)), δH 6 (Eq. ( 50)), and δH 5 (Eq. ( 45)). In the case of U 3a , this is best seen by comparing the first line of Eq. ( 17) with Eq. ( 50) of [START_REF] Zhong | Spin-averaged effective Hamiltonian of orders mα 6 and mα 6 (m/M ) for hydrogen molecular ions[END_REF], where a prefactor z a z e should be added.

C. Spin-spin scalar interaction

The effective Hamiltonian for this interaction is

H (0) ss(6) = U (0) 2d + U (0) 5b , U (0) 2d = 1 4m 2 p 2 e , 8π 3 δ 3 (r a ) µ e µ a U (0) 5b = - 1 3m Z 1 µ e µ 1 r 4 1 + Z 2 µ e µ 2 r 4 2 + Z 2 (r 1 r 2 )µ e µ 1 r 3 1 r 3 2 + Z 1 (r 1 r 2 )µ e µ 2 ) r 3 1 r 3 2 ( 33
)
and the second-order contribution is

∆E (0)2 nd -order ss = 2 H B Q(E 0 -H 0 ) -1 Q H (0) ss . ( 34
)
Again, the first-order and second-order terms contain divergences which have to be cancelled out. This was done in [START_REF] Korobov | Relativistic corrections of mα 6 (m/M ) order to the hyperfine structure of the H + 2 molecular ion[END_REF][START_REF] Korobov | Theoretical Hyperfine Structure of the Molecular Hydrogen Ion at the 1 ppm Level[END_REF] for hydrogen molecular ions, and the resulting corrections to the spin-spin contact interaction were evaluated numerically. Beyond the relativistic corrections considered here (of order (Zα) 2 E F , where E F is the Fermi splitting), there is a one-loop radiative contribution at the same order (α(Zα)E F ) [START_REF] Sapirstein | Quantum Electrodynamics[END_REF][START_REF] Eides | Theory of Light Hydrogenic Bound States[END_REF]. Other contributions to spin-spin scalar interactions include higher-order QED corrections (see [START_REF] Sapirstein | Quantum Electrodynamics[END_REF][START_REF] Eides | Theory of Light Hydrogenic Bound States[END_REF] and references therein) and effects involving the nuclear structure such as the Zemach [START_REF] Zemach | Proton structure and the hyperfine shift in hydrogen[END_REF] and recoil [START_REF] Arnowitt | The Hyperfine Structure of Hydrogen[END_REF][START_REF] Bodwin | Some corrections to the hydrogen hyperfine splitting[END_REF] corrections, and were taken into account in [START_REF] Korobov | Relativistic corrections of mα 6 (m/M ) order to the hyperfine structure of the H + 2 molecular ion[END_REF][START_REF] Korobov | Theoretical Hyperfine Structure of the Molecular Hydrogen Ion at the 1 ppm Level[END_REF].

D. Electron spin-orbit interaction

As explained above, for this interaction we make no distinction between nonrecoil and recoil contributions. The effective Hamiltonian is

H (6) so = U 1b + U 2b + U 5a + U 6b , U 1b = - 3Z 1 16m 4 p 2 e , 1 r 3 1 [r 1 ×p e ] s e - 3Z 2 16m 4 p 2 e , 1 r 3 2 [r 2 ×p e ] s e , U 2b = Z 1 4m 3 M 1 p 2 e , 1 r 3 1 r 1 ×P 1 s e + Z 2 4m 3 M 2 p 2 e , 1 r 3 2 r 2 ×P 2 s e , U 5a = Z 2 1 4m 2 M 1 1 r 4 1 [r 1 ×P 1 ] s e + Z 2 2 4m 2 M 2 1 r 4 2 [r 2 ×P 2 ] s e + Z 1 Z 2 4m 2 M 1 1 r 1 r 3 2 [r 2 ×P 1 ] s e + Z 1 Z 2 4m 2 M 2 1 r 3 1 r 2 [r 1 ×P 2 ] s e - Z 1 Z 2 4m 2 M 1 1 r 3 1 r 3 2 [r 1 ×r 2 ] (r 1 P 1 )s e + Z 1 Z 2 4m 2 M 2 1 r 3 1 r 3 2 [r 1 ×r 2 ] (r 2 P 2 )s e , U 6b = - Z 2 1 2m 2 M 1 1 r 4 1 [r 1 ×p e ] s e - Z 2 2 2m 2 M 2 1 r 4 2 [r 2 ×p e ] s e . (35)
The second-order contributions is ∆E 2 nd -order so = ∆E so + ∆E so-ret + ∆E

(1) so-so ,

∆E so = 2 H so Q(E 0 -H 0 ) -1 QH B , ∆E so-ret = 2 H so Q(E 0 -H 0 ) -1 QH ret , ∆E (1)
so-so = H so Q(E 0 -H 0 ) -1 QH so (1) . (36)
It is worth noting that both first and second-order terms are finite and do not require regularization. The spin-orbit interaction at this order was partially considered in [START_REF] Korobov | Spin-orbit corrections of order mα 6 to the fine structure of the (37,35) state in the 4 He p antiprotonic helium atom[END_REF] for the antiprotonic helium atom, but all terms were not included in that work. It should also be mentioned that the only other correction to the spin-orbit interaction at the mα 6 order corresponds to the effect of the electron's anomalous magnetic moment [START_REF] Eides | Theory of Light Hydrogenic Bound States[END_REF] and was therefore already included in [START_REF] Bakalov | High-Precision Calculation of the Hyperfine Structure of the HD + Ion[END_REF].

E. Spin-spin tensor interaction

The effective Hamiltonian for spin-spin tensor interaction is

H (2) ss(6) = U (2) 2d + U (2) 5b , U (2) 2d
= 1 4m 2 p 2 e , - r 2 a µ e µ a -3(µ e r a )(µ a r a ) r 5 a U (2) 5b = - 1 6m Z 1 r 2 1 µ e µ 1 -3(µ e r 1 )(µ 1 r 1 ) r 6 1 + Z 2 r 2 2 µ e µ 2 -3(µ e r 2 )(µ 2 r 2 ) r 6 2 - 1 6m Z 2 (r 1 r 2 )µ e µ 1 -3(µ e r 1 )(µ 1 r 2 ) r 3 1 r 3 2 + Z 1 (r 1 r 2 )µ e µ 2 -3(µ e r 2 )(µ 2 r 1 ) r 3 1 r 3 2 . ( 37
)
The second-order contribution is ∆E (2)2 nd -order ss = ∆E (2) ss + ∆E so-ss + ∆E

so-so N ,

∆E (2) ss = 2 H (2) ss Q(E 0 -H 0 ) -1 QH B , ∆E (2)
so-ss = 2 H (2) ss Q(E 0 -H 0 ) -1 QH so (2)
, ∆E

so-so N = 2 H so Q(E 0 -H 0 ) -1 QH so N (2)
Here also, all terms are finite and do not require regularization. The only other correction to the spin-spin tensor interaction at this order is the effect of the electron's anomalous magnetic moment.

VIII. NUMERICAL RESULTS AND CONCLUSION

The results of Secs. VII D and VII E can be used to calculate relativistic corrections to the electron spin-orbit and spin-spin tensor interaction coefficients of the hyperfine Hamiltonian in a one-electron molecular system. Here, we present corrections to the spin-orbit coefficient in both H + 2 [START_REF] Korobov | Hyperfine structure in the hydrogen molecular ion[END_REF] and HD + [START_REF] Bakalov | High-Precision Calculation of the Hyperfine Structure of the HD + Ion[END_REF] for a few transitions studied in recent or ongoing experiments [START_REF] Alighanbari | Precise test of quantum electrodynamics and determination of fundamental constants with HD + ions[END_REF][START_REF] Patra | Towards Doppler-free two-photon spectroscopy of trapped and cooled HD + ions[END_REF]6,[START_REF] Zhong | High-precision spectroscopy of hydrogen molecular ions[END_REF].

Our calculations rely on the "exponential" variational expansion [START_REF] Korobov | Coulomb three-body bound-state problem: Variational calculations of nonrelativistic energies[END_REF], where the wave function for a state of total orbital angular momentum L and parity Π = (-1) L is expanded in the following way:

Ψ π LM (R, r 1 ) = l1+l2=L Y l1l2 LM ( R, r1 )G Lπ l1l2 (R, r 1 , r 2 ), Y l1l2 LM ( R, r1 ) = R l1 r l2 1 Y l1 ( R) ⊗ Y l2 (r 1 ) LM , G Lπ l1l2 (R, r 1 , r 2 ) = N n=1 C n Re e -αnR-βnr1-γnr2 + D n Im e -αnR-βnr1-γnr2 . (39)
The complex exponents α n , β n , γ n are generated in a pseudorandom way. Matrix elements of all operators are calculated analytically; general methods of such calculations may be found e.g. in [START_REF] Drake | High Precision Calculations for Helium[END_REF][START_REF] Drake | Angular integrals and radial recurrence relations for two-electron matrix elements in Hylleraas coordinates[END_REF][START_REF] Harris | Singular and nonsingular three-body integrals for exponential wave functions[END_REF]. The matrix elements are reduced to finite sums of radial integrals of the general form

Γ l,m,n (α, β, γ) = ∞ 0 dR ∞ 0 dr 1 R+r1 |R-r1| dr 2 R l r m 1 r n 2 e -αR-βr1-γr2 , (40)
which are then calculated using recurrence relations.

From the numerical point of view, the calculation of first-order terms (U 1b , U 2b , and U 5a ) is straightforward; we used N = 2000 -3000 which was more than sufficient to get 4 significant digits.

Second-order contributions pose more difficult problems [START_REF] Korobov | Spin-orbit corrections of order mα 6 to the fine structure of the (37,35) state in the 4 He p antiprotonic helium atom[END_REF], especially the singular term ∆E so and the (less singular) term ∆E

(1) so-so . In the case of ∆E so , the intermediate wavefunction ψ (1) defined by

(E 0 -H 0 )ψ (1) = (H B -H B ) ψ 0 (41)
behaves like 1/r 1 (1/r 2 ) at small electron-nucleus distances. The regular trial functions [START_REF] Jentschura | Nonrelativistic QED approach to the Lamb shift[END_REF] would thus result in very slow convergence. In order to reduce this singularity, we use the transformation described in [START_REF] Korobov | Spin-orbit corrections of order mα 6 to the fine structure of the (37,35) state in the 4 He p antiprotonic helium atom[END_REF]:

H B = H B -(E 0 -H 0 )U -U (E 0 -H 0 ), (42)
where

U = c 1 r 1 + c 2 r 2 , c i = µ i (2µ i -m e ) 4m 3 e Z i . (43)
Here, 1/µ i = 1/m e + 1/M i . The second-order term may then be rewritten as follows:

H so Q(E 0 -H 0 ) -1 QH B = H so Q(E 0 -H 0 ) -1 QH B + U H so -U H so . (44)
With H B being replaced by H B in Eq. ( 41), the first-order wavefunction is now less singular and behaves like ln(r 1 ) (ln(r 2 )) at small distances. In the numerical evaluation, we use a "multilayer" basis set, where the first subsets (between 2 and 4) approximate the regular part of the intermediate wavefunction, and 8 others subsets contain growing exponents β n (γ n ) up to 10 4 in order to reproduce the ln(r 1 ) (ln(r 2 )) behavior at small distances, see an illustrative example in Table I. The total size of the intermediate basis set is typically around N ∼ 10000. The convergence will be analyzed in more detail in a future publication focusing on numerical results.

Subset [A1, A2] [A 1 , A 2 ] [B1, B2] [C1, C2] Ni 1 [5.

The ∆E

(1) so-so contribution is obtained from Eq. (B3) of the Appendix. The spin operator U 1 appearing in that equation is given in Sec. B 2. For the orbital operator T 1 , the calculation is separated into three terms a 0 , a -, a + corresponding to the possible values of the angular momentum L of intermediate states, L = L, L±1, see Eq. (B5) for definitions. The total contribution is given by Eq. (B6) (with k 1 = k 2 = k = 1). Since the first-order wavefunction [START_REF] Varshalovich | Quantum Theory of Angular Momentum[END_REF] with H so on the right-hand side is also singular at small electron-nucleus distances, we use a similar multilayer basis set as for the ∆E so contribution. An intermediate basis size up to N ∼ 20000 was used for the (L = 3, v = 9) state of HD + .

Our numerical results are given in Tables II (for H + 2 ) and III (for HD + ). From a study of convergence as a function of N , we estimate the numerical uncertainty of ∆E so and ∆E so-so to about 1 Hz. For the other contributions, all digits are significant. The total uncertainty of the spin-orbit interaction coefficient (denoted c e in H + 2 and E 1 in HD + ) is dominated by the yet unevaluated radiative correction of order mα 7 ln(α) [START_REF] Pachucki | Quantum electrodynamics effects on helium fine structure[END_REF][START_REF] Jentschura | Nonrelativistic QED approach to the Lamb shift[END_REF]. A tentative order of magnitude is α 3 ln(α)c e ∼ 100 Hz, but our preliminary calculations indicate that this correction is actually as large as 300-400 Hz. The calculation of this contribution is thus essential for further improvement of theoretical predictions of the hyperfine structure, and will be addressed in a forthcoming publication.

(L, v) c (BP ) e U 1b U 2b
U5a ∆Eso ∆Eso-ret ∆E

(1) so-so ∆c (in kHz). The leading-order (Breit-Pauli) value of ce (Ref. [START_REF] Korobov | Hyperfine structure in the hydrogen molecular ion[END_REF]) is given in column 2. Columns 3-8 are the first-order and second-order contributions listed in Eqs. [START_REF] Drake | High Precision Calculations for Helium[END_REF] and [START_REF] Drake | Angular integrals and radial recurrence relations for two-electron matrix elements in Hylleraas coordinates[END_REF], respectively. The total correction is given in column 9. The last column is our new value of ce.

(L, v) E (BP ) 1 U 1b U 2b
U5a ∆Eso ∆Eso-ret ∆E

(1) so-so ∆E II, for the spin-orbit coefficient E1 in HD + . The Breit-Pauli value in column 2 was obtained in Ref. [START_REF] Bakalov | High-Precision Calculation of the Hyperfine Structure of the HD + Ion[END_REF].

In conclusion, we have derived the complete effective Hamiltonian at the mα 6 and mα 6 (m/M ) for hydrogen molecular ions. The spin-independent and spin-spin scalar interaction terms were found to agree with previous calculations [START_REF] Korobov | Relativistic corrections of mα 6 (m/M ) order to the hyperfine structure of the H + 2 molecular ion[END_REF][START_REF] Korobov | Relativistic corrections of mα 6 order to the rovibrational spectrum of H + 2 and HD + molecular ions[END_REF][START_REF] Zhong | Spin-averaged effective Hamiltonian of orders mα 6 and mα 6 (m/M ) for hydrogen molecular ions[END_REF]. We then exploited this effective Hamiltonian to calculate corrections to the electronic spin-orbit hyperfine coefficient for a few states involved in experimentally studied transitions in H + 2 and HD + . The theoretical uncertainty has been reduced by more than a factor of 3, from about α 2 c e ∼ 1.5 kHz to about 300-400 Hz. Next steps are the calculation of radiative corrections at the next order, and of corrections to the spin-spin tensor interaction coefficients. It will then become possible to perform precise comparison with present and upcoming experimental data. Finally, the effective Hamiltonian we have derived may also be used to improve the hyperfine structure calculations in antiprotonic helium [START_REF] Korobov | Spin-orbit corrections of order mα 6 to the fine structure of the (37,35) state in the 4 He p antiprotonic helium atom[END_REF][START_REF] Hori | Buffer-gas cooling of antiprotonic helium to 1.5 to 1.7 K, and antiproton-to-electron mass ratio[END_REF].

Let us introduce the irreducible tensor operators

T k M = {O k1 a ⊗ Q(E 0 -H 0 ) -1 QO k2 b } kM , U k M = {S k1 a ⊗ S k2 b } kM .
Then, using the relationship (see Ref. [START_REF] Varshalovich | Quantum Theory of Angular Momentum[END_REF], Chapter 3, Sec. 3.3.2, Eq. ( 11))

{{A k1 ⊗ B k1 } 0 ⊗ {C k2 ⊗ D k2 } 0 } 00 = k Π k Π k1k2 {{A k1 ⊗ C k2 } k ⊗ {B k1 ⊗ D k2 } k } 00 ,
where Π n1n2... = (2n 1 + 1)(2n 2 + 1) . . ., one gets

(S k1 a • O k1 a )Q(E 0 -H 0 ) -1 Q(O k2 b • S k2 b ) = = (-1) k1+k2 Π k1k2 {S k1 a ⊗ O k1 a } 0 ⊗ {Q(E 0 -H 0 ) -1 QO k2 b • S k2 b } 0 00 = (-1) k1+k2 k Π k {T k ⊗ U k } 00 = k (-1) k1+k2+k (T k • U k ). (B2)
As a result, ∆E = k ∆E (k) ,

∆E (k) = (-1) k1+k2+k vLSJM (T k • U k ) vLS JM = (-1) k1+k2+k vL T k vL L L k L vLSJM (L k • U k ) vLS JM , (B3)
where L 0 = I, L 1 = L, L 2 = {L ⊗ L} 2µ , etc.

Orbital reduced matrix elements

For the operator T k acting on spatial degrees of freedom, one separates the calculation into different terms corresponding to the possible vales of the angular momentum L of intermediate states, L = L, L ± 1 (since min(k 1 , k 2 ) = 1 in all the cases under consideration here). The reduced matrix element of a given component L may then be expressed (see Ref. [START_REF] Varshalovich | Quantum Theory of Angular Momentum[END_REF], Chapter 13, Sec. 13.1.3, Eq. ( 10))

vL T k(L ) vL = (-1) k Π k k 1 k 2 k L L L n =0 vL O k1 a v n L v n L O k2 b vL E 0 -E n . ( B4
)
Let us define

a -= - 1 Π 2 L n =0 vL O k1 a v n L -1 v n L -1 O k2 b vL E 0 -E n , a 0 = 1 Π 2 L n =0 vL O k1 a v n L v n L O k2 b vL E 0 -E n , a + = - 1 Π 2 L n =0 vL O k1 a v n L + 1 v n L + 1 O k2 b vL E 0 -E n . (B5)
The prefactor of the spin operator L k • U k in Eq. (B3) is given by

(-1) k1+k2+k vL T k vL L L k L = (-1) k1+k2 Π 2 L Π k L L k L - k 1 k 2 k L L L -1 a -+ k 1 k 2 k L L L a 0 - k 1 k 2 k L L L + 1 a + .

(B6)

[1] J.-Ph. Karr, L. Hilico, J.C.J. Koelemeij, and V.I. Korobov, Hydrogen molecular ions for improved determination of fundamental constants. Phys. Rev. A 94, 050501(R) (2016).

  2µ e µ B s e = -(1 + a e )|e| m s e , µ a = µ a µ N

3 µ e µ a δ 3

 3 (r a ) -r 2a µ e µ a -3(µ e r a )(µ a r a ) r 5 a .

[r a ×p e ] r 3 a[r a ×P a ] r 3 a s e , H ss = µ e µ a r 3 a - 3 (µ e r a )(µ a r a ) r 5 a - 8πα 3 µ

 333353 s e -Z a (1+a e ) mM a e µ a δ(r a ),

2 .U 1 =U 2 =S ( 2 ) 5 3U 2 =

 212252 Irreducible spin operators a) With the electron spin-orbit Hamiltonian H so on both sides: U 0 = {s e ⊗ s e } 00 = -{s e ⊗ s e } 1µ = -the electron spin-orbit Hamilotnian H so and the nuclear spin-orbit Hamiltonian H so N :U 0 = {s e ⊗ I a } 00 = -1 √ 3 (s e • I a ), U 1 = {s e ⊗ I a } 1µ = i √ 2 [s e × I a], {s e ⊗ I a } 2µ = the electron spin-orbit Hamiltonian H so and the tensor spin-spin Hamiltonian H ss = {s e ⊗ I a } 2µ . We then get U 1 = {s e ⊗ S ss } 1µ = {s e ⊗ {s e ⊗ I a } 2 } 1µ = √ {{s e ⊗ s e } 0 ⊗ I a } 1µ + √ 15 6 {{s e ⊗ s e } 1 ⊗ I a } 1µ = -{s e ⊗ S ss } 2µ = -

TABLE I :

 I Variational parameters used in the calculation of singular second-order terms for the (L = 1, v = 0) state of HD + . [A1, A2] (resp. [A 1 , A 2 ] are the intervals in which the real (resp. imaginary) parts of exponents αn (see Eq. (39)) are generated, and [B1, B2] (resp. [C1, C2]) are the intervals for the real parts of βn (resp. γn). An indicative number of basis functions Ni in each subset is given the last column. The total basis size in this example is N = 11000.

		1, 5.5]	[0.8, 15.2]	[0.00, 1.80] [0.00, 1.65] 1830
	2	[5.1, 5.2]	[-0.6, 6.3]	[0.00, 1.56] [0.00, 1.59] 1170
	3	[5.1, 5.5] [-0.6, 15.2] [1.80, 10.0] [0.00, 1.65] 1290
	4	[5.1, 5.5] [-0.6, 15.2] [0.00, 1.80] [1.65, 10.0] 1290
	5	[5.1, 5.5] [-0.6, 15.2] [10.0, 10 2 ] [0.00, 1.65] 1070
	6	[5.1, 5.5] [-0.6, 15.2] [0.00, 1.80] [10.0, 10 2 ] 1070
	7	[5.1, 5.5] [-0.6, 15.2]	[10 2 , 10 3 ]	[0.00, 1.65]	900
	8	[5.1, 5.5] [-0.6, 15.2] [0.00, 1.80]	[10 2 , 10 3 ]	900
	9	[5.1, 5.5] [-0.6, 15.2]	[10 3 , 10 4 ]	[0.00, 1.65]	740
	10	[5.1, 5.5] [-0.6, 15.2] [0.00, 1.80]	[10 3 , 10 4 ]	740

TABLE II :

 II Relativistic corrections to the spin-orbit interaction coefficient ce for rovibrational states of H + 2

			e	ce(this work)
	(2,0) 42 162.530 1.542 -3.601 0.027 2.736	0.348	0.412 1.463	42 163.99
	(2,1) 39 571.598 1.451 -3.440 0.036 2.579	0.327	0.388 1.341	39 572.94

TABLE III :

 III Same as Table

			6) 1	E1(this work)
	(1,0) 31 984.645 1.170 -2.736 0.021 2.087	0.263	0.313 1.118	31 985.76
	(1,6) 22 643.474 0.834 -2.097 0.044 1.509	0.181	0.219 0.689	22 644.16
	(3,0) 31 627.353 1.156 -2.694 0.019 2.043	0.260	0.308 1.093	31 628.45
	(3,9) 18 270.577 0.680 -1.732 0.043 1.161	0.146	0.182 0.481	18 271.06

Acknowledgements

The authors thank L. Hilico for useful comments on the manuscript. V.I.K. acknowledges the support of the Russian Foundation for Basic Research under Grant No. 19-02-00058-a. Z.-X.Z. acknowledges support from the National Natural Science Foundation of China (Grants No. 91636216, No. 11974382, and No. 11474316), and from the Chinese Academy of Sciences (the Strategic Priority Research Program, Grant No. XDB21020200, and the YIPA program).

APPENDIX A: FOURIER INTEGRALS

In this Appendix, we summarize the three-dimensional integrals that were used in our derivations for the Fourier transformation from momentum to coordinate space. The master integral is 4π (2π) 3

and other useful integrals are 4π (2π) 3 dq q 2 δ ijq i q j q 2 e iqr = 1 2 A second-order contribution to the hyperfine splitting of a rovibrational state (v, L) may be written in the general form

where S a , S b , O a , O b are some irreducible orbital tensor operators, with S a , S b acting in the spin space and O a , O b in the orbital space. |vLSJM is a pure hyperfine state, with S the total spin, and J = L + S the total angular momentum.

The goal of this Appendix is to show how such quantities can be decomposed into irreducible tensor components, which are expressed as the scalar product of an irreducible orbital tensor operator with an irreducible spin operator of the same rank. Then, we will give the expressions of the spin operators and of the orbital reduced matrix elements.


Marcin Konopka

Piotr Cecot

Sébastien Ulrich
email: sebastien.ulrich@enscm.fr

Artur R Stefankiewicz

Tuning the Solubility of Self-Assembled Fluorescent Aromatic Cages Using Functionalized Amino Acid Building Blocks

Keywords: self-assembly1, molecular cages2, disulfides3, fluorescence4, dynamic covalent chemistry5

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Introduction

Due to the unique physicochemical properties and a wide range of applications, cage-type architectures enjoy unflagging interest. [START_REF] Hasell | Porous organic cages: soluble, modular and molecular pores[END_REF][START_REF] Markiewicz | Selective C70 encapsulation by a robust octameric nanospheroid held together by 48 cooperative hydrogen bonds[END_REF][START_REF] Beuerle | Covalent Organic Frameworks and Cage Compounds: Design and Applications of Polymeric and Discrete Organic Scaffolds[END_REF][START_REF] Mastalerz | Porous Shape-Persistent Organic Cage Compounds of Different Size, Geometry, and Function[END_REF] Of the many types of cage systems, those based on reversible covalent bonds This is a provisional file, not the final typeset article begin to clearly dominate, mainly due to their adaptive properties. [START_REF] Lehn | Perspectives in Chemistry-Aspects of Adaptive Chemistry and Materials[END_REF][START_REF] Briggs | A Perspective on the Synthesis, Purification, and Characterization of Porous Organic Cages[END_REF][START_REF] Ono | Dynamic Behavior of Covalent Organic Cages[END_REF] So far, molecular cages based on dynamic imine bonds (and/or acylhydrazones) and boronic esters have been the most frequently used among chemists. [START_REF] Beuerle | Covalent Organic Frameworks and Cage Compounds: Design and Applications of Polymeric and Discrete Organic Scaffolds[END_REF]Acharyya and Mukherjee, 2019;[START_REF] Kołodziejski | Dynamic polyimine macrobicyclic cryptandsself-sorting with component selection[END_REF] The dynamic output of imine chemistry is particularly opulent. [START_REF] Belowich | Dynamic imine chemistry[END_REF][START_REF] Schick | Transformation of Imine Cages into Hydrocarbon Cages[END_REF] Supramolecular architectures based on imine and/or acyl-hydrazone bonds have already found number of applications as biomolecular recognition receptors, [START_REF] Nial | DNA Intercalators in Cancer Therapy: Organic and Inorganic Drugs and Their Spectroscopic Tools of Analysis[END_REF]Ulrich, 2019) nanocapsules, [START_REF] Durot | Multiporphyrinic cages: architectures and functions[END_REF][START_REF] Eichstaedt | Self-Assembly and Ordering of Peptide-Based Cavitands in Water and DMSO: The Power of Hydrophobic Effects Combined with Neutral Hydrogen Bonds[END_REF][START_REF] Jędrzejewska | Peptide-based capsules with chirality-controlled functionalized interiorsrational design and amplification from dynamic combinatorial libraries[END_REF] sensors, [START_REF] Stefankiewicz | Highly Sensitive Magnetic Effects Induced by Hydrogen-Bonding Interactions in a High-Spin Metallosupramolecular Fe4II [2×2] Grid-Type Complex[END_REF][START_REF] Bravin | Supramolecular cages as differential sensors for dicarboxylate anions: guest length sensing using principal component analysis of ESI-MS and 1H-NMR raw data[END_REF] and self-healing materials. [START_REF] Roy | DYNAMERS: dynamic polymers as self-healing materials[END_REF][START_REF] Chao | Dynamic Covalent Polymer Networks Based on Degenerative Imine Bond Exchange: Tuning the Malleability and Self-Healing Properties by Solvent[END_REF]Drożdż et al., 2018a;Drożdż et al., 2018b) The use of disulfide bonds, reversible through inter alia thiolate substitution, in the generation of complex supramolecular architectures has received much attention, [START_REF] Otto | Dynamic Molecular Networks: From Synthetic Receptors to Self-Replicators[END_REF][START_REF] Black | Disulfide exchange: exposing supramolecular reactivity through dynamic covalent chemistry[END_REF][START_REF] Sobczak | Dynamic Covalent Chemistry under High-Pressure:A New Route to Disulfide Metathesis[END_REF] except those of cage topology that are still isolated cases in the literature. [START_REF] Sarma | Disulfides, Imines, and Metal Coordination within a Single System: Interplay between Three Dynamic Equilibria[END_REF][START_REF] Stefankiewicz | Template-directed synthesis of multi-component organic cages in water[END_REF]Stefankiewicz and Sanders, 2013;[START_REF] Naini | The exploration of Kemp's triacid (KTA) as the core for the synthesis of 3-fold symmetric 23-cyclophane, 22-cyclophane and novel linker directed designs[END_REF] The nature of dynamic systems based on imine and disulfide bonds is thus well known separately but multi-dynamic systems that employs them simultaneously are an enduring challenge. [START_REF] Sarma | Disulfides, Imines, and Metal Coordination within a Single System: Interplay between Three Dynamic Equilibria[END_REF][START_REF] Orrillo | Molecular Networks in Dynamic Multilevel Systems[END_REF][START_REF] Reuther | Tunable Orthogonal Reversible Covalent (TORC) Bonds: Dynamic Chemical Control over Molecular Assembly[END_REF] Recently, we have reported a method for obtaining a new type of molecular cage-like system using a set of two orthogonal reversible reactions (disulfides and acyl-hydrazones) that occur simultaneously. [START_REF] Drozdz | Generation of Multicomponent Molecular Cages using Simultaneous Dynamic Covalent Reactions[END_REF] This was the first report of multi-dynamic and multi-component cages selectively obtained in a one-pot process, and characterized in semi-aqueous media, by applying two distinct reversible covalent bonds. The reported cages were based on two simple building blocks: tetraphenylethylene-tetraaldehyde (TPE-Ald) being known for its fluorescence properties such as aggregation-induced emission (AIE), (Zhao et al., 2012;[START_REF] Mei | Aggregation-Induced Emission: Together We Shine, United We Soar![END_REF][START_REF] Feng | Macrocycles and cages based on tetraphenylethylene with aggregation-induced emission effect[END_REF] and cysteine hydrazide, a small, chiral molecule containing three different functional groups: amine, hydrazide and thiol. A multiplicity of factors was determined as contributing to the exceptional selectivity in the formation of cage-system, of which the presence of dimethyl sulfoxide (DMSO) in the reaction mixture was found essential for the effective cage generation. This is related to the participation of this solvent in the oxidation of thiols into disulfides which is promoted even under slightly-acidic conditions, as previously reported. (Tam et al., 1991;Atcher and Alfonso, 2013) Both thiol oxidation and disulfide exchange usually proceed under mild-basic conditions, whereas acyl-hydrazones require the presence of acid catalysts.

Therefore, the simultaneous formation of these two bonds, despite an obvious progress in this area, remains a challenging task and the use of DMSO as co-solvent helps in this regard. The presence of reversible covalent bonds in the cage structure is crucial as it allows generating thermodynamically stable products by self-correction of intermediate kinetic products, while maintaining dynamic features that are important for the responsiveness of those structures. The latter are extremely important when taking into account the potential use of such systems in the selective complexation of guest molecules or in the 'self-healing' processes. The first generation of our doubly-dynamic cages, despite their interesting structural features, required the use of well-defined solvent mixture (H2O-DMSO with predominant content of the latter from 75% to 94%). This somewhat limited the scope of investigation of e.g. fluorescence properties only to this solvent system and the potential applications one can think of.

Therefore, to expand the scope of our cages, we decided to modify the structure of one of the building blocks in order to generate two doubly-dynamic and multicomponent cage systems presenting similar structural properties but distinct hydrophobic/hydrophilic character. We chose to take advantage of the reactive α-amine present on amino acids and thus chose to functionalized the cysteine hydrazide building block at its N-terminal. It was of course necessary to ensure firstly that the preferential formation of cage structure would be retained, the principal intent simply being to modify the solubility of the cages, then to characterize the spectroscopic properties of the cages in a wider range of solvents. We describe here our syntheses of cages derived from modified cysteine hydrazide units and the properties of these new architectures in various media.

2

Results and discussion

Design of building blocks

Our investigations began with the design and selection of several structurally distinct molecular components for the efficient construction of doubly dynamic tetrapodal cage systems. As shown in Figure . 1, such cages can be formed in a self-assembly process between two tetratopic components constituting upper and lower planes of the cage and four ditopic molecules as bridging linkers. Both types of building blocks must have functional groups that allow facile formation of reversible covalent bonds such as acyl-hydrazones and disulfides. The TPE aldehyde (TPE-ALD) was retained as the aromatic core of the cage-like structures, because we wanted our new cages to have an analogous structure to those reported previously. [START_REF] Drozdz | Generation of Multicomponent Molecular Cages using Simultaneous Dynamic Covalent Reactions[END_REF] We decided to use the amino group in cysteine hydrazide to insert a structural extension via amide bond formation. Coupling reactions based on different amino acid systems are well described and run under mild conditions. We decided to use two organic acids to modify the cysteine hydrazide. We chose a more polar and hydrophilic component, 2-(2-methoxyethoxy)acetic acid (DEG) containing a short di-glycol fragment, in order to achieve enhanced aqueous solubility, and a non-polar group, 2-ethylhexanoic acid (EH), to increase hydrophobicity and generate cages soluble in organic solvents. It should be mentioned, that we used the racemic ethyl hexanoic acid to make the synthesis more cost effective and facile. We assumed here that one of the key features for this project is to maintain the R-configuration on the cysteine hydrazide moieties, and that the using of racemic EH acid will not affect the self-assembly process of cage formation.

Synthesis of building blocks

The syntheses (Figure 2) began with the preparation of a partially-protected cysteine hydrazide.

Commercially available Fmoc-STr-L-Cys-OH (1), was coupled with tert-butyl carbazate before the Fmoc group was removed in a standard reaction with piperimidine in DMF. N-hydroxysuccinimide activated esters (4 and 5) of diethyl-glycol acid and ethyl-hexyl acid were obtained by reaction with DCC and N-Hydroxysuccinimide (NHS) and H-L-Cys-STr-Hyd-Boc (3) was then combined with the activated esters in amide coupling reactions. In this way, two fully protected derivatives of cysteine hydrazide tethered with DEG and EH moieties were obtained (6 and 7). In the final step, the hydrazide and thiol groups were deprotected in both DEG and EH derivatives in a solution of TFA/TIS 9/1. Synthetic protocols and characteristics of all obtained compounds are available in the experimental section.

Self-assembly of cages

Cage formation using the new building blocks DEG-L-Cys-Hyd ( 9) and EH-L-Cys-Hyd (10) was first assessed in the original solvent system (DMSO/H2O 94/6). In both cases, LC-MS monitoring showed that complete conversion was reached after 3 days of equilibration at 50 °C, with no TPE-Ald left and the appearance of a new single peak in the mass spectra that corresponded to the expected cage This is a provisional file, not the final typeset article compounds, respectively DEG-CAGE and EH-CAGE. Cage formation using orthogonal dynamic covalent reactions, namely acyl-hydrazone and disulfide formation, was therefore confirmed with the new N-functionalized cysteine-derived building blocks (Figure 3). Then we varied the nature of the solvent. First, DEG-CAGE formation was studied in a solvent system with an increasing proportion of water, from DMSO/H2O 94/6 to 100% H2O. LC-MS analysis clearly demonstrated cage formation from DMSO/H2O 94/6 to DMSO/H2O 25/75 (Table 1 andESI). In samples with less than 25% DMSO, LC-MS indicated the presence of intermediates, which corresponded to acyl-hydrazone condensation products with no sign of higher structure generated by disulfide bond formation. Similarly, EH-CAGE formation was studied in organic solvents (from 100% DMSO to 100% CHCl3) and LC-MS showed effective cage formation up to 90% CHCl3 (Table 1 andESI). Here again, a minimum of 10% DMSO seems to be essential for complete cage formation, otherwise acyl-hydrazone intermediates are formed (ESI) and do not react further. At this stage, we explain the necessity of the presence of the indicated percentages of DMSO in the reaction mixture both in terms of solubility and the facilitated oxidation of thiols to disulfides. Both cages were also characterized by 1 H NMR. To obtain the sample the original post-self-assembly mixture was evaporated to dryness and then the residue was re-dissolved in the deuterated solvent. The recorded 1 H NMR spectra confirmed the complex structure of the tetrapodal cages that exist as a mixture of isomers (Figure 3c). Due to the high complexity of these molecules, we employed semiempirical molecular modeling (Recife Model 1) to get further insights into the structural features of the cages (Figure 3d). Based on this we estimated the sizes and volumes of each cage. Optimization of DEG-Cage has shown the retained internal cage cavity in comparison to the unmodified cage. The lengths and shape of side chains allow them to fit fully into the side grooves of the capsule, which causes the enhancement of molecule size and volume (approx. 4200 Å 3 of spherical volume). The EH-Cage side chain after optimization has no visible cavity inside the molecule.

The preferred configuration of ethyl-hexanoic chains forces the TPE backbones of the EH-cage molecule to twist slightly around the central axis, while aromatic adjust and stack on each other, which constitutes the flat shape and makes the EH-cage smaller than DEG-Cage (approx. 3200 Å 3 of spherical volume).

Fluorescence properties

The fluorescence spectra of both cage compounds, DEG-CAGE and EH-CAGE, were recorded in different solvent systems at the same concentration (0.5 mM). The spectra remained essentially unaltered in all solvents, showing an emission maximum around 510-520 nm. Both cages in all solvent mixtures employed showed significant emission enhancement which can be seen by comparing with the fluorescence spectrum of TPE-ALD (Figure 4). The observed emission enhancement is caused by rigidification and suppression of phenyl rings torsion of the two TPE units within a single cage molecule. (Shultz and Fox, 1989) Increasing the water content decreases about 5-fold the fluorescence emission intensity of DEG-CAGE, while, on the other hand, increasing the chloroform content increases about 2-fold the fluorescence emission intensity of EH-CAGE (Figure 4).

Conclusions

We have reported herein the design and synthesis of new building blocks, based on N-functionalized cysteine-derived amino acids, for the one-pot multi-component self-assembly of fluorescent aromatic cages. Our results establish a convenient synthetic strategy, and show that placing such side-groups does not hinder cage formationhighlighting the robustness of the approachand enables tuning the solubility of the corresponding cage compounds. These cages preserved interesting aggregationinduced emission properties in a now much wider range of solvent mixtures (from 90% CHCl3/DMSO to 75% H2O/DMSO), which we believe can be harnessed for sensing applications.

4 Materials and Methods.

Solvents and chemicals were purchased from commercial suppliers and used without further purification. Preparative purifications were performed by silica gel flash column chromatography (Merck® 40-60 µM). HPLC analyses were performed on a Waters HPLC 2695 (EC Nucleosil 300-5 C18, 125 × 3 mm) column, Macherey -Nagel) equipped with a Waters 996 DAD detector. The following linear gradients of solvent A (99.9% water and 0.1% TFA) into solvent B (99.9% acetonitrile and 0.1% TFA) were used: 0 to 100% of solvent B in 10 min; flow 1 mL/min. Retention times are given in minutes. LC/MS analyses were performed on a Shimadzu LCMS2020 (Phenomex Kinetex C18, 2.6 m × 7.5 cm, 100 Å) equipped with a SPD-M20A detector with the following linear gradient of solvent A (99.9% water and 0.1% HCOOH) into solvent B (99.9% acetonitrile, 0.1% HCOOH) and: 0 to 100% of solvent B in 10 min; flow 1 mL/min. Retention times are given in minutes. Fluorescence analyses were carried out on an AF-2500 HITACHI fluorescence spectrophotometer. UV-Vis absorption experiment was measured on UV-3100pC UVisco spectrophotometer. Samples of cages compounds were studied at 0.25 mM concentration. Excitation wavelength was set at 320 nm and emission spectra were recorded in the range 400 -600 nm. 1 H NMR, 13 C NMR spectra were recorded at 400 MHz for 1 H and 101 MHz for 13 C (Bruker Avance 400) in deuterated solvents. Chemical shifts are reported in ppm relative to the solvent residual peak. HR-MS analyses were carried out at the Laboratoire de Mesures Physiques, IBMM, Université de Montpellier using Micromass Q-Tof instruments. The TPE aldehyde (TPE-Ald) and Fmoc-L-Cys-STr-Hyd-Boc (2) were obtained according to the previously reported method. [START_REF] Drozdz | Generation of Multicomponent Molecular Cages using Simultaneous Dynamic Covalent Reactions[END_REF] 4.1 Synthesis protocols of essential chemicals and building blocks.

Synthesis of H-L-Cys-STr-Hyd-Boc (3)

Fmoc-L-Cys-STr-Hyd-Boc (2.3 g, 3.31 mmol) was dissolved in a solution of DMF/piperidine (8:2, v/v; 70 mM) and then was stirred for 1 h in r.t. After that the reaction mixture was concentrated in vacuo.

The crude residue was purified by flash chromatography on silica gel eluting with a gradient of DCM 100% 5 min., then 100% DCM to 90% DCM and 10% MeOH 30 min. Yield: 85%. 1 H NMR (400 MHz, CDCl3) δ 7.48 -7.40 (m, 6H), 7.32 -7.26 (m, 6H), 7.24 -7.19 (m, 3H), 3.01 (dd, J = 8.7, 3.9 Hz, 1H), 2.75 (dd, J = 13.0, 4.0 Hz, 1H), 2.57 (dd, J = 13.0, 8.8 Hz, 1H), 1.43 (s, 9H). 13 C NMR (101 MHz, CDCl3) δ 172. 00, 155.09, 144.60, 129.69, 128.13, 126.98, 81.84, 67.22, 53.48, 31.56, 28.24. ESI-MS: m/z calc. for [M+H] + 478.2164, found 478.2161.

Synthesis of DEG-AE (4)

The title compound was obtained using the previously reported method for analogous compound. [START_REF] Meißler | Enzyme-Triggered Antifouling Coatings: Switching Bioconjugate Adsorption via Proteolytically Cleavable Interfering Domains[END_REF]) NHS (0.8 g, 7.5 mmol) and 2-(2-methoxyethoxy)acetic acid (1.0 g, 7.5 mmol) were dissolved in 150 mL of dry THF, and then was cooled in ice bath to (0°C). DCC (1.55 g, 7.5 mmol) was added portion wise. This was stirred for one hour at r.t. and stored in a refrigerator overnight. The DCU precipitate was filtered off and the filtrate was concentrated under vacuum to give a colorless oil. The crude product was dissolved in a small amount of THF, the resulting suspension was filtered to remove the precipitate and the procedure was repeated until a clear solution was obtained. Solvent was then removed under high vacuum to give a colorless oil. Yield 89%. This is a provisional file, not the final typeset article

Synthesis of EH-AE (5)

The 2-ethylhexanoic acid (160 uL, 1.0 mmol) and N-hydroxycuccinimide (115 mg) were dissolved in DCM (5 mL). After 10 min. of stirring the EDC•HCl (192 mg, 1.0 mmol) was added and then the stirring was continued for 2 h at r.t. Title compound was purified by flash chromatography (gradient elution from 0% to 10% of MeOH in DCM). 1 H NMR (400 MHz, CDCl3) δ 2.83 (d, J = 5.2 Hz, 4H), 2.58 (tt, J = 8.8, 5.4 Hz, 1H), 1.83 -1.63 (m, 4H), 1.43 -1.30 (m, 4H), 1.03 (t, J = 7.5 Hz, 3H), 0.91 (t, J = 7.1 Hz, 3H). 13 C NMR (101 MHz,CDCl3) δ 171.61,169.32,44.92,31.72,29.27,25.76,25.64,22.62,13.98,11.67. ESI-MS: m/z calc. for [M+H] + 242.1373, found 242.1352.

Synthesis of DEG-L-Cys-STr-Hyd-Boc (6)

A solution of DEG-AE (231 mg, 1.0 mmol), H-L-Cys-STr-Hyd-Boc (525 mg, 1.1 mmol) and Et3N (0.28 mL, 2.0 mmol) in DMF (30 mL) was stirred for 24 hours at room temperature. The solvent was then evaporated under vacuum. Resulted oily residue was dissolved in 1 mL of acetone and precipitated in 1 M HCl (100 mL). The white, waxy precipitate was filtered off and dried. The crude product was purified by flash chromatography on silica gel eluting with a gradient of MeOH in DCM (from 0% to 10% MeOH). The titled product was obtained as a white solid. Yield 66%. 77, 169.38, 154.83, 144.44, 129.72, 128.17, 127.00, 71.60, 71.19, 70.24, 67.38, 59.06, 50.54, 32.76, 28.23. ESI-MS: m/z calc. for [2M+Na] + 1209.5012, found 1209.5124.

Synthesis of EH-L-Cys-STr-Hyd-Boc (7)

A solution of EH-AE (241 mg, 1.0 mmol), H-L-Cys-STr-Hyd-Boc (525 mg, 1.1 mmol) and Et3N (0.28 mL, 2.0 mmol) in DMF (30 mL) was stirred for 24 hours at room temperature. The solvent was then evaporated under vacuum. Resulted oily residue was dissolved in 1 mL of acetone and precipitated in 1 M HCl (100 mL). The white, waxy precipitate was filtered off and dried. The crude product was purified by flash chromatography on silica gel eluting with a gradient of MeOH in DCM (from 0% to 10% MeOH). The titled product was obtained as a white solid. Yield 56%. 1 H NMR (400 MHz, CDCl3) δ 7.44 -7.40 (m, 6H), 7.26 (s, 6H), 7.23 -7.18 (m, 3H), 4.18 (q, J = 7.1 Hz, 1H), 2.69 (dd, J = 13.2, 5.5 Hz, 1H), 2.59 (ddd, J = 13.1, 8.3, 2.9 Hz, 1H), 1.94 -1.81 (m, 1H), 1.61 -1.43 (m, 4H), 1.41 (s, 9H), 1.31 -1.18 (m, 4H), 0.87 -0.77 (m, 6H). 13 C NMR (101 MHz, CDCl3) δ 176. 83, 169.91, 154.77, 144.49, 129.64, 128.15, 126.96, 81.60, 67.24, 53.55, 49.23, 32.35, 31.05, 29.85, 28.22, 25.91, 22.87, 14.07, 12.12. ESI-MS: m/z calc. for [2M+Na] + 1229.6092, found 1229.5532.

Synthesis of L-Cys-Hyd (8)

H-L-Cys-(STr)-Hyd-Boc (500 mg, 1.05 mmol) was dissolved in TFA/TIS (95/5) solution (5 mL) and stirred for 4 hours at room temperature. After removal of 90% of the solvent, Et2O was added to the residue. The precipitate was triturated with Et2O and filtered. The crude material was then lyophilized twice to afford the product L-Cys-Hyd as a white solid. Yield 65%. 1 H NMR (400 MHz, CD3OD) δ 4.14 (m, 1H), 3.03 (qd, J = 14.6, 5.9 Hz, 2H). DEG-L-Cys-STr-Hyd-Boc (594 mg, 1.00 mmol) was dissolved in TFA/TIS (95/5 v.v.) solution (10 mL) and stirred for 4 hours at room temperature. After removal of 90% of the solvent, petroleum ether was added to the residue. A white precipitate formed, which was then centrifuged and washed several times with petroleum ether. The crude material was then lyophilized twice to afford the product DEG-L-Cys-Hyd as a white solid. Yield 78%. 1 H NMR (400 MHz, CD3OD) δ 4. 65 -4.53 (m, 1H), 4.06 (d, J = 2.9 Hz, 2H), 3. 2H),2H),3.41 (s,3H),2.93 (dq,J = 15.4,7.4,7.0 Hz,2H). 13 C NMR (101 MHz CD3OD) δ 173. 02, 170.77, 72.78, 71.92, 71.08, 59.20, 55.09, 49.00, 26.49. ESI-MS: m/z calc. for [M+H] + 252.1012, found 252.1522.

Synthesis of EH-L-Cys-Hyd (10)

EH-L-Cys-STr-Hyd-Boc (604 mg, 1.00 mmol) was dissolved in TFA/TIS (95/5 v.v.) solution (10 mL) and stirred for 6 hours at room temperature. After removal of 90% of the solvent, diethyl ether was added to the oily residue. A white precipitate formed, which was then centrifuged and washed several times with diethyl ether. The crude material was then lyophilized twice to afford the product EH-L-Cys-Hyd as a white solid. Yield 84%. 1 H NMR (400 MHz, CD3OD) δ 4.47 (t, J = 6.9 Hz, 1H), 2.89 (dd, J = 14.9, 6.7 Hz, 1H), 2.80 (dd, J = 13.1, 7.0 Hz, 1H), 2.22 (tt, J = 9.6, 5.0 Hz, 1H), 1.64 -1.51 (m, 2H), 1.51 -1.38 (m, 2H), 1.37 -1.21 (m, 4H), 0.89 (dt, J = 12.6, 7.0, 6.6 Hz, 6H). 13 C NMR (101 MHz, CD3OD) δ 179. 12, 171.09, 55.76, 39.95, 33.47, 30.74, 27.08, 23.74, 23.71, 14.32, 12.32 11 Tables Table 1: Screening of cage formation in different solvent systems using L-Cys-Hyd (8), DEG-L-Cys-Hyd (9), and EH-L-Cys-Hyd (10).

Entry

  13 C NMR (101 MHz, CD3OD): δ = 163.32, 55.76, 25.64. ESI-MS: m/z calc. for [M+H] + 136.0545, found 136.0541.

[

  two-column width] Figure 2. Synthesis of modified cysteine hydrazides. Reagents and conditions: I) tert-butyl carbazate, EDC•HCl, HOBt, Et3N, DCM, 0°C to rt; II) DMF/piperidine (8:2 v.v.), rt; III) DCC, NHS, THF or DCM, r.t; IV) Et3N, DMF, r.t.; V) TIS/TFA (90:10 v.v.) See experimental section for details.

Figure 3 .

 3 Figure 3. Characterization of described cages; a) Comparison of HPLC chromatograms (310 nm) of TPE-ALD, DEG-CAGE and EH-CAGE showing the formation of a single product by self-assembly, b) ESI-MS mass spectra of modified cages, calculated m/z for DEG-CAGE: [M+4H] 4+ 687.2265, found 687.70, calculated m/z for [M+2H] 2+ 1373.4458, found 1373.85; calculated m/z for EH-CAGE [M+4H] 4+ 707.3408, found 707.80, calculated m/z for [M+2H] 2+ 1413.6743, found 1413.95, c) 1 H NMR spectra of DEG-Cage (top) and EH-Cage (bottom). Significant peaks were marked according to the color pattern on the partial structures of the cages, d) Optimized models of cages, calculated approx. sizes for DEG-Cage r = 10 Å, sphV = 4200 Å 3 (left) and for EH-Cage r = 9.15 Å, sphV = 3200 Å 3 (right).

Figure 4 .

 4 Figure 4. Comparison of fluorescence plots of new cages with reference to the unmodified cage (L-Cys-Hyd, red plot) and TPE-Ald (violet plot) (0.25 mM solutions, excitation wavelength 320 nm, range 400 -600 nm): a) plots of DEG-CAGE witch various ratios of DMSO to the H2O; b) plots of EH-CAGE witch various ratios of DMSO to the CHCl3.

  the hydrazide used in reaction with TPE-Ald.

  . ESI-MS: m/z calc. for [M+H] + 262.1573, found 262.1608. . performed all organic synthesis, most experiments, analysis and co-wrote the paper. P.C. performed molecular modeling and optimization of the tetrapodal cages. S.U. performed some analysis, interpreted the results and co-wrote the paper. A.R.S. interpreted the results and co-wrote the paper.Stefankiewicz, A.R., and Sanders, J.K. (2013). Diverse topologies in dynamic combinatorial libraries from tri-and mono-thiols in water: sensitivity to weak supramolecular interactions. Chem. Commun. 49, 5820-5822.Tam, J.P., Wu, C.R., Liu, W., and Zhang, J.W.(1991). Disulfide bond formation in peptides by dimethyl sulfoxide. Scope and applications. J. Am. Chem. Soc. 113, 6657-6662. General scheme of the molecular cages formation process. Three functional groups involved in the self-assembly process were marked with coloured atom labels (aldehyde: red; hydrazide: blue; thiol: orange). Structural modifications that change the physical properties of the cage have been shown in background colour (light-blue for diethyl glycol moiety and light-yellow for ethyl-hexyl moiety). The colour-gradient under the structure of each cage shows the contribution of individual fragments in the solubility property of the entire system.

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		Chem. 22, 23726-23740.
	10 List of figures captions
	Figure 1.
	5	Conflict of Interest
	There are no conflicts to declare.
	6	Authors Contributions
	M.K7	Funding
	ARS thank the National Science Centre (grant SONATA BIS 2018/30/E/ST5/00032) for financial
	support. SU thanks the ANR (ANR-17-CE07-0042-01) for funding. MK and PC thank grant
	INNCHEM no. POWR.03.02.00-00-I023/17 co-financed by the European Union for financial support.
	8	Supplementary Material
	Electronic Supplementary Information (ESI) available: NMR, MS and HPLC chromatograms.
	9	Notes and references

Acharyya, K., and Mukherjee, P.S. (2019). Organic Imine Cages: Molecular Marriage and Applications. Angew. Chem. Int. Ed. 0. Atcher, J., and Alfonso, I. (2013). The effect of DMSO in the aqueous thiol-disulphide dynamic covalent chemistry of model pseudopeptides. RSC Adv. 3, 25605.

This is a provisional file, not the final typeset article