Ludovic Pricoupenko

Implicit ladder summation in the Hartree-Fock-Bogoliubov approach

Keywords: numbers: 05, 30, Jp,04, 20, Cv

The fully variational Hartree Fock Bogoliubov approach for bosons is studied in the limit of zero range forces in two-and three-dimensions. The equation of state obtained in two-dimensions is expressed in a parametric form. It is shown that the Λ potential permits to perform an implicit summation of the ladder diagrams without leaving the variational scheme, restoring thus the consistency of this approximation.

I. INTRODUCTION

Modeling interatomic forces by a point-like isotropic potential appeared fruitful in the context of ultracold atoms as a way to study the dilute and low temperature regime for bosons or two-spin component fermions. In this respect, the Bethe Peierls model where pairwise interactions between particles are replaced by contact conditions is especially relevant to finding non trivial properties in the two-and few-body problems, like confinement induced resonances [1][2][3] or also in universal collisional properties as in the four-fermion problem [4]. Nevertheless the Bethe Peierls formalism is not adapted to usual formalisms of the many-body problem where the interaction is described by a standard two-body vertex [START_REF] Abrikosov | Methods of Quantum Field Theory in Statistical Physics[END_REF][START_REF] Blaizot | Quantum Theory of Finite Systems[END_REF][START_REF] Fetter | Quantum Theory of Many-Particle Systems[END_REF]. The Fermi-Huang pseudo-potential which encapsulates the Bethe Peierls contact condition is a way to implement directly the zero-range limit in a perturbative approach of the many-body problem and was successfully used in the description of the dilute Bose gas [START_REF] Huang | [END_REF]. However when applied to the fully variational Hartree Fock Bogoliubov (HFB) approach, the Fermi-Huang pseudopotential describes a molecular Bose gas instead of the usual atomic phase [9]. This problem is solved by using the Λ potential which in exact treatments, provides a description strictly equivalent to the Bethe Peierls model, while it permits to improve approximate formalisms [10]. This potential is a function of the free parameter Λ which offers the opportunity to suppress the gap in the HFB description of the homogeneous Bose gas, making thus the formalism compatible with the Hugenholtz Pines theorem [11]. In Ref. [12], the formalism was applied to the two-dimensional (2D) Bose gas. It provides an equation of state (EOS) which coincides with the result of Popov in the dilute limit [13,14] while giving at finite areal density an approximation which compares nicely with exact Monte-Carlo computations and perturbative results [15][16][17]. Moreover the formalism solves the problem of the unexpected stabilization of a vortex in a non rotating trap, a phenomenon which occurs when HFB equations are truncated by neglecting the effect of the anomalous average of the noncondensate field operator [19,20]. However, the derivation of the HFB equations using the Λ potential is not straightforward because the ansatz made for the density matrix doesn't belong to the correct Hilbert space (it doesn't support all the nontrivial singularities linked to the Bethe Peierls contact condition).

The present paper gives a simple derivation of the fully variational HFB equations for the three-dimensional (3D) and the 2D Bose gas by using the Λ -ǫ potential introduced in Ref. [21]. This potential enables to relax the strict zero range limit for the variational calculation while keeping the structure of the original Λ potential and the exact zero range limit is achieved afterward. This way, HFB equations are obtained without any approximation following closely the standard and general variational scheme described for example in Ref. [START_REF] Blaizot | Quantum Theory of Finite Systems[END_REF]. Emphasis is put on the 2D case where a parametric form of the non trivial EOS is derived. Finally, the ladder approximation of the Bethe Salpeter equation is solved in the low energy limit. This calculation shows that the particular choice made for the parameter Λ in the HFB equations permits to implicitly sums the ladder diagram in the diagonal self-energy at a collisional energy consistent with the evaluation of the off-diagonal self energy.

II. INTERACTION POTENTIAL OF VANISHING RANGE

A. Contact condition

This paper deals with dilute gases of neutral bosons moving in a D-dimensional space (only the cases D = 2 or D = 3 are considered) where interactions between particles can be described by a short range pairwise potential of radius denoted by b D . At sufficiently low density (denoted n) and temperature, i.e. nb 3 D ≪ 1 and λ DB ≫ b D where λ DB is the de Broglie wavelength, the transition matrix in two-body collisions of particles in the gas can be approximated by its lowest order in the binary colli-sional energy (denoted E):

T D (E + i0 + ) = Ω D 2 2µ ×        a 3 (1 + ia 3 k 0 ) ; (D = 3) -1 ln(-ia 2 k 0 e γ /2) ; (D = 2), (1)
where k 0 is the relative momentum (E =2 k 2 0 2µ ), µ is the reduced mass, a D is the D-dimensional scattering length and Ω D is the full D-dimensional space angle (Ω 3 = 4π and Ω 2 = 2π). In Eq. ( 1) in the two-dimensional case, γ is the Euler's constant, and the 2D scattering length a 2 is by definition a positive parameter. The zero range potential approach is a way to exactly recover these expressions of the transition matrix. It can be formulated for a system composed of N -particles as follows. The many-body wave-function in the configuration space is singular at the contact (r i = r j ) of each pair of interacting particles labeled by (ij) while out of these singularities it solves the Schrödinger's equation without pairwise interacting potential. For each interacting pair (ij) as the relative coordinates (denoted r ij ) tends to zero the singular behavior is parameterized by the scattering length with:

Ψ(r 1 , . . . r N ) = A i⇌j Ψ ×            1 a 3 - 1 r ij + O(r ij ); (D = 3) ln r ij a 2 + O(r ij ); (D = 2).
(2) In Eq. ( 2) the limit is taken for fixed values of r k (k = i, j) and of the center of mass of the pair (ij) and the strength's singularity denoted by A i⇌j Ψ is a function of these coordinates. In this formulation often called Bethe Peierls model in the literature, the pairwise potential is thus replaced by asymptotic conditions on the wavefunction. While its simplicity is appealing, the Bethe Peierls model is not adapted to the standard techniques used in many-body theories.

B. Pseudo-potential

Another way to implement the zero-range potential approach is to use the Λ-potential introduced in Ref. [10]. In this formalism, the true finite range pairwise potential is replaced by a pseudo-potential which is in fact a way to perform implicitly a subtractive regularizing scheme achieved at the negative energy

E Λ = - r 1 , r 2 |V Λ |ψ = T D (E Λ )R Λ [ψ] δ(r 12 ), (3)
where T D (E Λ ) is evaluated at negative energy in Eq. ( 1) by using the usual analytical continuation k 0 = iΛ and R Λ [ . ] is a regularizing operator:

R Λ [ • ] = lim r12→0    (∂ r12 + Λ) [r 12 • ] ; (D = 3) 1 -ln e γ 2 Λr 12 r 12 ∂ r12 [ • ] (D = 2).
(4) In Eq. ( 4), the subtraction cannot be performed at Λ = 0 in 2D while in 3D the Λ potential coincides with the Fermi-Huang pseudo-potential for Λ = 0.

In Ref. [21], it has been shown that in two-and fewbody problems it can be fruitful to relax the strictly zero range limit by replacing the δ-distribution in Eq. ( 3) with a family of functions δ ǫ of finite support of range ǫ while keeping the subtractive structure of the Λ potential and to perform the zero range limit ǫ → 0 in a second step. It appears convenient to choose a family of Gaussian functions such that lim ǫ→0 r|δ ǫ = δ(r) which have the interesting property in the momentum representation to have an expression independent of the dimension D:

k|δ ǫ = χ ǫ (k) = exp - k 2 ǫ 2 4 . (5)
Using this family of functions, the Λ -ǫ potential can be written in the momentum representation as:

k 1 , k 2 |V Λ ǫ |k 3 , k 4 [ • ] = (2π) D δ(K 12 -K 34 ) T D (E Λ )χ ǫ (k 12 ) lim ǫ→0 r Λ ǫ [χ ǫ (k 34 ) • ] . (6)
In Eq. ( 6) k i is the momentum of the particle i, K ij (respectively k ij ) is the total (respectively the relative) momentum of the pair (ij) and r Λ ǫ is a regularizing operator:

r Λ ǫ [ • ] =          ∂ ǫ + π 2 Λ ǫ • ; (D = 3) 1 + ǫ 2 ln(e γ Λ 2 ǫ 2 /2)∂ ǫ • ; (D = 2). (7)
One may recognize in Eq. ( 6) a form similar to a separable potential. This structure is preserved in the configuration space:

r 1 , r 2 |V Λ ǫ |r 3 , r 4 [ • ] = δ(R 12 -R 34 ) T D (E Λ ) r 12 |δ ǫ lim ǫ→0 r Λ ǫ [ δ ǫ |r 34 • ] , (8)
where R ij (respectively r ij ) denotes the center of mass (respectively the relative) coordinates of the pair (ij) and

r|δ ǫ = (2πǫ 2 ) -D/2 exp -r 2 /ǫ 2 .
It is worth pointing out that the way the zero range limit is performed in the Λ -ǫ potential in Eqs. [START_REF] Blaizot | Quantum Theory of Finite Systems[END_REF][START_REF] Huang | [END_REF] totally differs from the usual method of the renormalization of the coupling constant (which tends to zero in the limit of a large ultraviolet momentum cutoff) often used in the many-body problem, even if the resulting equations coincide when the zero-range limit is well defined. This issue is considered in the appendix A.

III. HFB FORMALISM FOR BOSONS

A. Variational approach in the zero range limit

The asymptotic conditions in Eq. ( 2) define the domain of the Hilbert space in the zero-range limit. In principle, in a variational approach one must preserve this structure by using an ansatz in the correct Hilbert space. However in the many-body problem, this constraint appears too stringent for being adopted. In this respect, the introduction of the Λ -ǫ potential greatly simplifies the implementation of the variational schemes: one performs the variation of the functional at finite ǫ avoiding thus all the difficulties linked to the singularities of Eq. (2) and the zero range limit where ǫ = 0 is only achieved when one uses the equations resulting from the variational principle. In the present section, this techniques is applied to the Hartree Fock Bogoliubov approach.

B. Variation of the functional

Using the expressions of the Λ -ǫ potential in Eqs. [START_REF] Blaizot | Quantum Theory of Finite Systems[END_REF][START_REF] Huang | [END_REF] the HFB equations are obtained at finite ǫ along the same lines as in Ref. [START_REF] Blaizot | Quantum Theory of Finite Systems[END_REF]. This section gathers the major steps of this derivation.

The gas is composed of N identical bosons of mass m. The field operator denoted by ψ verifies the usual bosonic commutation rules: [ ψ(r), ψ † (r ′ )] = (2π) D δ(rr ′ ) and [ ψ(r), ψ(r ′ )] = 0. The standard HFB approximation is a symmetry breaking approach where the atomic field ψ is split into a classical field Φ and a quantum fluctuation

φ = ψ -Φ with, φ = 0. ( 9
)
The density operator is chosen as a Gaussian ansatz in terms of the field φ:

Dǫ = exp(-Kǫ /k B T ) Z ǫ , (10)
where Z ǫ is the partition function and the variational Hamiltonian Kǫ is quadratic in terms of { φ, φ † }:

Kǫ = d D r 1 d D r 2 h ǫ (r 1 , r 2 ) φ † (r 1 ) φ(r 2 ) + 1 2 ∆ ǫ (r 1 , r 2 ) φ † (r 1 ) φ † (r 2 ) + 1 2 ∆ * ǫ (r 1 , r 2 ) φ(r 1 ) φ(r 2 ) . ( 11
)
This ansatz is an implicit function of the classical field Φ. The choice of a Gaussian density operator permits to use Wick's theorem. Consequently all the averages of products of the operator φ(r) can be expressed in terms of the diagonal one-particle density matrix

ρǫ (r 1 , r 2 ) = φ † (r 2 ) φ(r 1 ) ǫ (12)
and of the off-diagonal one-particle density matrix

κǫ (r 1 , r 2 ) = φ(r 2 ) φ(r 1 ) ǫ . (13)
In Eqs. (10)(11)(12)(13) and in what follows, quantities or averages where an index ǫ is added means that they are considered for a finite value of ǫ while the index ǫ is dropped when the zero range limit (ǫ → 0) is achieved. For example, this is the case for the depletion denoted by ñ(R) = ρ(R, R) and for the total particle density

n(R) = |Φ(R)| 2 + ñ(R). ( 14
)
In the subsequent equations the pair wavefunction is used:

κ ǫ (r 1 , r 2 ) = ψ(r 2 ) ψ(r 1 ) ǫ (15)
and also the Λ-regularized anomalous averages

κΛ (R 12 ) = lim ǫ→0 r Λ ǫ d D r 12 r 12 |δ ǫ κǫ (r 1 , r 2 ) (16) κ Λ (R 12 ) = Φ(R 12 ) 2 + κΛ (R 12 ). ( 17
)
In Eq. ( 17) κΛ can be also evaluated from Eq. ( 4)

by κΛ = R Λ [κ].
The variational principle applied to the Grand-potential E ǫ -T S ǫ -μN ǫ where E ǫ is the total energy, N ǫ is the number of particles and S ǫ = -k B ln( Dǫ ) ǫ is the entropy gives [START_REF] Blaizot | Quantum Theory of Finite Systems[END_REF]:

h ǫ (r 1 , r 2 ) = - 2 2m (∆ r δ) (r 12 ) -μδ(r 12 ) + V ext (R 12 ) + 2 d D r 3 d D r 4 r 1 , r 3 |V Λ ǫ |r 2 , r 4 ρ ǫ (r 4 , r 3 ) (18)
and

∆ ǫ (r 1 , r 2 ) = d D r 3 d D r 4 r 1 , r 2 |V Λ ǫ |r 3 , r 4 κ ǫ (r 3 , r 4 ). ( 19
)
After a straightforward integration one obtains:

h ǫ (r 1 , r 2 ) = - 2 2m (∆δ) (r 12 ) + [V ext (R 12 ) -μ] δ(r 12 ) + Σ 11 (R 12 ) r 12 |δ ǫ (20)
and

∆ ǫ (r 1 , r 2 ) = Σ 12 (R 12 ) r 12 |δ ǫ , (21)
where the diagonal and off-diagonal self-energies are defined by:

Σ 11 (R) = 2T D (E Λ )n(R), Σ 12 (R) = T D (E Λ )κ Λ (R). ( 22
)
The equation verified by the condensate field is also obtained easily at finite ǫ by minimization of the Grand potential. In presence of an external potential V ext (r)

acting on each particle of the gas, in the zero range limit it reads:

(H 1p r -μ)Φ(r) + T D (E Λ ) 2ñ(r) + |Φ(r)| 2 Φ(r) + T D (E Λ )κ Λ (r)Φ * (r) = 0. ( 23
)
In Eq. ( 23), the one-particle Hamiltonian H 1p r is defined by:

H 1p r = - 2 2m ∆ r + V ext (r). ( 24
)
The variational ansatz Kǫ is diagonalized by using a special Bogoliubov transformation on the particle field:

φ(r) = n bn u n (r) + b † n v * n (r) , (25)
where the bosonic operator bn (resp. b † n ) destroys (resp. creates) a quasi-particle of quantum number n. The modal amplitudes {u n , v n } in Eq. ( 25) are the eigenvectors of the linear system:

H 1p -μ + Σ 11 u n + Σ 12 v n = ω n u n H 1p -μ + Σ 11 v * n + Σ * 12 u * n = -ω n v * n . (26)
In Eq. ( 26) the coordinates (r) have been omitted for conciseness. In this basis the variational ansatz can be written as:

Kǫ = n ω n b † n b n . (27)
From the commutation relations of the operators ( bn , b † n , φ, φ † ), one can deduce the closure relations

n [u n (r 1 )u * n (r 2 ) -v * n (r 1 )v n (r 2 )] = δ(r 12 ) (28) n [v n (r 1 )u * n (r 2 ) -u * n (r 1 )v n (r 2 )] = 0. ( 29
)
The two fields ρ and κ can be expressed in terms of the mode amplitudes at thermal equilibrium :

ρ(r 1 , r 2 ) = n v * n (r 1 )v n (r 2 ) (1 + f n ) + u n (r 1 )u * n (r 2 )f n (30) κ(r 1 , r 2 ) = n 1 2 + f n v * n (r 1 )u n (r 2 ) + v * n (r 2 )u n (r 1 ) , (31)
where f n = b † n b n is the Bose occupation factor:

f n = 1 exp( ωn kB T ) -1 . ( 32
)
In the zero range limit, the one-particle density ρ(r 1 , r 2 ) is regular at r 12 = 0 and is equal to the depletion, while the pair wavefunction κ(r 1 , r 2 ) verifies the Bethe-Peierls contact condition in Eq. ( 2) as the relative coordinates r 12 tend to zero. The simplest way to show this property is to derive the equation verified by κ(r 1 , r 2 ). Using Eqs. (26,28-31) one obtains:

H 1p r1 + H 1p r2 κ(r 1 , r 2 ) + r 1 , r 2 |V Λ |κ = 2μκ(r 1 , r 2 ) -[ Σ 11 (r 1 ) + Σ 11 (r 2 )] κ(r 1 , r 2 ) -Σ 12 (r 1 )ρ(r 2 , r 1 ) -Σ 12 (r 2 )ρ(r 1 , r 2 ). ( 33
)
In the right hand side of Eq. ( 33) there is no δdistribution while the Λ potential acts on the pair wavefunction in the left hand side of this equation. Hence, the pair wavefunction verifies the contact condition Eq. ( 2) and using Eq. ( 4) one founds the strength's singularity in terms of the anomalous average:

A κ (R) = 2µ 2 Ω D T D (E Λ )κ Λ (R) ∀Λ > 0. ( 34
)
The off-diagonal self-energy evaluated from Eq. ( 22) is then proportional to A κ :

Σ 12 (R) = 2 Ω D 2µ A κ (R) (35)
and is thus not an explicit function of the parameter Λ.

C. A judicious choice for Λ(R)

In an exact approach, the description of the system does not depend on a particular choice made for the free parameter Λ [which can be also a function of R 12 in Eq. ( 8)]. However, the expression of the diagonal self-energy in Eq. (22) shows that HFB equations are Λdependent. Interestingly, this makes possible to solve the so-called gap problem in HFB (or Gaussian variational approach) which is not compatible with the Hugenholtz-Pines theorem [11] and was first shown to occur in Ref. [18]. For this purpose, the natural choice for the free-parameter Λ is given by imposing that the anomalous mode (u 0 , v 0 ) = (Φ, -Φ) is a zero energy solution of Eq. (26). In an inhomogeneous systems, this leads to a value of Λ which depends on the position. This specific function Λ(R) is denoted by Λ ⋆ (R) and is obtained by solving the equation:

κΛ ⋆ (R) = 0 ∀ R. ( 36
)
Using the Λ freedom of the off-diagonal self energy, one finds

T D (E Λ ⋆ )(R) = T D (E Λ )(R) 1 + κΛ (R) Φ 2 (R) . ( 37
)
In the 3D case Eq. (37) gives [10]:

aΛ ⋆ (R) = κΛ=0 (R) κΛ=0 (R) + Φ 2 (R) , (38)
while in the 2D case there doesn't exist such a general expression.

D. Homogeneous gas

In what follows equations are used for the homogeneous gas with the choice Λ = Λ ⋆ , assuming thus that Eq. ( 36) is satisfied. The thermodynamic limit is also supposed to be achieved. The bosonic field operator is expanded on the plane wave basis:

ψ(r) = c 0 + d D k (2π) D c k e ik•r , (39)
where Φ = c 0 and c k annihilates a plane wave of momentum k. The operators c k , c † k ′ verify the usual commutation rules:

[c k , c † k ′ ] = (2π) D δ(k -k ′ ) and [c k , c k ′ ] = 0.
The special Bogoliubov transformation takes the form:

φ(r) = d 3 k (2π) 3 b k u k e ik•r + b † k v * k e -ik•r , ( 40
)
where the operators (b k , b † k ) respectively create and destroy a quasi-particle of momentum k and verify also the bosonic commutation rules

[b k , b † k ′ ] = (2π) D δ(k -k ′ ), [b k , b k ′ ] = 0

and the modal amplitudes verify

|u k | 2 -|v k | 2 = 1. ( 41
)
Equations (23,36) give the expression of the chemical potential for the homogeneous gas in terms of the condensate density and of the depletion:

μ = T D (E Λ ⋆ )(Φ 2 + 2ñ). (42)
Hence using Eqs. (22,42) one obtains for Λ = Λ ⋆ the relation μ = Σ 11 -Σ 12 , which is the Hugenholtz Pines theorem [11]. The solutions of Eq. ( 26) are chosen real:

u k = 1 √ 2    2 k 2 2m + T D (E Λ ⋆ )Φ 2 ω k + 1    1 2 v k = - 1 √ 2    2 k 2 2m + T D (E Λ ⋆ )Φ 2 ω k -1    1 2
(43)

ω k = 2 k 2 2m 1 2 2 k 2 2m + 2T D (E Λ ⋆ )Φ 2 1 2
.

At zero temperature the depletion is given by

ñ = d D k (2π) D |v k | 2 (44)
and in the zero-range limit, the off-diagonal one-particle density matrix is:

κ(r 1 , r 2 ) = d D k (2π) D -Σ 12 2 ω k exp(ik • r 12 ). ( 45
)
HFB solutions are found for a given value of the particle density:

n = Φ 2 + d D k (2π) D v 2 k . ( 46
)
1. 3D Bose gas

The 3D homogeneous case has been studied in Ref. [10]. At zero temperature, Eqs. (38,43,46) together with the following expression of the regular part of the pairing field [obtained for example by using Eq. (A1)]:

κΛ=0 = Σ 12 d 3 k (2π) 3 m 2 k 2 - 1 2 ω k (47) lead to (n -Φ 2 )a 3 = 8 3 √ π a 3 (3n -2Φ 2 ) 3 2 . ( 48
)
Hence, for each value of the atomic density, the condensate fraction is found by solving equation ( 48). In the dilute regime (na

3 ≪ 1) one has n ≃ Φ 2 , T 3 (E Λ ⋆ ) ≃ T 3 (0)
and as expected the variational approach coincides with the usual perturbative Bogoliubov theory [10]. The transition matrix T 3 (E Λ ⋆ ) is an increasing function of the density and diverges at the critical value n c = π 192a 3 where Λ ⋆ = 1/a and Φ = 0. In this regime the mean field approach breaks-down, aΛ ⋆ is close to unity and the probability density for a quasi-particle to have a wavenumber k [i.e. v 2 k k 2 /(2π 2 ñ)] is peaked near k = 1/a: collisions between non condensed atoms are not well described in this formalism. Clearly, inelastic processes and correlations not included in the HFB approximation must be taken into account to study this regime.

2D Bose gas

In the 2D homogeneous case, the standard HFB formalism for bosons can be used only at zero temperature where a true condensate exists. This situation was studied in Ref. [12] where it was shown that the present HFB equations permit to obtain an EOS which appears recently to be very good when compared to exact perturbative results beyond the Bogoliubov approximation. In what follows, this EOS is derived in a parametric form. For this purpose, one first calculates the asymptotic behavior of the off-diagonal one-particle density matrix as r 12 tends to zero:

κ(r 1 , r 2 ) = mΣ 12 2π ln e γ 2 mT 2 (E Λ ⋆ )|Φ| × r 12 + O(r 12 )
(49) which provides the Λ-regularized part:

κΛ = mΣ 12 4π ln mT 2 (E Λ ⋆ )|Φ| 2 2 Λ 2 . ( 50
)
The consistent choice for Λ ⋆ obtained from Eq. ( 36) is thus

Λ ⋆ = -2π|Φ| 2 ln (e γ Λ ⋆ a 2 /2) . ( 51
)
The depletion is given by ñ

= mT 2 (E Λ ⋆ )|Φ| 2 4π 2 = Λ ⋆ 2 4π . (52)
Finally using Eq. ( 42), the constraint Λ = Λ ⋆ imposed by the Hugenholtz Pines theorem provides the EOS

n = t 2 1 -2 ln e γ t 2 4πa 2 2 ; μ = 2 t 2 1 - 1 ln( e γ t 2 ) ma 2 2 , (53)
defined through the dimensionless parameter t = a 2 Λ ⋆ (0 < t < 2e -γ ). The energy per particle is obtained from

E/N = n 0 μdn E N = 2 t 2 ma 2 2 × 5 4 -ln e γ t 2 1 -2 ln e γ t 2 . ( 54
)
The low areal density regime corresponds to the limit of vanishing values of the dimensionless parameter t → 0 + and in this regime one recovers the result of Popov [13]:

n = mμ 4π 2 ln 4 2 e (2γ+1) mμa 2 2 for na 2 2 ≪ 1, (55)
which is also the prediction of the Bogoliubov theory [14].

For finite areal densities, Eq. ( 53) gives an estimate compatible with Monte-Carlo simulations [15,16], the perturbative expansion one order beyond the Bogoliubov theory [17] and another approximate method [22].

An implicit ladder summation

In terms of Feynman diagrams, the diagonal selfenergy Σ 11 in Eq. ( 22) is treated at the one-loop level [START_REF] Fetter | Quantum Theory of Many-Particle Systems[END_REF]. That is why it explicitly depends on Λ [unlike the off-diagonal self-energy as shown in Eq. ( 35)]. This section shows that the choice Λ = Λ ⋆ permits to implicitly perform the summation of the ladder diagrams enabling thus to evaluate the T -matrix appearing in Σ 11 at the correct energy.

In the standard many-body technique, the collision between two quasi-particles is described by the effective potential Γ, which takes into accounts the effect of the medium [START_REF] Fetter | Quantum Theory of Many-Particle Systems[END_REF]:

k 1 , k 2 |Γ|k 3 , k 4 , (56)
where k = (ω, k) denotes a D + 1 dimensional vector.

In subsequent equations, the bare vertex for the Λ -ǫ potential is denoted by:

k 1 , k 2 |Γ b ǫ |k 3 , k 4 = k 1 , k 2 |V Λ ǫ |k 3 , k 4 . ( 57
)
The ladder summation is obtained through the ladder approximation of the Bethe Salpeter equation [START_REF] Fetter | Quantum Theory of Many-Particle Systems[END_REF]:

k 1 , k 2 |Γ ǫ |k 3 , k 4 = k 1 , k 2 |Γ b ǫ |k 3 , k 4 + i d D+1 q (2π) D+1 × k 1 , k 2 |Γ b ǫ |k 1 -q, k 2 + q G(k 1 -q) × G(k 2 + q) k 1 -q, k 2 + q|Γ ǫ |k 3 , k 4 , (58)
where q = (q 0 , q). In what follows, this equation is solved at zero temperature by using the one-body Green's function consistent with the HFB approximation:

G(k) = u 2 k ω -ω k + i0 + - v 2 k ω + ω k -i0 + . ( 59
)
The structure of Eq. ( 58) together with the definition of Γ b ǫ in Eqs. [START_REF] Blaizot | Quantum Theory of Finite Systems[END_REF]57), give the form of the dressed vertex:

k 1 , k 2 |Γ ǫ |k 3 , k 4 [ • ] = (2π) D δ(K 12 -K 34 ) g eff χ ǫ (k 12 ) lim ǫ→0 r Λ ǫ [χ ǫ (k 34 ) • ] (60)
and one wishes to compute the strength g eff . Integration over q 0 gives:

i dq 0 2π G(k 1 -q)G(k 2 + q) = u 2 k1-q u 2 k2+q -Ω + ω k1-q + ω k2+q + v 2 k1-q v 2 k2+q Ω + ω k1-q + ω k2+q , (61)
where Ω = ω 1 + ω 2 . In the limit of vanishing energy: k 1 → 0 , k 2 → 0 and Ω → 0 one has:

g eff = T D (E Λ ) 1 -T D (E Λ )J , (62)
where

J = -lim ǫ→0 r Λ ǫ d D q (2π) D χ ǫ (q) 2 u 4 q + v 4 q 2 ω q . ( 63
)
One then may use the identities:

u 4 q + v 4 q = 1 + 2u 2 q v 2 q and u q v q = - Σ 12 2 ω q . ( 64
)
The bit associated with u 2 q v 2 q in Eq. ( 63) gives an unphysical infrared divergence due to the fact that the effective potential is evaluated here in the strict zero energy limit of the scattering processes. In what follows this contribution on the value of J is denoted by J div . A natural infrared cut-off for J div consistent with the ladder approximation in the medium is given by the inverse inter-particle distance q min ∼ n 1/D . Owing to the inequality q min ≫ 1/ξ where ξ = /(2mΣ 12 ) is the healing length in the Bose gas one finds that T D (E Λ )J div ≡ O(q -6 min ξ -6 ) can be neglected in Eq. (62). The approximation u 4 q + v 4 q ∼ 1 can thus be performed, and one may recognize in Eq. (63) the expression of the Λ-regularized part of the off-diagonal one-particle density matrix at zero temperature considered for finite values of ǫ:

κǫ (r 1 , r 2 ) = d D q (2π) D χ ǫ (q) -Σ 12 2 ω q exp(iq • r 12 ). ( 65
)
One thus obtains J = κΛ /( Σ 12 ), and using Eqs. (37,62) the strength appearing in the effective potential is given by:

g eff = T D (E Λ ) Φ 2 + κΛ Φ 2 = T D (E Λ ⋆ ). (66)
Assuming that the spectrum is gapless, this result is valid for all Λ. For Λ = Λ ⋆ then Γ ǫ = Γ b ǫ in Eq. ( 58), meaning that when used at the one-loop order the bare vertex Γ b ǫ incorporates the ladder summation for Λ = Λ ⋆ , which is the desired result.

IV. CONCLUSIONS

In this paper the variational HFB equations in the zero range potential approach have been obtained following the standard derivation of Ref. [START_REF] Blaizot | Quantum Theory of Finite Systems[END_REF]. This way, results of Refs. [10,12] where the Λ potential was used are recovered in a simple manner. The EOS for the 2D Bose gas obtained in the HFB approximation is expressed in a parametric form. It is explicitly shown that the specific choice made for the parameter Λ permits to perform the ladder summation while keeping the variational scheme intact. These results have been obtained by using a representation of the Λ potential where the strict zero range limit is relaxed in intermediary calculations. The natural continuation of this work is to use the Λ -ǫ potential vertex of Eq. ( 57) in standard techniques of many-body theory borrowed from quantum field theory, including the treatment of the fermionic gas. The Λ freedom is a gift of the zero-range potential approach which permits to improve approximate treatments of the many-body problem. The best choice for the parameter Λ in the general time dependent HFB formalism or also in the variational Balian Vénéroni approach [START_REF] Blaizot | Quantum Theory of Finite Systems[END_REF] remains an open question. Another interesting issue is the possible use of a fully variational approach for the 2D Bose gas at finite temperature where the low energy classical modes of the field operator could be treated consistently in the variational scheme.

Appendix A: Contact condition obtained from the renormalization of the coupling constant

In the literature, numerous effective interactions are used for implementing the zero-range potential limit. As an example, the contact interaction on a discrete lattice where the coupling constant is adjusted to reproduce the low energy scattering properties is well adapted for numerical studies and leads to equations very similar to the one obtained using the contact conditions of Eq. ( 2) [23]. This appendix focuses on a standard techniques of the renormalization of the vertex interaction in the 3D continuous space where the strength of the force vanishes in the zero range limit. It is shown below that in the zero-range limit of the effective interaction, one recovers in the two-body problem the contact condition of the Bethe Peierls model obtained from the Λ -ǫ potential in the momentum representation.

Bethe Peierls contact condition in the momentum representation

In Ref. [21], the operator r Λ ǫ was introduced without loss of generality through the identity:

lim ǫ→0 r Λ ǫ [ δ ǫ |ψ ǫ ] = lim ǫ→0 δ ǫ |ψ ǫ - 2 Ω D 2µ A ψ φ Λ ǫ , (A1)
where |ψ ǫ is a two-body state considered in the center of mass frame, A ψ is its strength's singularity in the limit where ǫ tends to zero, and |φ Λ ǫ is a reference state equal to the action of the two-body Green's function at the energy E Λ = -2 Λ 2 2µ on the ket |δ ǫ :

k|φ Λ ǫ = - 2µ 2 × χ ǫ (k) k 2 + Λ 2 . (A2)
Equation (A1) illustrates the fact that the Λ potential is a regularization techniques based on a subtractive scheme.

Using the operator r Λ ǫ , the Bethe Peierls contact condition in Eq. ( 2) can be written without referring to any representation as:

lim ǫ→0 r Λ ǫ [ δ ǫ |ψ ǫ ] = Ω D 2 2µ A ψ T D (E Λ ) . ( A3
)
Inserting the closure relation in the left hand side of Eq. (A3), the contact condition can be expressed in the momentum representation as:

lim ǫ→0 d D k (2π) D χ ǫ (k) k|Ψ ǫ + Ω D A ψ χ ǫ (k) k 2 + Λ 2 = -A ψ f D (iΛ) (A4) where Λ > 0, f D (iΛ) = -2µT D (E Λ )/(Ω D
2 ) is the scattering amplitude in the Bethe Peierls model evaluated at negative energy and

A Ψ = -1 Ω D lim k→∞ k 2 lim ǫ→0 k|Ψ ǫ . (A5)
For example, in the Λ -ǫ potential model a scattering state of incoming momentum k 0 reads:

k|ψ ǫ = (2π) 3 δ(k -k 0 ) + Ω D A ψ χ ǫ (k) k 2 0 + i0 + -k 2 (A6)
where A ψ = -f D (k 0 ).

Λ 22µ where Λ is an arbitrary positive parameter[21]. It can be introduced by its action on a two-body wavefunction ψ as:

Acknowledgments

Yvan Castin, Maxim Olshanii and Peter Schuck are acknowledged for discussions. Laboratoire de Physique Théorique de la Matière Condensée is UMR 7600 of CNRS and its Cold Atoms group is associated with IFRAF.

Link with the renormalization of the coupling constant

In what follows, Eq. (A4) is derived by using the model potential:

where v Q is the bare coupling constant and g Q (r) is a family of functions converging toward a δ-distribution in the limit of arbitrarily high values of the ultraviolet momentum cutoff Q (Q → ∞). For a two-body scattering state with an incoming plane wave of momentum |k 0 , the transition matrix of the model is

The Fourier transform of the function g Q (r) is denoted by

so that the transition matrix reads:

In the limit of small momenta as compared to the ultraviolet momentum cutoff Q (|k| ≪ Q and |k 0 | ≪ Q) or equivalently in the limit of large ultraviolet momentum cutoff, the following identity holds:

A12) which makes possible to directly identify the scattering amplitude of the model at low collisional energy in Eq. (A10):

Comparing with Eq. (A6) one obtains the relation:

Using the fact that

, (A15) one obtains a closed equation for f Q (k 0 ):

The integral in the right hand side of this last equation is performed easily by using the standard analytical continuation k 0 = iq (where q > 0) and finally:

The value of the bare coupling constant v Q is chosen such that the scattering amplitude f Q (k 0 ) coincides with the Bethe Peierls scattering amplitude f 3 (k 0 ) in the limit of an arbitrarily large momentum cutoff Q, thus yielding:

Using the fact that |a 3 | ≪ Q, the bare coupling constant can thus be expressed as:

From Eqs. (A18) and (A14) one obtains:

Finally, the identity lim Q→∞ k<Q

(A21) permits to rewrite Eq. (A20) in the same form as the contact condition of Eq. (A4).


Adamma Aghaizu
email: aaghaizu@sgul.ac.uk

Elisabeth Jane Adams
email: elisabeth.adams@yahoo.co.uk

Katy M Turner
email: katy.turner@bristol.ac.uk

Sally Margaret Kerry

reader in Phillip Hay
email: phay@sgul.ac.uk

epidemiologist Ian Simms
email: ian.simms@hpa.org.uk

reader in general practice Pippa Oakeshott
email: oakeshot@sgul.ac.uk

What is the

BACKGROUND

Many developed countries have set up Chlamydia trachomatis screening programmes, but it is uncertain whether these are cost effective. [START_REF] Kalwij | Screening and treatment of Chlamydia trachomatis infections[END_REF] The English national chlamydia screening programme has been criticized for both its cost (over £100 million) and the lack of outcome data, particularly on prevention of pelvic inflammatory disease (PID). 2;3 There are two crucial reasons why a single chlamydia test is likely to have only a small effect in reducing the incidence of PID over 12 months. First it will only prevent the estimated 30% of PID which is due to chlamydia. [START_REF] Low | Effectiveness of chlamydia screening: systematic review[END_REF] Secondly it will not prevent PID due to incident chlamydial infection occurring during the year. [START_REF] Oakeshott | Randomised controlled trial of screening for Chlamydia trachomatis to prevent pelvic inflammatory disease: the POPI (prevention of pelvic infection) trial[END_REF] Economic evaluations demonstrate the cost effectiveness of chlamydia screening to rely mainly on the prevention of PID and potential complications -tubal factor infertility, ectopic pregnancy and chronic pelvic pain. [START_REF] Adams | The cost effectiveness of opportunistic chlamydia screening in England[END_REF] At present, few data exist on the healthcare seeking behaviour of women with PID and the associated healthcare costs. Historically acute PID was treated mainly in hospital [START_REF] Westrom | Decrease in Incidence of Women Treated in Hospital for Acute Salpingitis in Sweden[END_REF] , but recent changes in PID diagnosis and management have led to wide variation in cost estimates. 6;8-10 The POPI (prevention of pelvic infection) chlamydia screening trial is unique in prospectively screening and following up 94% of 2529 female London students for 12 months and obtaining medical records of those diagnosed with PID. [START_REF] Oakeshott | Randomised controlled trial of screening for Chlamydia trachomatis to prevent pelvic inflammatory disease: the POPI (prevention of pelvic infection) trial[END_REF] Using data from the POPI trial, the aims of this study were to examine the type and number of healthcare settings visited by women with PID, to calculate the cost of an episode of clinical PID, and to estimate how many cases of PID could be prevented per year in London at 2009 screening levels in the English national chlamydia screening programme. We also conducted an exploratory study comparing EQ5-D scores at baseline and after 12 months in women with and without PID.

METHODS

The POPI Trial

The design, recruitment methods and results of the POPI trial have been described elsewhere. 5;11 Briefly, between 2004-6, an ethnically diverse sample of 2529 16-27 year old sexually active, female students were recruited from 20 London universities and colleges. Participants completed a questionnaire and provided self-taken vaginal swabs and consent for follow up and access to their medical records after 12 months. Samples randomly allocated to the intervention group were tested for chlamydia immediately and women with positive tests were referred for treatment.

Samples in the deferred screening group were frozen at -80°C and tested after 12 months. Based on follow up questionnaires backed by a medical record search, three genitourinary medicine physicians blind to group allocation and baseline chlamydia status, used modified Hager's [START_REF] Hager | Criteria for Diagnosis and Grading of Salpingitis[END_REF] and CDC criteria [START_REF]Sexually transmitted diseases treatment guidelines[END_REF] to classify cases into probable, possible or not PID.

Only women aged <25 years are eligible for the English national chlamydia screening programme. For this analysis we therefore excluded 270 women aged 25 or older and restricted it to the 2259 women aged 16 to 24 (mean 20.3 years SD 2.2). Of these women, 28% (623/2243) were from black ethnic minorities, 44% (991/2245) reported two or more sexual partners in the previous year, and 6% (135/2252) were chlamydia positive at baseline. Follow up at 12 months was 94% (2115/2259).

Healthcare seeking behaviour for PID associated symptoms

For each identified PID episode, data on clinical consultations across all healthcare settings were extracted from patient and general practice questionnaires and medical records. We included all cases of PID presenting during 12 months follow up. We defined an episode of PID as all consultations related to symptoms, treatment and follow up of PID within 13 months of baseline screening. Healthcare settings were categorised into general practice, genitourinary medicine clinics, hospital outpatient, hospital inpatient and other (such as walk-in or primary care centres).

PID costing

For each visit we extracted the following information from the records: the staff involved and their time, the drugs, pathology and consumables. Where there were no data on the staff grade or time, assumptions were made by the lead clinicians (PO & PH). We excluded one woman from the cost analysis as she was treated as an outpatient at a hospital in the USA and no medical records were available. The total cost per episode of PID was the sum of the clinical staff cost, drugs, pathology and consumables used, and any inpatient care in a given episode. The clinical cost per minute included the indirect and overhead cost for a clinical member of staff. 14;15 The drug costs were taken from the published Drug Tariff for September 2008. [START_REF][END_REF] The costs of pathology were from local data and other estimates. 15;17 The costs for first and follow-up visit to an outpatient genitourinary medicine clinic (£139, £82 respectively) were taken from the national tariff. [START_REF]Payment by Results: guidance and tariff for[END_REF] In a sensitivity analysis we also estimated the actual time and consumable costs. The costs of a visit to an accident and emergency department (£75) and outpatient gynaecology clinic visit (£145) and a three day non-elective inpatient episode for a late miscarriage associated with PID (HRG code M09, £786),were also taken from the national tariff. [START_REF]Payment by Results: guidance and tariff for[END_REF] As in previous analyses, the costs of partner notification were not included. [START_REF] Adams | The cost effectiveness of opportunistic chlamydia screening in England[END_REF] All costs were estimated in £UK2008/9, using published estimates from that time period where available.

Scaling up to the London population

There were approximately 505,299 16-24 year old women in London in 2008/9 [START_REF]Population esimates submitted by Strategic Health Authorities[END_REF] of whom 82% (414,345) were estimated to be sexually active 20 . For various chlamydia prevalences 5;21 and screening rates we estimated the number of PID cases due to chlamydia at baseline and those preventable by a single chlamydia test using the new risk reduction estimate observed in women aged 16-24 in the POPI trial. [START_REF] Oakeshott | Randomised controlled trial of screening for Chlamydia trachomatis to prevent pelvic inflammatory disease: the POPI (prevention of pelvic infection) trial[END_REF] We also completed a further analysis using the predicted 2010 39% coverage rate. [START_REF]Major step forward in screening[END_REF] The preventable PID associated cost was estimated by multiplying the number of preventable cases by the newly estimated cost of PID. We made no assumptions about the duration of chlamydia infection or reinfection rates.

Health related quality of life in participants with and without PID

Baseline and follow up questionnaires included questions for the generic health related quality of life measure EQ5-D. This tool has two parts; the EQ5D descriptive system with the five dimensions mobility, self-care, usual activities, pain, and anxiety/depression and the EQ-5D visual analogue scale where the 0 and 100 endpoints are the worst and best possible health state. Weighted UK preference values were linked to the self-reported health state scores for a 0-1 index where 0 is death and 1 is perfect health. [START_REF] Euroqol | EuroQol -a new facility for the measurement of healthrelated quality of life[END_REF] Exploratory analyses were performed comparing scores at baseline and follow up in women with and without PID using t-tests.

RESULTS

Incidence of PID

There were 36 cases of PID in the 2115 women who were followed up for 12 months: overall incidence 1.7% (36/2115, 95% C.I. 1.2 to 2.3). The incidence of PID in women who were chlamydia positive at baseline was 1.5% (1/66) in the screened group and 8.8% (6/68) in the deferred screening controls. The difference in PID rates between screened and unscreened chlamydia positives was 7.3% (95% C.I. -0.82 to 16.5%).

Healthcare settings for PID

A case note review indicated the 35 women diagnosed with PID in the UK made a total of 69 visits to a healthcare setting for related symptoms and management (Table 1). Most women attended a general practice surgery or genitourinary medicine clinic. There was an average of 2.0 (SD 0.9, range 1 to 4) attendances at any healthcare setting per PID episode. One woman was admitted for 3 days of inpatient hospital care.

Cost of clinical care for an episode of PID

The average costs of attendances in each setting and for the first attendance at that setting are given in Table 2 (details in the Appendix). Aside from one case, the first visit in a given setting was more expensive than subsequent visits, due to a longer estimated clinical visit and greater pathology costs. The estimated average cost per attendance across all settings was £83 (£68 if the genitourinary medicine tariff was used). Given an average of 2.0 attendances per woman, this works out to £163 (£29-£960) healthcare costs per case (£135 using the estimated costs in genitourinary medicine clinics rather than the tariff). Doubling the estimated amount of clinician time per visit increased the average PID episode cost to £191 (SD £160).

London-wide rate of PID and annual cost of PID due to chlamydia infection at baseline

Assuming a 7.3% difference in PID rates between screened and unscreened chlamydia positives in the POPI trial, Table 3 shows the estimated number and cost of PID cases in London prevented by a single chlamydia test over 12 months using different prevalences of chlamydia and different rates of screening. Assuming a 6% chlamydia prevalence, 21.5% screening coverage in 2009 and 8.8% progression to PID rate, we would expect an estimated 471 cases of PID due to chlamydia at baseline annually in London of which at least 391 (95% C.I. -44 to 882) cases costing £63,733 (95% C.I -£7,172 to £143,766) could probably be prevented each year (highlighted Table 3). The current cost of a screen in London is £45 [START_REF]Guidance for the commissioners on the cost of providing chlamydia screening in primary care and the community: a review of costs in practice across England in[END_REF] which scales to a total screening cost of £4,008,788 at 21.5% coverage.

EQ5-D and EQ Visual Analogue Scale scores in women with and without PID

Twenty eight women with PID (78%) and 1641 without PID (79%) completed both baseline and follow up questionnaires including the EQ5-D section. Women who developed PID had significantly lower scores than those who did not at both baseline and follow up (Table 4).

DISCUSSION

Principal findings

Around 90% of women with PID were managed in the community, mainly in general practice or genitourinary medicine clinics, and there was only one hospital admission.

The average cost of an episode of PID was £163. At 21.5% 2009 annual screening levels around 400 (95% C.I. -44 to 882) cases of PID due to chlamydia at baseline and costing approximately £64,000 (95% C.I. -£7,000 to £144,000) could probably be prevented each year in London.

Strengths and weaknesses

This is the first study to calculate the cost of PID in London and provides new accurate information on the healthcare seeking behaviour and current management of women with PID. The data are from the largest community based prospective study of PID in sexually active women aged 16-24 to date, providing crucial data on the rate of progression to PID in women with treated and untreated chlamydial infection. The follow up rate was very high and we were able use medical records to estimate the costs of each attendance within an episode of PID. The definition of PID, based on analysis of clinical records and questionnaires by experienced genitourinary medicine physicians is likely to be more robust than diagnoses taken from general practice [START_REF] Simms | The rate of diagnosis and demography of pelvic inflammatory disease in general practice: England and Wales[END_REF] and genitourinary medicine clinic databases [START_REF]GUMCAD (KC60) data for[END_REF] .

The main limitation is that the number of PID cases was small due to the low prevalence in this community-based population and we could not include asymptomatic cases. The results are also likely to underestimate the potential benefits of chlamydia screening. Evaluating the effect of a single chlamydia test for women omits potential savings (and costs) due to repeated screening, sexual health promotion, and screening men which should ultimately reduce the prevalence and incidence of chlamydia. [START_REF] Adams | The cost effectiveness of opportunistic chlamydia screening in England[END_REF] The effectiveness of chlamydia screening may be underestimated in the POPI trial because nearly half the untreated chlamydia positives in the deferred screening group were tested for chlamydia independently during the trial. [START_REF] Oakeshott | Randomised controlled trial of screening for Chlamydia trachomatis to prevent pelvic inflammatory disease: the POPI (prevention of pelvic infection) trial[END_REF] The cost of an episode of PID may be underestimated due to incomplete data and the lack of patient costs, indirect costs and partner notification costs. Also, our estimates focus on the costs of acute PID, and we did not include the potential long-term costs of PID. Previous studies have shown that potential longterm consequences such as ectopic pregnancy and infertility, can comprise a substantial proportion of the overall costs of PID. [START_REF] Adams | The cost effectiveness of opportunistic chlamydia screening in England[END_REF] Another weakness is that the diagnosis of PID lacks specificity and sensitivity and the definition of PID has a large impact on reported incidence. 26 However, our PID incidence was the same as in the Scholes trial 27 and similar to that in general practice [START_REF] Simms | The rate of diagnosis and demography of pelvic inflammatory disease in general practice: England and Wales[END_REF] . Secondly, our estimates of the incidence of PID in London and potential cases prevented by screening involve many assumptions and have wide confidence intervals which cross 0, yet may be the best available at present. A single chlamydia test over 12 months will not prevent all cases of PID. This is further reflected in the estimated range of 'cost savings'. The EQ5-D is a blunt instrument for quality of life, and it was not completed by women at the time they had PID. However the finding that women who went on to develop PID had lower quality of life at baseline is novel.

Finally, women in the POPI trial may not be representative and did not include women who had never been sexually active. However, the proportion reporting two or more sexual partners in the previous year was similar to London women in the national survey for attitudes and sexual lifestyles 2000 population based study [START_REF] Fenton | Sexual behaviour in Britain: reported sexually transmitted infections and prevalent genital Chlamydia trachomatis infection[END_REF] and chlamydia positivity rates were similar to national chlamydia screening programme London rates.

Comparison with other studies

The £163 cost of a PID episode in our study was slightly greater than the £139 (UK £2008 equivalent) used in the cost effectiveness analysis by Adams and colleagues [START_REF] Adams | The cost effectiveness of opportunistic chlamydia screening in England[END_REF] . This could be because we had clinical data which showed that most women had more than one consultation for a PID episode. However, this is only the cost for acute PID. If the costs include the long-term complications such as tubal infertility, the average life time cost may range from $1060 -$3180. [START_REF] Yeh | A refined estimate of the average lifetime cost of pelvic inflammatory disease[END_REF] The cost of screening per quality adjusted life year has previously been estimated for the current NCSP strategy; £27,262 assuming a progression to PID rate of 10%. [START_REF] Adams | The cost effectiveness of opportunistic chlamydia screening in England[END_REF] As our PID cost estimate is similar to the one used by Adams et al (£163 versus £137(47)), estimates for the screening cost per quality adjusted life year are also likely to be similar if used in the previous model with similar assumptions.

Estimates of the burden of PID in London from the POPI trial and genitourinary medicine clinic (KC60) surveillance data were similar at around 7000 cases per year. There is considerable public health burden of PID associated morbidity. However, few studies have attempted to explore the health related quality of life. One study found that women who had had PID reported lower mental and physical health scores than women who had not had PID. [START_REF] Smith | Quality of life utilities for pelvic inflammatory disease health states[END_REF] However our study appears to be the first to assess EQ5-D both before and after an episode of PID. It suggests that some women who go on to develop PID may already have a lower quality of life perhaps associated with a more at risk lifestyle [START_REF] Oakeshott | Randomised controlled trial of screening for Chlamydia trachomatis to prevent pelvic inflammatory disease: the POPI (prevention of pelvic infection) trial[END_REF] and clinicians could consider the possibility of additional health or psychosocial problems. We were unable to assess the effect potential long-term sequelae may have on quality of life.

Implications

Most women with PID were managed in general practice or genitourinary medicine clinics. This suggests that some patients with PID may be choosing to be managed in the community, and general practitioners should perhaps be more open towards diagnosing PID. It also implies that using a single source of data such as primary care or hospital may be inadequate for PID surveillance in England. Our results

suggest that to prevent one case of PID over 12 months, 13 women with chlamydia need to be treated. The cost of clinical PID was relatively low compared to costs of screening, but all chlamydia screening strategies are likely to improve health and cost money [START_REF] Adams | The cost effectiveness of opportunistic chlamydia screening in England[END_REF] . In 2009 it probably cost around £4 million in chlamydia screening (at £45 per test) to prevent 400 cases of clinical PID in London associated with £64,000 in healthcare costs, but policy makers need to be aware of other potential benefits of regular screening. These include sexual health promotion and reducing STI incidence. Meanwhile these improved estimates of the cost and incidence of PID can

  Assuming a 1.7% incidence of PID (as in the POPI trial) and 414,345 sexually active women aged 16-24 living in London gives an estimate of 7044 (95% C.I. 4973 to 9530) cases of PID due to any cause in London in 2008/9. For comparison there were 6182 diagnoses of PID in London genitourinary medicine clinics in 2008.[START_REF]GUMCAD (KC60) data for[END_REF] Since these data are not stratified by age, if we assume that, as with chlamydia diagnoses 1 , around 40% of cases were in women aged 25 or more, this suggests there were approximately 3709 cases of PID in women aged 16-24 attending genitourinary medicine clinics. If a similar number were diagnosed in general practices, this suggests, that there were around 7418 episodes of PID in London in 2008 which is in line with the POPI trial estimate.

Table 1 Healthcare setting attendance for 35 women 1 diagnosed with PID during the study period based on questionnaires and review of medical records.

 1

		General Practice	Genitourinary Medicine Clinic	Hospital outpatient*	Hospital inpatient**	Other***
	Where first attended for PID	17	15	1	0	2
	Total women attending
	type of site	21	16	5	1	3
	Total attendances at
	healthcare settings	31	27	6	1	4
	1 A further woman was excluded from the detailed cost analysis as she was
	diagnosed and treated for PID as a hospital outpatient in the USA and no medical
	records were available
	* Including accident and emergency and gynaecology
	**Hospital inpatient stay of 3 days for late miscarriage and PID
	***Other: Walk-in and primary care centre

Table 2 Cost per attendance at different healthcare settings (UK£ 2008/9) for 35 women diagnosed with PID during the study period based on questionnaires and review of medical records

 2

		General	Genitourinary Medicine	Hospital	Hospital	Other****
		Practice	Clinic* (SD)	outpatient**	inpatient***	(SD)
		(SD)			(SD)	(SD)
	Source of costs	Estimated	Tariff	Estimated	Tariff	Tariff	Estimated
	Average cost	41 (15)	139 (n/a)	110 (21)	89 (31)	786 (n/a)	53 (24)
	per first
	attendance at
	the setting
	Average cost	35 (16)	116 (29)	79 (46)	87 (29)	786 (n/a)	42 (8)
	per all
	attendances
	*Note: Genitourinary medicine clinic costs are estimated two ways: either using the
	national tariff (first visit is £139, follow-up £82), or by estimating the time, drugs,
	consumables and test costs.
	**Includes both accident and emergency and outpatient gynaecology estimates
	which are tariff based
	***Inpatient stay of 3 days for late miscarriage and PID
	****Other: walk-in and primary care centre.

Table 3 Estimated number and cost of PID cases prevented by a single chlamydia test over 12 months for 414,345 sexually active women aged 16-24 in London with different prevalences of chlamydia and different rates of screening

 3

					Expected PID		Expected PID	Preventable
		Assumed chlamydia prevalence	Number of women aged 16-24 screened	Number of chlamydia positive women	cases over 12 months in unscreened women with chlamydia	Assumed screen coverage	cases over 12 months in screened women with chlamydia	PID cases Risk difference (7.3%, 95% C.I. -0.82 to	Total preventable PID-associated healthcare costs (at £163 per case)*
					(8,8%)		(1.5%)	16.5)**
	NATSAL 2000* chlamydial prevalence	3%	414,345	12431	1094	100%	187	908 (-102 to 2,052)	£148,004 (-£16,626 to £334,476)
	NCSP 2009 chlamydial prevalence	6 %	414,345	27261	2399	100%	409	1990 (-224 to 4,498)	£324,370 (-£36,512 to £733,174)

NCSP 2009 6% chlamydial prevalence and 21.5% coverage 6% 89,084 5345 471 21.5% 81 391 (-44 to 882) £63,733 (-£7,172 to £143,766)

  Risk difference calculated by number of PID cases expected in unscreened women minus cases expected in screened women.

	NCSP 2010* expected 39% coverage	6%	161,595	9696	853	39%	146	708 (-80 to1,600)	£115,404 (-£13,040 to £260,800)
	* Assume population estimate remains the same.

**

APPENDIX:

Detailed costing Item Unit Unit cost Comment/ Reference

	BASHH -recommended PID treatment (blended cost) 30	10.97 =(A+B) + average(C, D) + E
	A. i.m. ceftriaxone 250mg	1 dose	2.94	Rocephin + 1% lignocaine 16
	Setting B. injection kit Personnel stat.	kit	Cost/ min*	Mean (range) 0.28 mins/visit	Comment/ Reference To administer injection, includes antiseptic wipe, needle, syringe, plaster 15
	General Practice C. oral doxycycline Receptionist D. oral ofloxacin	0.47 100mg/ 14 days 400mg/ 14 days	1.61 0.5 12.63	Assumed to be the same as for a 100mg caps, in packs of 8 (scale by 16 7/8) for 2 packs receptionist in genitourinary medicine clinics 400mg tabs, pack of 5 & pack of 10 16
	Nurse E. oral metronidazole	0.57 400mg/ 14 days	13 (10-20) 0.63	Per minute of client contact, with qualification costs 14 400mg tabs, pack of 21, scale by 2/3 16
	Additional costs	Unit		Per minute of patient contact, with Unit cost Comment/ Reference
	Doctor Azithromycin	1g	2.70	10 (5-15) 8.91	qualification costs, but excluding direct staff costs (i.e. nurse) 14 4x 250mg capsules, Zithromax 16
	Genitourinary Medicine Clinic Cefalexin Ciprofloxacin Receptionist	0.47 500mg 14 days 500 mg * 10d	4.8 (0.5-5) 2.19 0.87	250mg capsules 16 LSHP SH tariff 15 16
	Nurse, day ward (Band 5) Clotrimazole Coamoxiclav	0.73 500g and cream 250mg/125mg;	15 5.21 4.09	Canesten combi 16 Per minute of patient contact, with qualification costs 14 16
	Nurse (team leader) & Health Adviser (Band 6) 21 tabs 1.12 15 mg * 30d Culture for GC Codeine Sample	17(5-20) 1.39 7.14	Per minute of patient contact, with 16 qualification costs 14 Estimate from St George's
	Diclofenac	Medical Consultant	50mg	2.77	0.29 16 (5-20)	Per patient related minute, with Assume diclofenac potassium 25mg qualification costs tabs x2, 1 dose
	Erythromycin	500mg bd 14	5.95	Erymax (erythromycin 250mg
	Fluconazole	days 150 mg	0.47	gastro-resistant capsules), in packs of 28 16 16
	HIV and syphilis test	test		8.00	Estimate from St George's
	Levonelle		1500mg	5.11	Levonorgestrel 1.5mg tabs 16
	Chlamydia & Gonorrhoea laboratory processing Urine Pot Mefanamic acid Vaginal swab (for CT/GC) Transport Tube Microscopy (in-house) Trimethoprim Lab/pathology request form Gloves Urine pregnancy test	test item 500mg item item Sample 200g bd 3 item days pair test	10.61 2.29 0.22 1.50 5.00 0.08 0.25 0.25 0.04 0.80	Estimate from St George's 15 Assume 3xday/7 days; (for 28 pills, 500mg) 16 15 15 Assumption (10 min Band 5 & consumables) 15 16 Packs of 14 tabs 15 15 Assumption
	Speculum		item		0.79	15
	KY		dose		0.29	15
	High vaginal swab (for	item		0.22	15
	microscopy)
	Slide (for microscopy)	item		0.10	15
	Treatment: blended cost (see	treatment	10.97	Average, see calculation below
	below)
	Notification-phone call	per min	0.15	Assumption
	Notification-text message	per text	0.10	Assumption
					1 7

Funding: This study was supported by the BUPA Foundation (grant no 684/GB14B).

be used to inform further modelling and economic analyses of the impact of chlamydia screening.

Ethical approval: This study was approved by Wandsworth Research Ethics Committee (reference 03.0012).

Competing interest statement

All authors declare that the answer to the questions on your competing interest form are all No and therefore have nothing to declare.

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T Carette
email: tcarette@ulb.ac.be

M R Godefroid

Ab initio calculations of the 33 S 3p 4 3 P J and 33 S -/ 37,35 Cl 3p 5 2 P o J hyperfine structures

Keywords: numbers: 32.10.Hq, 31.15.A-, 32.10.Fn negative ions, sulfur, chlorine, hyperfine structures

Ab initio calculations of the 33S3p43PJ and 33S/37, 35Cl 3p52PoJ hyperfine structures

Introduction

As the experimental techniques evolve, the window of isotope effects in negative ions is becoming wider and wider [1,2,3,4,5,6,7]. It offers new possibilities for understanding negative ions, correlation effects and the effect of nuclear properties on the electronic structure. In order to keep to this agenda, we need accurate models.

The hyperfine interaction, as arising mainly from the non-spherical nature of the electronic wave function near the nucleus, is an operator that is particularly sensitive to both core and valence correlation effects. It is also sensitive to contributions that do not affect the energy much. When trying to calculate it, we face the difficulty of getting simultaneously a good description of the valence correlation -crucial for negative ions -, core-valence and core correlation. It is therefore a useful tool for testing new approaches.

Experimental studies of the hyperfine interaction in atomic negative ions are scarce. A pioneer study was made by Mader and Novick on the 3 He -1s2s2p 4 P o metastable state [8]. Much more recently, the hyperfine structure of the only known E1 transition between bound states of a negative ion, the Os -, was studied [9]. In between, Trainham et al [1] performed a study of the 33 S -3p 5 2 P o 3/2 hyperfine structure. The S - ion offers a good possibility for testing the methods on a middleweight system.

Recently, the isotope shift on the electron affinity of sulfur between the zero-spin isotopes 32 and 34 was measured and calculated [6]. A similar study involving the isotope 33 of sulfur, of spin I = 3/2, would require the knowledge of the hyperfine structure of the involved states [5]. However, if the 33 S -ground state hyperfine structure is known, the one of the 33 S 3p 4 3 P o J multiplet is not, to the knowledge of the authors. We present in Section 3 large scale MCHF calculations on the iso-electronic systems S -and Cl, testing the possibility of using the valence correlation orbitals [10] for describing core effects. This choice of orbital set was successfully adopted for including a core-valence contribution in the calculation of the isotope shift on the electron affinity of sulfur [6] and chlorine [7]. In Section 4, we calculate the hyperfine structure of the neutral sulfur lowest multiplet and point out a remarkable interplay of core and valence correlation in a MCHF description. We compare our theoretical results with experiment in Section 5 and conclude in Section 6 that some computational strategies provide a better definition for the core region.

Theory

The MCHF / CI approach

The multiconfiguration Hartree-Fock (MCHF) variational approach is based on the following ansatz

Ψ(γLSM L M S ) = i c i Φ(γ i LSM L M S ) ( 1 )
where the Φ(γ i LSM L M S ) are configuration state functions (CSF) that are built on a single orthonormal basis set [START_REF] Fischer | Computational Atomic Structure : An MCHF Approach[END_REF]. A CSF is a symmetry-adapted linear combination of Slater determinants corresponding to a given term with its own coupling tree. Energy functionals integrated on the electron angular coordinates for an ansatz of the type (1) are efficiently obtained using the Racah algebra [START_REF] Fischer | Hartree-Fock method for atoms. A numerical approach[END_REF]. Large expansions of CSFs can be automatically generated using existing programs [START_REF] Sturesson | [END_REF], allowing to reduce the considered excitations of the main configuration to classes of states. In the MCHF theory, all c i are optimized and some -if not all -orbital radial functions are varied until self-consistency. We use the ATSP2K implementation of this method [14].

In this work, we first perform MCHF calculations to determine the set of spectroscopic and virtual one-electron states in the MCHF expansion. We then do subsequent larger configuration interaction (CI) calculationsi.e. optimizing only the {c i } mixing coefficients in (1). At all stages, the variational principle is applied using the non-relativistic isotope-independent Hamiltonian. An orbital active set is defined as the set of all orbitals characterized by quantum numbers with n ≤ n max and l ≤ l max , and is denoted n max l max .

Our scheme is based on the concept of "multi-reference-interacting" (MR-I) set of CSFs detailed elsewhere [6]. It is defined as the set of all CSFs that interact to first order with at least one component of a small set of configurations, the multi-reference (MR). The LS angular momenta of the occupied sub-shells are coupled by decreasing n and l. For a given orbital basis set, a MR-I space is a subset of all single and double (SD) excitations of the MR. Even though this scheme is largely optimized for having the fastest convergence of the energy, it also allows one to efficiently obtain accurate wave functions [15]. Using this strategy, the most important components of the MCHF expansion (1) contributing to the state's hyperfine structure are included with the right mixing coefficient.

Hyperfine interaction

The hyperfine structure of a LSJ level is caused by the magnetic and electric interaction of the electrons with the atomic nucleus of angular momentum quantum number I [START_REF] Lindgren | Case Stud[END_REF]. The total atomic angular momentum quantum number is denoted F . To first order, the hyperfine energy correction has the form

W (J, J) = A J C 2 + B J 3C(C + 1) -4I(I + 1)J(J + 1) 8I(2I -1)J(2J -1) . ( 2
)
where A J and B J are respectively the magnetic dipole (M1) and electric quadrupole (E2) hyperfine constants and C = F (F + 1) -J(J + 1) -I(I + 1). The theory underlying the computation of hyperfine structures using multiconfiguration wave functions can be found in references [17,18] in which the non relativistic hyperfine interaction is expressed in terms of the J-independent orbital (a l ), spin-dipole (a sd ), contact (a c ) and electric quadrupole (b) electronic hyperfine parameters

a l ≡ ΓLSM L M S | N i=1 l (1) 0 (i)r -3 i |ΓLSM L M S , (3)
a sd ≡ ΓLSM L M S | N i=1 2C (2) 0 (i)s (1) 0 (i)r -3 i |ΓLSM L M S , (4)
a c ≡ ΓLSM L M S | N i=1 2s (1)
0 (i)r -2 i δ(r i )|ΓLSM L M S , (5) b
≡ ΓLSM L M S | N i=1 2C (2)
0 (i)r -3 i |ΓLSM L M S , (6)
and calculated for the magnetic component M L = L and M S = S [17]. The first three parameters (3), (4), and (5) contribute to the magnetic dipole hyperfine interaction constant through

A J = A l J + A sd J + A c J , (7)
with

A l J = G µ µ I I K l J a l , (8)
A sd J = 1 2 G µ g s µ I I K sd J a sd , (9)
A c J = 1 6 G µ g s µ I I K c J a c , (10)
where the K J 's are the appropriate non-relativistic angular factors [19]. The last one (b) constitutes the electronic contribution to the electric quadrupole hyperfine interaction

B J = -G q Q K b J b . (11)
Expressing the electronic parameters a l , a sd and a c in atomic units (units of a -3 0 ) and µ I in nuclear magnetons (units of µ N ), the magnetic dipole hyperfine structure constants A J are calculated in units of frequency (MHz) by using G µ = 95.41067. Similarly, the electric quadrupole hyperfine structure constants B J are expressed in MHz when adopting atomic units (units of a -3 0 ) for b, barns for Q and G q = 234.96475.

The distinction between core and valence electrons

When studying an atomic system, it is common to distinguish core and valence electrons based on the mono-configuration approximation. This separation can usually be made quite efficiently by choosing a rare-gas-like core. In our case, the core is neon-like (1s 2 2s 2 2p 6 ) and the valence is composed of the n = 3 electrons (3s 2 3p 5 for S -and Cl, and 3s 2 3p 4 for S). From that distinction, a lot of concepts based on a first-order picture of the correlation arise: core polarization, valence, core-valence and core-core correlation (V, CV, CC), etc. This terminology remains vague and the contributions those terms refer to, often depend on the method [10,20,[START_REF] Lindgren | Atomic Many-Body Theory[END_REF][START_REF] Migdalek | [END_REF]. In MCHF theory it is in general impossible to have a clean partition of those different contributions due to systematic rearrangements of the {c i } MCHF eigenvector and orbital shapes from one model to another. The variational procedure is not perturbative and the core-valence distinction is based on a perturbative picture of the correlation. Still, the core-valence separation has a physical background so that it should not be discarded. Hence, we need a reliable definition of the core in order to interpret the so-called "core effects". In this work, the uncorrelated core is defined as being the N c -electron core state built on the Hartree-Fock orbital set optimized for the total system. In case of multiple solutions in the HF model, we select the one satisfying the Koopmans requirement [START_REF] Fischer | Hartree-Fock method for atoms. A numerical approach[END_REF].

3. Hyperfine Structures of the 3p 5 2 P o J levels of S -and Cl

Definition of the orbital set: MCHF calculations

To explore the impact of the choice of the MCHF model defining the orbital set on the results of the open-core CI models, we use two references: the single main configuration, denoted SR, and the set of all valence single and double (SD)-excitations in the n = 3 shell, denoted MR. The latter is formally written

MR = {1, 2} 10 {3s, 3p} 5 {3} 2 . ( 12
)
In this notation, distinct sets of electrons are dispatched in series of shells and/or subshells. Here, all ten slots of the n = 1, 2 shells are filled, five electrons are shared between the spectroscopic orbitals 3s and 3p and the two remaining electrons occupy either the spectroscopic orbitals or the virtual 3d subshell. After having generated the CSF list, we perform HF frozen-core MCHF calculations on the SR-I and MR-I sets obtained by activating all the electrons in orbital basis sets ranging from 4f to 13h and 10h for SR and MR, respectively. The so-obtained one-electron radial functions are denoted SR-I-C and MR-I-C. In addition, we perform valence MCHF calculations on the MR-I nl , n = 4 -13, l ≤ 5 (h orbitals). The resulting basis set is denoted MR-I-V. In all calculations, the [Ne] core is frozen to its HF shape. Tables 1 and2 display, respectively for Cl and S -, the weights of the first configurations in the various MCHF calculations with n max l max = 10h. The weight of a configuration is defined as

w = i c 2 i 1/2 , ( 13
)
where the sum runs over the CSFs belonging to the configuration. We observe that the order of the few most important configurations is very similar in all cases. Still, some differences appear mainly due to the fact that some radial functions P nl (r) become inner orbitals in the MR-I-C and SR-I-C models. This is illustrated in the first four columns of Table 3 in which we compare the mean radius of the spectroscopic and correlation orbitals of the MR-I-C model with the ones of the MR-I-V model for both Cl and S -. In open-core calculations, most of the n = 5 orbitals become inner orbitals and P 4d , P 4f contract significantly. The SR-I-C orbitals do not differ strongly from the MR-I-C ones and are therefore not presented here. 0.1137 2p 6 3s 1 3p 4 3d 1 4p 1 0.0557 2p 6 3s 1 3p 4 3d 1 4p 1 0.0550 3s 1 3p 4 3d 1 4p 1 0.0755 2p 6 3s 1 3p 4 4s 1 4p 1 0.0496 2p 6 3s 1 3p 4 4s 1 4p 1 0.0500 3s 1 3p 4 4s 1 4p 1 0.0703 2p 6 3s 1 3p 4 3d 1 4f 1 0.0431 2p 6 3s 1 3p 4 3d 1 4f 1 0.0441 3s 1 3p 4 3d 1 4f 1 0.0624 2p 6 3s 1 3p 4 3d 1 5f 1 0.0424 2p 6 3s 1 3p 4 3d 1 5f 1 0.0436 3s 2 3p 4 4p 1 0.0606 2p 6 3p 5 3d 2 0.0342 2p 6 3p 5 3d 2 0.0420 3s 2 3p 4 4f 1 0.0494 2p 6 3s 1 3p 4 4p 1 5s 1 0.0328 2p 6 3s 1 3p 4 4p 1 5s 1 0.0328 3s 2 3p 4 5p 1 0.0463 2p 4 3s 2 3p 5 5p 2 0.0309 2p 4 3s 2 3p 5 5p 2 0.0305 3p 5 3d 2 0.0418 2p 4 3s 2 3p 5 4d 1 5d 1 0.0296 2p 4 3s 2 3p 5 4d 1 5d 1 0.0295 3s 2 3p 3 4f 2 0.0355 2p 6 3s 2 3p 3 3d 1 5g 1 0.0276 2p 6 3s 2 3p 3 3d 1 5g 1 0.0281 3s

Fully correlated CI calculations

The results of Tables 1 and2 indicate that, even if the variational contents of openand closed-core calculations are a priori very different [START_REF] M R Godefroid | Atomic structure variational calculations in spectroscopy[END_REF], it is here possible to find extensions of the models developed above that remain comparable. Indeed, if we choose as multi-references the five or nine first configurations in the sorted lists

MR 5 = {3s 2 3p 5 , 3s 2 3p 3 3d 2 , 3s 1 3p 5 3d 1 , 3s 2 3p 3 4p 2 , 3s 1 3p 4 3d 1 4f 1 } (14)
MR 9 = MR 5 ∪ {3s 1 3p 4 4s 1 4p 1 , 3s 1 3p 4 3d 1 4p 1 , 3p 5 3d 2 , 3s 2 3p 4 4f 1 } (15)
for both systems, the selected MRs account for approximately the same correlation effects in all S -and Cl orbital sets. It allows a significant comparison between the CI calculations performed on the MR 5 -and MR 9 -I-C multi-reference-interacting sets performed with the orbital sets arising from either the SR-I-C, MR-I-C or MR-I-V MCHF calculations. Figure 1 and the two top plots of Figure 2 present the convergence of the a l , a sd , a c and b hyperfine parameters (3-6) calculated with the MR 5 -I-C model in the three orbital basis sets (SR-I-C, MR-I-C and MR-I-V). All parameters are given in atomic units (a -3 0 ). In all cases the MR-I-V 13h and SR-I-C 13h results differ by about 2 10 -2 a -3 0 , the final A J constants themselves differing by less than half a percent. The convergence of the calculations with the orbital sets obtained from the MR-I-V model is slow compared to the one of the calculations based on the SR-and MR-I-C orbital sets. It is remarkable to note that, to the contrary of the magnetic dipole parameters (a), the electric quadrupole parameter (b) converges as rapidly in the SR-I-C and MR-I-V basis sets.

Comparing the S -and Cl trends, it is the similarities that first strike. However, the y-axis scales being the same between the left (S -) and corresponding right (Cl) plots, we observe easily that (i) the S -hyperfine parameters are shifted to smaller values compared to the Cl parameters, and (ii) the S -hyperfine structure calculations converge slightly faster that the Cl ones. These two observations can be easily understood by the mere diffuseness of the negative ion electron charge distribution. Indeed, when an electron attaches to a neutral atom, the valence shells spread and the core-valence separation becomes larger. Then, even though the negative ion have more electrons, the core-valence overlap is only slightly larger in the negative ion than in the neutral, even if there are more valence electrons in the former than in the latter [7,[START_REF] Carette | Isotope effects in atomic spectroscopy of negative ions and neutral atoms: a theoretical contribution[END_REF]. Hence, the negative ion core is more spherical and the hyperfine constants tend to be smaller than in comparable systems. Consequently, even if negative ions are highly correlated systems, they are characterized by a comparatively smaller core-valence correlation. In particular, the hyperfine structure of negative ions is slightly less sensitive to correlation effects than one would first expect.

Even though the MR 5 and MR 9 sets do not account for exactly the same correlation effects depending on the active set, the approach based on the orbitals obtained at the SR-I-C 13h level are superior. Tables 4 and5 show the total energies and a l , a sd , a c and b parameters deduced from the MR 5 -, MR 9 -I-C 12h and MR 5 -I-C 13h models using the SR-I-C orbitals. The final results are obtained by reporting the impact of the 13 th shell on the MR 5 -I-C model on the MR 9 based values.

Hyperfine Structures of the 3p 4 3 P J levels of S

Definition of the orbital set: MCHF calculations

For the 3p 4 3 P state of neutral sulfur, we first select the references from which the MCHF expansions are generated. Similarly to Cl and S -, we use the mono-reference (SR) and 0.0394 2s 2 2p 6 3s 1 3p 3 3d 1 4f 1 0.0436 3s 1 3p 3 4s 1 4p 1 0.0580 2s 2 2p 6 3s 1 3p 3 3d 1 5p 1 0.0394 2s 2 2p 6 3s 1 3p 3 3d 1 5p 1 0.0393 3s 1 3p 3 3d 1 4f 1 0.0562 2s 2 2p 6 3p 4 3d 2 0.0389 2s 2 2p 4 3s 2 3p 4 4p 2 0.0389 3s 1 3p 3 3d 1 4p 1 0.0547 2s 2 2p 6 3s 2 3p 2 5p 2 0.0386 2s 2 2p 6 3s 2 3p 2 5p 2 0.0387 3p 4 3d 2 0.0498 2s 2 2p 6 3s 1 3p 3 5s 1 5p 1 0.0338 2s 2 2p 6 3s 1 3p 3 5s 1 5p 1 0.0339 3s 2 3p 3 4f 1 0.0489 2s 2 2p 6 3s 2 3p 2 4p 1 5p 1 0.0324 2s 2 2p 6 3s 1 3p 3 3d 1 6f 1 0.0328 3s 2 3p 3 4p 1 0.0486 2s 2 2p 6 3s 1 3p 3 3d 1 6f 1 0.0312 2s 2 2p 6 3s 2 3p 2 4p 1 5p 1 0.0324 3s 1 3p 3 3d 1 5f 1 0.0356 2s 2 2p 4 3s 2 3p 4 4d 1 5d 1 0.0289 2s

0.0252 2s 1 2p 5 3s 2 3p 4 4s 1 4p 1 0.0253 2s 1 2p 5 3s 2 3p 4 4s 1 4p 1 0.0250 3s 1 3p 3 4p 1 4d 1 0.0247 a multireference MR = {1, 2} 10 {3s, 3p} 4 {3} 2 . ( 16
)
The weights of the most important configurations in the SR-I-C, MR-I-C and MR-I-V calculations using the active set 10h are presented in Table 6. The corresponding mean radius of the orbitals are shown in the fifth and sixth columns of Table 3. We observe that the eigenvector composition differences between the open-core and valence calculations are much more pronounced than in the S -and Cl cases. For instance, the 1s 2 2s 2 2p 4 3s 2 3p 4 4p 2 configuration gains a significant weight in both opencore calculations (SR-and MR-I-C), the 4p orbital being localized in the inner region of the atom. This important core reorganization is accompanied by the extinction of the closed-core configuration 3s 2 3p 2 4p 2 in comparison with the MR-I-V eigenvector composition, counterbalanced by the turning on of another closed-core configuration, 3s 2 3p 2 5p 2 . In the open-core models, the cumulative weight of these two configurations 2p 4 3s 2 3p 4 4p 2 and 2p 6 3s 2 3p 2 5p 2 reaches 0.055, which is comparable to the weight of the 3s 2 3p 2 4p 2 in the closed core model (= 0.068). It indicates that the correlation of the 3p 4 and 2p 6 electrons is similar so that, even if we fix the core to its HF shape, high-order correlation effects mix the core and valence electrons through the 2p 4 3p 4 4p 2 configuration, (3p, 4p) being variational. Similar observations are made in the context of the calculation of the electron affinity of Chlorine [7], in which case the negative ion neon-like core can strongly mix with the valence 3p 6 electrons compared to the neutral MCHF solution.

Fully correlated CI calculations

The large differences between the roles played by each correlation orbital in the opencore and valence calculations prevent us from comparing the two models as in the cases of S -and Cl. Although the results of the different open-core CI models are coherent, we limit our discussion to the SR-and MR-I-C results.

Driven by the eigenvector composition analysis presented in Table 6, we choose the multi-reference

MR 5 = {3s 2 3p 4 , 3s 2 3p 2 3d 2 , 3s 1 3p 4 3d 1 , 3s 1 3p 3 3d 1 4f 1 , 3p 4 3d 2 }. (17)
Then, for evaluating the impact of the addition of a core-polarization configuration in the reference, we build the MR

MR 6 = MR 5 ∪ {2s 1 2p 5 3s 2 3p 4 4s 1 4p 1 } , (18)
and finally we define

MR 8 = MR 6 ∪ {2s 2 2p 4 3s 2 3p 4 4p 2 , 2s 2 2p 6 3s 1 3p 3 3d 1 5p 1 }. (19)
We show in Table 7 the hyperfine parameters for the sulfur 3p 4 3 P ground state using these MR p -I-C models (p = 5, 6, 8) for the largest possible active set and using the SR-I-C radial functions. The final results are obtained by reporting the impacts of the 13 th correlation shell on the MR 5 -I-C model and the impacts of the extension of the MR 6 reference to the MR 8 reference using the active set 9h , on the hyperfine parameter values obtained in the MR 6 -I-C 12h calculation.

Comparison with experiment

The three considered isotopes, 33 S, 35 Cl and 37 Cl, have a spin I = 3/2 and respectively a magnetic dipole moment of +0.6438212 (14) µ N , +0.8218743(4) µ N and +0.6841236(4) µ N [START_REF] Stone | [END_REF]. Their nuclear quadrupole moments are still best determined using theoretical inputs [26], as we will see below. The non-relativistic A J constants computed using the final set of hyperfine parameters of Tables 4 , 5 and 7 are shown in Table 8. We estimate the relativistic corrections by running mono-reference non-relativistic and corresponding relativistic CI calculations using the Pauli approximation with the SR-I-C 9h orbital set, as described in references [15,19].

At the non-relativistic level, the neutral sulfur A 1 ( 3 P ) hyperfine constant is characterized by a strong cancellation between the spin-dipole (A sd ) and orbit (A l ) contributions. Indeed, in the MR 6 -I-C 12h CI calculation performed with the SR-I-C orbitals, we find

We realize from Table 4, 5 and 7 that the contact term (A c ) is by far the less converged contribution, bringing the largest source of uncertainty (∼ 1 MHz). It is unclear to which extent the 33 S -theory-experiment excellent agreement is accidental. As far as the B J constants are concerned, it is more relevant to tabulate the electric quadrupole moments (Q) obtained for S -and Cl from the formula [17]

Q = - (B 3/2 ) exp (b) th G q ( 21
)
where G q = 234.96475 for obtaining Q in barns when b is in a -3 0 and B J in MHz. The available experimental data are:

We compare our Q values with previous works in Table 9. Since, to our knowledge, there is no measurement of the neutral sulfur hyperfine structure, we use our value for the b(S 3 P ) parameter for estimating the B J constants of 33 S with each Q( 33 S) value. a Reference [2] for 33 S and [28] for the chlorine isotopes b Reference [29] c Reference [30]

Conclusion

We perform MCHF-CI and RCI calculations of the hyperfine constants of the 3p 5 2 P o J multiplet of 33 S -and 35,37 Cl and the 3p 4 3 P J multiplet of 33 S. We obtain good agreement with previous theoretical works [2,28] for the nuclear electric quadrupole moments of 33 S and 37,35 Cl, and with the A(3p 5 2 P o 3/2 ) experimental values [1,27]. It appears that the contact contribution, the main source of uncertainty in our non-relativistic calculations, is ten times smaller in S -than in Cl. We interpret this as an effect of an increased separation of the core and valence regions in negative ions.

We show that, for sufficiently large active sets, orbitals optimized in closed-core MCHF calculations reproduce the results of proper open-core MCHF calculations. This approach has a significant advantage: the core-valence distinction in frozen and closedcore MCHF calculations is much cleaner. It allows to minimize the high-order core and valence mixing and hence get a better comparison between calculations performed on different systems (e.g. S and S -).

Figure 1 .

 1 Figure 1. For both S -(left plots) and Cl (right plots), values of non-relativistic magnetic dipole hyperfine interaction parameters a l , a sd and a c obtained by CI calculations in the MR 5 -I-C model, in atomic units (a -3 0 ), with the three explored orbital basis sets SR-I-C, MR-I-C and MR-I-V, as a function of the active space nl , l ≤ 5.

Figure 2 .

 2 Figure 2. For both S -(left plots) and Cl (right plots), values of non-relativistic electric quadrupole hyperfine interaction parameter b (in a -3 0 ) and energy E (in E h ) obtained by CI calculations in the MR 5 -I-C model with the three explored orbital basis sets SR-I-C, MR-I-C and MR-I-V, as a function of the active space nl , l ≤ 5.

A

  l = 105.04 MHz A sd = -110.74 MHz A c = 6.28 MHz .

B 3 / 2 (B 3 / 2 (

 3232 33 S -) = 26.2437 Cl) = 43.245 245(55) MHz[27]

Table 1 .

 1 Sorted weights of the first configurations in the MR-I-V, MR-I-C and SR-I-C wave functions of Cl. The active set is 10h . The 1s and 2s sub-shells are closed in all those configurations. 3s 1 3p 4 3d 1 4f 1 0.0527 2p 6 3s 1 3p 4 3d 1 4f 1 0.0538 3s 1 3p 4 3d 1 4f 1 0.0691 2p 6 3s 1 3p 4 3d 1 4p 1 0.0411 2p 6 3p 5 3d 2 0.0416 3s 1 3p 4 4s 1 4p 1 0.0584 2p 6 3s 1 3p 4 4s 1 4p 1 0.0403 2p 6 3s 1 3p 4 3d 1 4p 1 0.0407 3s 1 3p 4 3d 1 4p 1 0.0537 2p 6 3p 5 3d 2 0.0346 2p 6 3s 1 3p 4 4s 1 4p 1 0.0407 3s 2 3p 4 4f 1 3s 1 3p 4 4p 1 5s 1 0.0310 2p 6 3s 1 3p 4 4p 1 5s 1 0.0311 3s 2 3p 3 4f 2 0.0370 2p 6 3s 2 3p 3 3d 1 5g 1 0.0269 2p 6 3s 2 3p 3 3d 1 5g 1 0.0273 3s 2 3p 4 4p 1 0.0322 2p 4 3s 2 3p 5 4d 1 5d 1 0.0268 2p 4 3s 2 3p 5 4d 1 5d 1 0.0267 3s 2 3p 3 3d 1 5g 1 0.0306 2p 4 3s 2 3p 5 5d 2

	SR-I-C		MR-I-C		MR-I-V
	config.	w	config.	w	config.	w
	2p 6 3s 2 3p 5	0.9712	2p 6 3s 2 3p 5	0.9584	3s 2 3p 5	0.9567
	2p 6 3s 2 3p 3 3d 2	0.1413	2p 6 3s 2 3p 3 3d 2	0.1877	3s 2 3p 3 3d 2	0.1905
	2p 6 3s 1 3p 5 3d 1	0.0806	2p 6 3s 1 3p 5 3d 1	0.1121	3s 1 3p 5 3d 1	0.1145
	2p 6 3s 2 3p 3 4p 2	0.0649	2p 6 3s 2 3p 3 4p 2	0.0653	3s 2 3p 3 4p 2	0.0786
	2p 6 0.0475
	2p 6 3s 2 3p 4 4f 1	0.0311	2p 6 3s 2 3p 4 4f 1	0.0323	3p 5 3d 2	0.0417
	2p 6 0.0257	2p 4 3s 2 3p 5 5d 2	0.0253	3s 2 3p 4 5p 1	0.0292
	2p 6 3s 2 3p 3 4f 2	0.0252	2p 6 3s 2 3p 3 4f 2	0.0248 3s 1 3p 4 4p 1 4d 1 0.0271
	2p 4 3s 2 3p 5 5p 2	0.0249	2p 4 3s 2 3p 5 5p 2	0.0245 3s 2 3p 3 4s 1 4d 1 0.0257

Table 2 .

 2 Sorted

	SR-I-C		MR-I-C		MR-I-V
	config.	w	config.	w	config.	w
	2p 6 3s 2 3p 5	0.9675	2p 6 3s 2 3p 5	0.9522	3s 2 3p 5	0.9448
	2p 6 3s 2 3p 3 3d 2	0.1372	2p 6 3s 2 3p 3 3d 2	0.1931	3s 2 3p 3 3d 2	0.1935
	2p 6 3s 2 3p 3 4p 2	0.0843	2p 6 3s 1 3p 5 3d 1	0.1124	3s 1 3p 5 3d 1	0.1140
	2p 6 3s 1 3p 5 3d 1	0.0765	2p 6 3s 2 3p 3 4p 2	0.0848	3s 2 3p 3 4p 2

weights of the first configurations in the SR-I-C, MR-I-C and MR-I-V wave functions of S -. The active set is 10h . The 1s and 2s sub-shells are closed in all those configurations.

Table 3 .

 3 2 3p 3 4s 1 4d 1 0.0343 2p 4 3s 2 3p 5 5d 2 0.0263 2p 6 3s 2 3p 4 5f 1 0.0272 3s 1 3p 4 4p 1 4d 1 0.0342 2p 6 3s 2 3p 4 5f 1 0.0260 2p 4 3s 2 3p 5 5d 2 0.0259 3s 1 3p 4 3d 1 5f 1 0.0335 Mean radius (in a 0 ) of the n = 1 to 5 orbitals of Cl, S -and S in the MR-I-V 10h (valence) and MR-I-C 10h (open-core) models. 1s, 2s, 2p are Hartree-Fock orbitals.

	Cl		S -		S
	nl valence open-core	valence open-core	valence open-core
	1s 0.09130	0.09130	0.09715	0.09715	0.09715	0.09715
	2s 0.44171	0.44171	0.47585	0.47585	0.47577	0.47577
	2p 0.40572	0.40572	0.44106	0.44106	0.44104	0.44104
	3s 1.54735	1.55045	1.76214	1.76836	1.71095	1.71268
	3p 1.82220	1.82778	2.28135	2.29572	2.02884	2.03898
	3d 1.76250	1.75997	2.15941	2.16075	1.95878	1.96343
	4s 2.03477	1.92840	2.44030	2.39304	2.40713	0.71754
	4p 2.41053	2.13745	3.34104	2.80812	2.81064	0.70561
	4d 2.23120	1.35932	2.86818	1.64193	2.45573	1.42118
	4f 1.82616	1.30900	3.03237	1.17530	2.98537	1.20579
	5s 2.14697	0.69456	2.71559	0.74676	2.28372	2.13212
	5p 2.16166	0.68502	2.88173	0.75013	2.42314	2.38464
	5d 2.08468	0.54938	2.04493	0.76611	1.70728	0.60688
	5f 2.06913	0.65051	2.31986	1.55795	2.31174	0.63876
	5g 1.85751	1.78797	3.40818	2.12699	3.83176	1.94307

Table 4 .

 4 Best estimates of the energy (in E h ) and hyperfine parameters (in a -3 0 ) of Cl (see text). The final values are the results of the MR 9 -I-C 12h in which the impact of the 13 th correlation shell on the MR 5 -I-C model is reported.

		MR 5		MR 9
		12h	13h	12h	final
	E	-460.12061 -460.12145 -460.12315
	a l	7.15768	7.15724	7.14014	7.13971
	a sd	-1.50471	-1.50541	-1.50369 -1.50439
	a c	0.49079	0.48733	0.52870	0.52525
	b	2.84768	2.84738	2.83912	2.83882

Table 5 .

 5 Best estimates of the energy (in E h ) and hyperfine parameters (in a -3 0 ) of S - (see text). The final values are the results of the MR 9 -I-C 12h in which the impact of the 13 th correlation shell on the MR 5 -I-C model is reported.

		MR 5		MR 9
		12h	13h	12h	final
	E	-398.15235 -398.15324 -398.15553
	a l	4.30388	4.30366	4.28953	4.28931
	a sd	-0.91284	-0.91324	-0.91211 -0.91251
	a c	-0.00209	-0.00306	0.03581	0.03484
	b	1.71294	1.71293	1.70529	1.70528

Table 6 .

 6 Sorted weights of the first configurations in the SR-I-C, MR-I-C and MR-I-V wave functions of S. The active set is 10h . The 1s shell is closed in all those configurations.

	SR-I-C		MR-I-C		MR-I-V
	config.	w	config.	w	config.	w
	2s 2 2p 6 3s 2 3p 4	0.9726	2s 2 2p 6 3s 2 3p 4	0.9587	3s 2 3p 4	0.9566
	2s 2 2p 6 3s 2 3p 2 3d 2	0.1139	2s 2 2p 6 3s 1 3p 4 3d 1	0.1574	3s 1 3p 4 3d 1	0.1607
	2s 2 2p 6 3s 1 3p 4 3d 1	0.1106	2s 2 2p 6 3s 2 3p 2 3d 2	0.1547	3s 2 3p 2 3d 2	0.1569
	2s 2 2p 6 3s 1 3p 3 3d 1 4f 1 0.0419	2s 2 2p 6 3p 4 3d 2	0.0497	3s 2 3p 2 4p 2	0.0677
	2s 2 2p 4 3s 2 3p 4 4p 2

  2 2p 4 3s 2 3p 4 4d 1 5d 1 0.0285 3s 2 3p 3 5p 1 0.0313 2s 2 2p 6 3s 2 3p 3 4f 1

		0.0271	2s 2 2p 6 3s 2 3p 3 4f 1	0.0281	3p 5 4p 1	0.0292
	2s 2 2p 4 3s 2 3p 4 5d 2	0.0260	2s 2 2p 4 3s 2 3p 4 5d 2	0.0256	3s 1 3p 2 3d 3	0.0282
	2s 2 2p 4 3s 2 3p 4 4d 2	0.0260	2s 2 2p 4 3s 2 3p 4 4d 2	0.0256	3s 2 3p 2 4f 2

Table 7 .

 7 Best estimates of the energy (in E h ) and hyperfine parameters (in a -3 0 ) of the neutral sulfur 3 P state. The final results are obtained by reporting the impacts of the 13 th correlation shell on the MR 5 -I-C model and the impacts of the extension of the MR 6 reference to the MR 8 reference using the active set 9h , on the values of hyperfine parameters obtained in the MR 6 -I-C 12h calculation.

			MR 5		MR 6		MR 8
		9h	12h	13h	9h	12h	9h	final
	E	-398.07547 -398.08254 -398.08329 -398.07655 -398.08363 -398.07952
	a l	5.13968	5.12937	5.12869	5.14143	5.13020	5.13748	5.12557
	a sd	1.08020	1.07860	1.07918	1.08205	1.08039	1.08447	1.08339
	a c	0.74528	0.85674	0.85825	0.80130	0.91908	0.82046	0.93976
	b	-2.05860	-2.05035	-2.04957	-2.05942	-2.05064	-2.05571 -2.04615

Table 8 .

 8 Comparison of theoretical and experimental A J hyperfine constants (in MHz) for the lowest multiplet of 33 S, 33 S -, 35 Cl and 37 Cl. Non-relativistic estimations (NR) computed from the final results of Tables 4, 5 and 7. Relativistic corrections estimated with a CI-RCI approach[15,19].

		33 S		33 S -	35 Cl		37 Cl
		A 2	A 1	A 3/2	A 1/2	A 3/2	A 1/2	A 3/2	A 1/2
	Non-relativistic	133.58	0.33 92.48 483.30 168.52 846.09 202.45 1016.46
	+rel. corrections 130.09 -4.62 91.43 496.28 167.52 872.05 201.25 1113.22
	Experiment			91.49(9) a	170.69 b		205.05 b
	a Reference [1].

b Reference

[27]

.

Table 9 .

 9 Nuclear electric quadrupole moments (Q, in barns) estimated with equation (21) for 33 S -, 35 Cl and 37 Cl. From those Q, we calculate the B J constants of 33 S lower multiplet with our value of b(S 3 P ). Non-relativistic estimations (NR) computed from the final values of Tables 4, 5 and 7. Relativistic corrections estimated with a CI-RCI approach[15,19].

			33 S		35 Cl	37 Cl
		Q	B 2 ( 33 S)	B 1 ( 33 S)	Q	Q
	Non-relativistic	-0.0655(6)	-31.49	15.74 -0.082265	-0.064833
	+rel. corrections	-0.0657(6)	-31.60	15.80 -0.081764	-0.064438
	Sundholm and Olsen a -0.0678(13)	-32.60	16.30 -0.08165(80) -0.06435(64)
	Alonso et al b				-0.0850(11)
	Yakobi et al c				-0.0811(12)