Jan Erik Linderberg

Yngve Ohrn
email: ohrn@qtp.ufl.edu

Yngve Öhrn

O n l y Propagators and their implications

Keywords: propagator, density, response propagator, density, response, Initiation

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Löwdin suggested in September 1961 that time was ripe for an attack on the dispersion energy contribution to the cohesive energy of molecular solids, in particular rare gas crystals, and assigned the task to JL who had just returned from a year with the Quantum Theory Project. The treatment of fluctuating dipole fields as the source of long range interactions was most readily tackled in terms of the concepts of propagators and associated elements of the field theoretical tool box. Decent results were found with moderately elaborate calculations [2].

The present authors were reunited at Uppsala in 1963 and were inspired by John Hubbard's paper on electron correlation in narrow energy bands [3] to apply his Green function decoupling procedure to the generalized Hückel model introduced by Pariser and Parr and by Pople [4]. Our results were exciting and were gracefully communicated by Coulson [5]. Additional results seemed to corroborate that we were onto to something useful [6].

Methods from field theory were not generally embraced in the molecular electronic structure community. Slater was particularly skeptical towards the techniques of second quantization that had become "stylish" [7] and Löwdin was concerned about the N-representability issue. There were efforts under way at Cambridge to further the use of second quantization, somewhat ironically published in a tribute to Slater [8].

McLachlan spearheaded the applications in molecular orbital theory [9] and his paper with Ball [10] remains a basic reference for time-dependent studies of electronic properties. We were not alone and continued our exploration of the options offered by second quantization, propagators and novel approximation methods. Both situations operate with a finite basis of spin orbitals U:{ u 1 (x), u 2 (x), …u M (x)} and the associated power set P(U), which has a one-to-one correspondence to the set of all Slater determinants that can be formed from U. The empty set is included i P(U) and represents the state without electrons, often called the vacuum. There are 2 M elements in the power set and they are either denoted by the list of the particular spin orbitals for a given element or by the occupation number of the individual basis orbitals, a M-digit binary number where each position corresponds to a spin orbital in U. Elements of the power set P(U) span a subspace of the full Fock space that can accommodate an arbitrary number of electrons.

Transformations or mappings between elements of the power set are composed of elementary creation (addition, a r † ) or annihilation (removal, a r ) operators for individual spin orbitals. A consistent algebra requires the anticommutation relations a r a s + a s a r = 0; a r † a s † + a s † a r † = 0; a r a s † + a s † a r = δ rs + S rs .

Overlap integrals S occur for most primitive bases and may be removed by linear transformations. General many electron operators occur as linear combinations of products, the hamiltonian, as an example, is H = Σ rs a r † h rs a s + ½Σ rr'ss' (rs|r's') a r † a r' † a s' a s .

Mulliken's notation is used for the Coulomb interaction between spin orbital densities 04/12/10 3 u r * (x)u s (x) and u r' * (x')u s' (x'). The formalism exposes the structural similarity between methods where the integrals, h rs and (rs|r's'), are derived from specified spin orbitals and proper integrations and methods where they are inferred from other sources. The evaluation of a matrix representation of the hamiltonian from the elements of the power set requires the standard rules as put forth by Löwdin and others [11].

The second quantization algebra offers an approach to avoid the explicit and a mean field theory obtains when the operator product a r' † a s' is replaced by its expectation value, the reduced density matrix γ s'r' , in the second form. Self-consistent fields It was shown above that a mean field model for an electronic system results from a replacement, in the operator equation of motion, of an operator product with its mean value: γ sr = 〈 a r † a s 〉. A second step is then to relate the expectation value to the field.

Hartree introduced the criterium of self-consistency and this remains in electronic structure theory as the Hartree-Fock scheme when proper exchange is accounted for. The Fock matrix with elements f rs = h rs + Σ r's' {(rs|r's')-(rs'|r's)}γ s'r' is hermitean and its eigenvalues determine the canonical spin orbitals. These are used with the build up principle to characterize the desired state. Neither spatial nor spin symmetry is explicitly invoked and the general solution is rarely desirable. It is self-consistent to assume spin invariance in a system with an even number of electrons and it is also consistent to assume spherical symmetry for an atomic or ionic system with "magic" number of electrons. Other cases will result in non-consistent situations. occupations. A proper ensemble was readily described in second quantization terms. Thus it holds that averages 〈 a r † a s 〉 = q r δ rs imply the ensemble operator

Γ = Π r [(1 -q r )a r a r † + q r a r † a r ]; Tr Γ = 1.
The standard Hartree-Fock state has occupation numbers q r either zero or one. The ensemble gives an average energy as a function of the occupation numbers, E =Tr ΓH = Σ r h rr q r +½Σ rs [ (rr|ss) -(rs|sr) ] q r q s, and optimization of the spin orbitals results in Fock equations with fractional occupation numbers, is so desired. Derivatives of the energy, ε r = ∂E/∂q r , are the orbital energies and eigenvalues of the Fock matrix. The ensemble operator is not an eigenoperator of the number operator N op = Σ r a r † a r and the dispersion comes to

〈 [ N op -< N op >] 2 〉 = Π r ( 1 -q r ) q r
showing that the ensemble has contributing of varying electron number, a grand canonical ensemble. Such a construct serves to define a suitable set of basis spin orbitals for situations when spatial and spin symmetry features are desirable. Kaijser demonstrated the usefulness of such a basis for the determination of transition moments and life times for the manifold of low lying states of the Ti II ion [13]. Viinikka [14] also applied the grand canonical ensemble construct to define a basis that performed well in the study of the multiplet structure of core hole states in transition metal ions. Recent developments in basis set design have found the ensemble average with fractional occupation numbers to have advantages [15].

The grand canonical as well as the canonical ensemble construct were used by Poul Jørgensen [16], one of many postdoctoral associates that have been exchanged 04/12/10 6 between the JL and YÖ groups over the years. He compared the detailsof the theoretical basis with that of the Slater construction of the Fock operator. He also applied [17] the grand canonical ensemble construction for a time-dependent Hartree-Fock treatment of the triplet-triplet absorption spectra of alternant hydrocarbons.

Detailed calculations on transition metals [18] showed that there was no indication that the orbital energy of the 4s-level should be below the 3d-level. It was interesting to note that a simple function could represent the dependence on the occupation number of the orbital energy: ε(q) = ε(0)[ 1 -q/q c ] 3/2 . The maximum occupation, q c , gives a slightly negative ion. This form can be compared to Jørgensen's differential ionization concept [19], Mulliken's electronegativity [20], and the concept of absolute hardness [21].

Calculations with fractional occupation numbers were also used to approximate excitation energies. Slater [22] showed that for an energy expression that is a continuous function of occupation numbers one may attempt to calculate energy differences from Taylor series expansions:

E(…q r -1…q s + 1…) -E(…q r …q s …) = ∂E/∂q s -∂E/∂q r + … while observing that the second order terms are eliminated when the derivatives are evaluated at the midpoint, ( q r -½, q s + ½), termed the transition state. Applications confirmed that this concept offered a viable alternative to the determination of separate self-consistent solutions to the two states.

Electron propagator approximations

Hubbard's model for narrow energy bands in solids [3]

: 〈 [ A j , A k † ] + 〉= δ jk ,〈 [ [ A j , H ], A k † ] + 〉= ε j δ jk } is
defined and one infers that a useful approximation for the propagators follows:

« A j ; A k † » E = δ jk / ( E -ε j ).
Further analysis of consistency requires that the expectation values that are used to define M and K are derived from the propagators, i. e. by a contour integral in the complex Eplane:

〈 A k † A j 〉 = (2πi) -1 ∫ C dE « A j ; A k † » E
Fulfillment of these conditions can rarely be realized beyond the Hartree-Fock level and it is necessary to exercise caution.

Numerous calculations have demonstrated that a moderate extension of the operator manifold provides excellent interpretations of photoelectron spectra, including shake-up and shake-off processes, albeit not fully self-consistent. program suite called GREENFUNC, which survived in bits and pieces for over a decade in electronic structure software around the world. Early publications [24,25,26] set the standard and were followed by many others from QTP. This development in the propagator effort is covered in another publication in this issue of Molecular Physics.

Polarization propagator approximations

Lindhard [27] was concerned with the stoppage of particles in matter and derived the response of an electron gas to an external electric field by means of a self-consistent determination of the induced effects in the gas. His wave length and frequency dependent dielectric function exhibited the collective plasmon excitations as well as single particle features. This was the first detailed application of the linearized time-dependent Hartree method and remains a landmark in electronic structure theory. Dirac had delineated the equation of motion for the reduced density matrix in the time domain [28] and applications to non-homogeneous systems [10,29] appeared as complements to the elaborate diagrammatic and formal perturbation theory expansions that were the rule in many-electron theory during quite a few years.

Knowledge of response functions such as generalized polarizabilities and dielectric functions offers a route to correlation features through the fluctuationdissipation theorem. Thus Nozières and Pines [30] expressed the Coulomb interaction energy in the electron gas through integrals over the dielectric function. Dispersion energy contributions to the cohesion in crystals could similarly be obtained by accommodating periodicity and exchange [2].

Much emphasis has been centered on the possibility of the direct determination of 04/12/10 9

Λ =〈 [ A, A † ] 〉 and Ω =〈 [ [ A, H ], A † ] 〉are such that Ω is a non-negative
hermitian form when the ground state/ensemble is stable towards distortions and an energy minimum. The proper diagonal expressions for Λ and Ω offer pairs of adjoint operators, A † and A, that can be normalized so that

1 =〈 [ [ A, H ], A † ] 〉=〈 [ [ A † , H ], A ] 〉; λ =〈 [ A, A † ] 〉> 0.
Eigenvalues λ are proportional to wave lengths of excitations created by A † and the associated propagator is « A; A † » E = λ 2 /( E λ -1 ). The λ spectrum is bounded from above by the inverse of the first excitation energy of the system but approximations in the matrix evaluations causes the bound to be approximate as well.

An attractive feature of the random phase type approximations is that transition moments and oscillator strengths can be calculated in either the dipole length or the dipole velocity form if the spin orbital basis is such that the length operator D = Σ d rs a r † a s and the velocity operator V = Σ v rs a r † a s are related by the commutation relation

iV = [ D, H ].
This holds for a complete basis and may be enforced in the Pariser-Parr-Pople model [31] and, occasionally, in general cases. ] ≠ 0, which has consequences for the choice of reference system orientation. Transition moments involving magnetic fields need operator representations that are consistent with translational symmetry [32].

Consistency implies that average values are determined from the propagators, vide infra, and can be achieved in simple situations [33]. The related quest for a explicit ground state representative resulted in the conclusion that the standard random phase approximation is consistent with an antisymmetrized geminal power state [34]. Further discussion of this form is presented in an another contribution to this issue.

Reflexions by old men

Momentous occurrences in the April quarter of 1968, among them the murders of Martin Luther King Jr. and Robert F. Kennedy, interspersed a very active scientific collaboration of the present authors at the Quantum Theory Project. We started a project that was to go on for a few years and resulted in "the little yellow book" [35] and were much concerned with the matters of fractional occupation numbers in variants of self-consistent theory.

Our paper on the underpinnings of the Pariser-Parr-Pople model [12] exposed the problem of total energy calculations from different but equivalent expressions. These difficulties arise from truncations in operator spaces and the lack of a proper variational functional beyond the Hartree-Fock approximation. Nozières's textbook [36] Density functional theory relies on the concept of a universal relation between the electron density and the total electronic energy. The precise functional has as yet eluded the practioners but quite satisfactory approaches are in use. This will also be dealt with by others in this issue. Time-dependent extensions lead to equations that are formally equivalent with the ones discussed above under the polarization propagator heading.

Several recent applications seem to deviate from the proper path by mixing elements such as exchange with the pure density functional formalism. The present authors perceive that electronic structure theory has reached a plateau where the next mountain range requires new ideas, concepts, and mathematics. The discretization of three-space by Gaussian basis sets was initiated by Boys and Preuss more than 50 years ago and the effect has been tremendous. We ask ourselves, however, about the essential information content in huge integral arrays and associated numerical processes that demand computational efforts that increase as some large power of the number of atoms in our systems. Most of the information is provided by very modest considerations and it is the small, albeit important, deviation that has necessitated the very large scale machinery. Löwdin maintained that theoretical advances could be as spectacular as the hardware improvement, it remains to be seen.

Among the formal tools that might have potential for substantial changes in the approach to electronic structure theory is the so called adiabatic connection.

E(ξ) = 〈 H(ξ) 〉; dE(ξ)/d ξ = 〈 dH(ξ)/d ξ 〉.
This holds for eigenstates of the hamiltonian and for variationally stable states and is known as the Hellmann-Feynman theorem [37]. The adiabatic concept allows the integration to give

E(ξ 1 ) -E(ξ 0 ) = ∫d ξ 〈 dH(ξ)/d ξ 〉
and this is useful when a suitable expression is available for the integrand. This has been realized in Hückel theory where formal charges and bond orders serve as derivatives [38].

Changes in atomic parameters or bonds could then be estimated and indications be drawn about reactivities, induced spin densities and similar properties.

Both Hellmann and Feynman were concerned with the forces between atoms and explored the energy variation with changes in nuclear positions within the Born-Oppenheimer picture. Their results are accordingly relevant for considerations based on the virial theorem. Slater [39] deduced that the kinetic energy in the ground state of a diatomic molecule at internuclear distance R equals T = -E(R) -R∂E(R)/∂R while the potential energy is V =2E(R) + R∂E(R)/∂R. The first relation expresses the total energy in terms of an integral over the expectation value os a one-electron operator that can be evaluated from the electron propagator. Succes of such a procedure depends on the consistency of the approximation of the propagator.

So is also the case when the fluctuation-dissipation theorem is applied to the evaluation of the electron interaction energy in perturbation theory. Löwdin suggested in September 1961 that time was ripe for an attack on the dispersion energy contribution to the cohesive energy of molecular solids, in particular rare gas crystals, and assigned the task to JL who had just returned from a year with the Quantum Theory Project. The treatment of fluctuating dipole fields as the source of long range interactions was most readily tackled in terms of the concepts of propagators and associated elements of the field theoretical tool box. Decent results were found with moderately elaborate calculations [2].

The present authors were reunited at Uppsala in 1963 and were inspired by John Hubbard's paper on electron correlation in narrow energy bands [3] to apply his Green function decoupling procedure to the generalized Hückel model introduced by Pariser and Parr and by Pople [4]. Our results were exciting and were gracefully communicated by Coulson [5]. Additional results seemed to corroborate that we were on to something useful [6].

Methods from field theory were not generally embraced in the molecular electronic structure community. Slater was particularly skeptical towards the techniques of second quantization that had become "stylish" [7] and Löwdin was concerned about the N-representability issue. There were efforts under way at Cambridge to further the use of second quantization, somewhat ironically published in a tribute to Slater [8].

McLachlan spearheaded the applications in molecular orbital theory [9] and his paper with Ball [10] remains a basic reference for time-dependent studies of electronic properties. We were not alone and continued our exploration of the options offered by second quantization, propagators and novel approximation methods. Transformations or mappings between elements of the power set are composed of elementary creation (addition, a r † ) or annihilation (removal, a r ) operators for individual spin orbitals. A consistent algebra requires the anticommutation relations a r a s + a s a r = 0; a r † a s † + a s † a r † = 0; a r a s † + a s † a r = δ rs + S rs .

Overlap integrals S occur for most primitive bases and may be removed by linear transformations. General many electron operators occur as linear combinations of products, the hamiltonian, as an example, is H = Σ rs a r † h rs a s + ½Σ rr'ss' (rs|r's') a r † a r' † a s' a s .

Mulliken's notation is used for the Coulomb interaction between spin orbital densities and a mean field theory obtains when the operator product a r' † a s' is replaced by its expectation value, the reduced density matrix γ s'r' , in the second form. The essential Σ-Π separation of the Pariser-Parr-Pople model appears rather directly in the second quantization formulation [12] and details the role of the Σframework as a dielectric medium which screens the interaction within the Π-system. A link is also established to the form of a Heisenberg spin-hamiltonian in a weak coupling limit.

Self-consistent fields

It was shown above that a mean field model for an electronic system results from a replacement, in the operator equation of motion, of an operator product with its mean value: γ sr =〈 a r † a s 〉. A second step is then to relate the expectation value to the field.

Hartree introduced the criterion of self-consistency and this remains in electronic structure theory as the Hartree-Fock scheme when proper exchange is accounted for. The Fock matrix with elements f rs = h rs + Σ r's' {(rs|r's')-(rs'|r's)}γ s'r' is hermitean and its eigenvalues determine the canonical spin orbitals. These are used with the build up principle to characterize the desired state. Neither spatial nor spin symmetry is explicitly invoked and the general solution is rarely desirable. It is self-consistent to assume spin invariance in a system with an even number of electrons and it is also consistent to assume spherical symmetry for an atomic or ionic system with "magic" number of electrons. Other cases will result in non-consistent situations.

Slater chose to evaluate the reduced density matrix elements from an ensemble average based on an open shell configuration with a fixed number of electrons and obtained a Fock matrix with spin orbitals appropriate for a central field. The open shell spin orbitals were useful in further applications in the solid state but could not be related to ionization processes. Applications to transition metal atoms uncovered another awkwardness. A starting central field would indicate that the 4s-shell was filled and the least bound spin orbitals were the 3d's. The field from this occupation would lead to the opposite and a consistent solution was not obtained. A remedial approach introduced fractional occupations and acceptable solutions. Slater's derivation was not based on an ensemble average but came directly from an expression that is valid only for integer occupations. A proper ensemble was readily described in second quantization terms.

Thus it holds that averages 〈 a r † a s 〉 = q r δ rs imply the ensemble operator

Γ = Π r [(1 -q r )a r a r † + q r a r † a r ]; Tr Γ = 1.
The standard Hartree-Fock state has occupation numbers q r either zero or one. The ensemble gives an average energy as a function of the occupation numbers, E =Tr ΓH = Σ r h rr q r +½Σ rs [ (rr|ss) -(rs|sr) ] q r q s, and optimization of the spin orbitals results in Fock equations with fractional occupation numbers, if so desired. Derivatives of the energy, ε r = ∂E/∂q r , are the orbital energies and eigenvalues of the Fock matrix. The ensemble operator is not an eigenoperator of the number operator N op = Σ r a r † a r and the dispersion comes to

〈 [ N op -< N op >] 2 〉 = Π r ( 1 -q r ) q r
showing that the ensemble has contributions of varying electron number, a grand canonical ensemble. Such a construct serves to define a suitable set of basis spin orbitals for situations when spatial and spin symmetry features are desirable. Kaijser demonstrated the usefulness of such a basis for the determination of transition moments and life times for the manifold of low lying states of the Ti II ion [13]. Viinikka [14] also applied the grand canonical ensemble construct to define a basis that performed well in the study of the multiplet structure of core hole states in transition metal ions. Recent developments in basis set design have found the ensemble average with fractional occupation numbers to have advantages [15].

The grand canonical as well as the canonical ensemble construct were used by Poul Jørgensen [16], one of many postdoctoral associates that have been exchanged between the JL and YÖ groups over the years. He compared the details of the theoretical basis with that of the Slater construction of the Fock operator. He also applied [17] the grand canonical ensemble construction for a time-dependent Hartree-Fock treatment of Detailed calculations on transition metals [18] showed that there was no indication that the orbital energy of the 4s-level should be below the 3d-level. It was interesting to note that a simple function could represent the dependence on the occupation number of the orbital energy: ε(q) = ε(0)[ 1q/q c ] 3/2 . The maximum occupation, q c , gives a slightly negative ion. This form can be compared to Jørgensen's differential ionization concept [19], Mulliken's electronegativity [20], and the concept of absolute hardness [21].

Calculations with fractional occupation numbers were also used to approximate excitation energies. Slater [22] showed that for an energy expression that is a continuous function of occupation numbers one may attempt to calculate energy differences from Taylor series expansions:

E(…q r -1…q s + 1…) -E(…q r …q s …) = ∂E/∂q s -∂E/∂q r + …
while observing that the second order terms are eliminated when the derivatives are evaluated at the midpoint, ( q r -½, q s + ½), termed the transition state. Applications confirmed that this concept offered a viable alternative to the determination of separate self-consistent solutions to the two states.

Electron propagator approximations

Hubbard's model for narrow energy bands in solids [3] led him to a decoupling of the equations of motion for the electron propagator or Green function G rs (E) = <<a r ;a s † >> E .

His approximation was reformulated by means of a matrix representation in an extended operator basis [23]. Later efforts were developed in operator spaces spanned by general

:〈 [ A j , A k † ] + 〉= δ jk ,〈 [ [ A j , H ], A k † ] + 〉= ε j δ jk } is
defined and one infers that a useful approximation for the propagators follows:

« A j ; A k † » E = δ jk / ( E -ε j ).
Further analysis of consistency requires that the expectation values that are used to define M and K are derived from the propagators, i. e. by a contour integral in the complex Eplane:

〈 A k † A j 〉 = (2πi) -1 ∫ C dE « A j ; A k † » E
Fulfillment of these conditions can rarely be realized beyond the Hartree-Fock level and it is necessary to exercise caution.

Numerous calculations have demonstrated that a moderate extension of the operator manifold provides excellent interpretations of photoelectron spectra, including shake-up and shake-off processes, albeit not fully self-consistent.

Applications of electron propagator theory became possible with the seminal work of George Purvis, who joined the group of YÖ at QTP in 1970. He developed a program suite called GREENFUNC, which survived in bits and pieces for over a decade in electronic structure software around the world. Early publications [24,25,26] set the standard and were followed by many others from QTP. This development in the propagator effort is covered in another publication in this issue of Molecular Physics. Polarization propagator approximations Lindhard [27] was concerned with the stoppage of particles in matter and derived the response of an electron gas to an external electric field by means of a self-consistent determination of the induced effects in the gas. His wavelength and frequency dependent dielectric function exhibited the collective plasmon excitations as well as single particle features. This was the first detailed application of the linearized time-dependent Hartree method and remains a landmark in electronic structure theory. Dirac had delineated the equation of motion for the reduced density matrix in the time domain [28] and applications to non-homogeneous systems [10,29] appeared as complements to the elaborate diagrammatic and formal perturbation theory expansions that were the rule in many-electron theory during quite a few years.

Knowledge of response functions such as generalized polarizabilities and dielectric functions offers a route to correlation features through the fluctuationdissipation theorem. Thus Nozières and Pines [30] expressed the Coulomb interaction energy in the electron gas through integrals over the dielectric function. Dispersion energy contributions to the cohesion in crystals could similarly be obtained by accommodating periodicity and exchange [2].

Much emphasis has been centered on the possibility of the direct determination of excitation energies and transition moments from the poles and residues of the polarization propagator. The basic approximation has several flavors: linearized time-dependent Hartree-Fock, Random Phase Approximation, Equation of Motion method to mention the most common ones. Their formal equivalence derives from an operator algebra adapted to be similar to the one for the electron propagator. A number conserving operator is

1 = 〈 [ [ A, H ], A † ] 〉 = 〈 [ [ A † , H ], A ] 〉; λ =〈 [ A, A † ] 〉 > 0.
Eigenvalues λ are proportional to wave lengths of excitations created by A † and the associated propagator is « A; A † » E = λ 2 /( E λ -1 ). The λ spectrum is bounded from above by the inverse of the first excitation energy of the system but approximations in the matrix evaluations causes the bound to be approximate as well.

An attractive feature of the random phase type approximations is that transition moments and oscillator strengths can be calculated in either the dipole length or the dipole velocity form if the spin orbital basis is such that the length operator D = Σ d rs a r † a s and the velocity operator V = Σ v rs a r † a s are related by the commutation relation

iV = [ D, H ].
This holds for a complete basis and may be enforced in the Pariser-Parr-Pople model [31] and, occasionally, in general cases. Limited basis sets may destroy fundamental commutation rules, e. g. [ D x , D y ] ≠ 0, which has consequences for the choice of reference system orientation. Transition moments involving magnetic fields need operator representations that are consistent with translational symmetry [32].

Consistency implies that average values are determined from the propagators, vide infra, and can be achieved in simple situations [33]. The related quest for an explicit ground state representative resulted in the conclusion that the standard random phase approximation is consistent with an antisymmetrized geminal power state [34]. Further discussion of this form is presented in another contribution to this issue.

Reflexions by old men

Momentous occurrences in the April quarter of 1968, among them the murders of Martin Luther King Jr. and Robert F. Kennedy, interspersed a very active scientific collaboration of the present authors at the Quantum Theory Project. We started a project that was to go on for a few years and resulted in "the little yellow book" [35] and were much concerned with the matters of fractional occupation numbers in variants of self-consistent theory.

Our paper on the underpinnings of the Pariser-Parr-Pople model [12] exposed the problem of total energy calculations from different but equivalent expressions. These difficulties arise from truncations in operator spaces and the lack of a proper variational functional beyond the Hartree-Fock approximation. Nozières's textbook [36] deals with higher order approximations and the ensuing Ward identities. No entirely satisfactory model has as yet been devised and the applications of propagators to total energy questions cannot maintain a variational bound to the results. This is similar to the variants of the coupled-cluster methods but these are being developed to offer reliable estimates to total energies as exemplified in adjoining contributions to this issue.

Density functional theory relies on the concept of a universal relation between the electron density and the total electronic energy. The precise functional has as yet eluded the practioners but quite satisfactory approaches are in use. This will also be dealt with by others in this issue. Time-dependent extensions lead to equations that are formally equivalent with the ones discussed above under the polarization propagator heading. Several recent applications seem to deviate from the proper path by mixing elements such as exchange with the pure density functional formalism. The present authors perceive that electronic structure theory has reached a plateau where the next mountain range requires new ideas, concepts, and mathematics. The discretization of three-space by Gaussian basis sets was initiated by Boys and Preuss more than 50 years ago and the effect has been tremendous. We ask ourselves, however, about the essential information content in huge integral arrays and associated numerical processes that demand computational efforts that increase as some large power of the number of atoms in our systems. Most of the information is provided by very modest considerations and it is the small, albeit important, deviation that has necessitated the very large-scale machinery. Löwdin maintained that theoretical advances could be as spectacular as the hardware improvement, it remains to be seen.

Among the formal tools that might have potential for substantial changes in the approach to electronic structure theory is the so-called adiabatic connection. It is based on the equality of the derivative of the expectation value of the hamiltonian with respect to a parameter and the expectation of the derivative of the operator itself:

E(ξ) = 〈 H(ξ) 〉; dE(ξ)/d ξ = 〈 dH(ξ)/d ξ 〉.
This holds for eigenstates of the hamiltonian and for variationally stable states and is known as the Hellmann-Feynman theorem [37]. The adiabatic concept allows the integration to give E(ξ 1 ) -E(ξ 0 ) = ∫d ξ 〈 dH(ξ)/d ξ 〉

and is useful when a suitable expression is available for the integrand. This has been realized in Hückel theory where formal charges and bond orders serve as derivatives [38]. Changes in atomic parameters or bonds could then be estimated and indications be drawn about reactivities, induced spin densities and similar properties.

Both Hellmann and Feynman were concerned with the forces between atoms and explored the energy variation with changes in nuclear positions within the Born-Oppenheimer picture. Their results are accordingly relevant for considerations based on the virial theorem. Slater [39] deduced that the kinetic energy in the ground state of a diatomic molecule at an internuclear distance R equals T = -E(R) -R∂E(R)/∂R while the potential energy is V =2E(R) + R∂E(R)/∂R. The first relation expresses the total energy in terms of an integral over the expectation value of a one-electron operator that can be evaluated from the electron propagator. Success of such a procedure depends on the consistency of the approximation of the propagator.

So is also the case when the fluctuation-dissipation theorem is applied to the evaluation of the electron interaction energy in perturbation theory. The premise is that expectation values 〈a r † a t † a u a s 〉 =δ st 〈 a r † a u 〉 +Σ n 〈0′a r † a s ′n〉〈n′a t † a u ′0〉 are obtainable from the corresponding response function or propagator and that these are calculated as functions of a formal interaction strength parameter. The integration from zero to unit strength offers an estimate of the total energy contribution. A simple application to the linear chain Hubbard model gives a modest improvement over the basic mean field approximation [40].

Propagators have been part of these authors' lives for the better part of the fifty years since the Quantum Theory Project was initiated. Their emphasis on processes rather than stationary states appeals to the physico-chemical realization that knowledge is obtained by probing the response of the material world. They do connect the pure state

  schemes emanating from Boys's, Roothaan's and Hall's early development represent the

  Exponential operators provide compact representations of unitary transformations of the basis and unusual symmetries, e.g. particle-hole relations in alternant hydrocarbon molecules in the Pariser-Parr-Pople model, can be exposed. The developments of the coupled-cluster variants in current usage would probably have been more awkward in direct configuration space formulations. The essential Σ Π  separation of the Pariser-Parr-Pople model appears ratherdirectly in the second quantization formulation[12] and details the role of the Σframework as a dielectric medium which screens the interaction within the Π-system. A link is also established to the form of a Heisenberg spin-hamiltonian in a

Page 9 of 31 URL

 31 transition moments from the poles and residues of the polarization propagator. The basic approximation has several flavors: linearized time-dependent Hartree-Fock, Random Phase Approximation, Equation of Motion method to mention the most common ones. Their formal equivalence derives from an operator algebra adapted to be similar to the one for the electron propagator. A number conserving operator is considered of the form A † = Σ a r † [ c rs + c rtus a t † a u + …] a s and the sesquilinear forms

  theory through the works by Gell-Mann and Brueckner and by Hubbard in the mid 50's.Thus he wanted to establish the connection to the configuration space methods that he was so familiar with and to this end he assigned one of us (JL) to give a seminar to the Quantum Chemistry Group at Uppsala on second quantization in the fall of 1957. The sources were the classical papers by Born and Jordan, the detailed treatment by Fock and the treatises by Dirac and Corson. Löwdin followed up with formal field theory but it was when Stig Lundqvist returned from a sabbatical with Brueckner that the concepts and techniques became more familiar topics at the Group. The lecture notes taken by Lars Hedin was the basis for the Technical Report from the Group [1] and Lundqvist's lectures at the first Winter Institute at Gainesville in December 1960 and Sanibel Island in January 1961. The authors spent a brief period together at Gainesville in early 1961 when YÖ began a two-year stay at the University of Florida.

  Exponential operators provide compact representations of unitary transformations of the basis and unusual symmetries, e.g. particle-hole relations in alternant hydrocarbon molecules in the Pariser-Parr-Pople model, can be exposed. The developments of the coupled-cluster variants in current usage would probably have been more awkward in direct configuration space formulations.

Page 22 of 31 URL

 31 triplet absorption spectra of alternant hydrocarbons.

Page 7 of 31 URL: http://mc.manuscriptcentral.com/tandf/tmph

	Molecular Physics
	equations of motion for the electron propagator or Green function G rs (E) = <<a r ;a s † >> E .
	His approximation was reformulated by means of a matrix representation in an extended
	operator basis [23]. Later efforts were developed in operator spaces spanned by general
	operator products. Useful forms are A † = Σ a r † (c r + c rst a s † a t + …) , they define two
	sequilinear forms, M =〈 [ A, A † ] + 〉 and K =〈 [ [ A, H ], A † ] + 〉, which may be
	brought to diagonal form. The metric M is hermitian and non-negative while the effective F o hamiltonian K is hermitian. Linear dependencies may occur from an initial choice of
	r
	operators and reflect properties of the state or ensemble used for the averages. A P r e e canonical set of operators { A j †
	R e v i e w
	O n
		l y
	led him to a decoupling of the
	04/12/10	7

tmph Molecular Physics By Jan Linderberg at 11:02 am, Apr 12, 2010

  The premise is that expectation values 〈a r † a t † a u a s 〉=δ st 〈 a r † a u 〉+Σ n 〈0 a calculated as functions of a formal interaction strength parameter. The integration from zero to unit strength offers an estimate of the total energy contribution. A simple application to the linear chain Hubbard model gives a modest improvement over the basic mean field approximation[40].Propagators have been part of these authors lives for the better part of the fifty years since the Quantum Theory Project was initiated. Their emphasis on processes rather than stationary states appeals to the physico-chemical realization that knowledge is obtained by probing the response of the material world. They do connect the pure stateNearly fifty years of collaboration on the development of formulations of the many-electron theory for molecular, atomic and condensed matter in terms of propagators is surveyed.

		Page 16 of 31
	Propagators and their implications
	Yngve Öhrn and Jan Linderberg
	Quantum Theory Project, Departments of Chemistry and Physics, University of Florida,
	Gainesville, Florida U. S. A.
	YÖ: ohrn@qtp.ufl.edu, JL: jan@chem.au.dk
	F o F o F o o F (Received 20 July 2005; final version received 17 August 2006)
	r r r r
	P P P P
	e e r e e r e e r e e r
	R e v i e w R e v i e w R e v i e w R e v i e w
	O n O n O n O n
		l y l y l y l y
	⎢ r † a s n ⎢ 〉〈n a ⎢ t † a u 0 ⎢ 〉are
	obtainable from the corresponding response function or propagator and that these are
	04/12/10 04/12/10 August 2, 2010	13 16 1

URL: http://mc.manuscriptcentral.com/tandf/Keywords: propagator; density; response; Initiation Field theoretical methods were familiar to Per-Olov Löwdin from his early work with relativity and electromagnetic theory. He was alerted to its impact in many electron

  The formalism exposes the structural similarity between methods where the integrals, h rs and (rs|r's'), are derived from specified spin orbitals and proper integrations and methods where they are inferred from other sources. The evaluation of a matrix representation of the hamiltonian from the elements of the power set requires the standard rules as put forth by Löwdin and others[11].

	Page 19 of 31	Molecular Physics
	u r * (x)u s (x) and u r'
	The second quantization algebra offers an approach to avoid the explicit F o construction of a matrix representative of the hamiltonian and still be able to infer
	r
	essential properties. It holds, in an orthonormal basis, that P r e e [a
		R e v i e w
		O n
			l y
	August 2, 2010	4

URL: http://mc.manuscriptcentral.com/tandf/tmph * (x')u s' (x'). r ,H] = Σ s h rs a s + Σ r'ss' (rs|r's')a r' † a s' a s ; [ [a r ,H], a s † ] + = h rs + Σ r's' {(rs|r's')-(rs'|r's)}a r' † a s'

  Useful forms are A † = Σ a r † (c r + c rst a s † a t + …) , they define two sequilinear forms, M =〈 [ A, A † ] + 〉 and K =〈 [ [ A, H ], A † ] + 〉, which may be brought to diagonal form. The metric M is hermitian and non-negative while the effective hamiltonian K is hermitian. Linear dependencies may occur from an initial choice of operators and reflect properties of the state or ensemble used for the averages. A canonical set of operators { A j †

	Page 23 of 31	Molecular Physics
	operator products.
	F o
	r
	P
	e e r
		R e v i e w
		O n
			l y
	August 2, 2010	8

URL: http://mc.manuscriptcentral.com/tandf/tmph

Page 24 of 31 URL: http://mc.manuscriptcentral.com/tandf/tmph

	Molecular Physics
	F o
	r
	P
	e e r
	R e v i e w
	O n
		l y
	August 2, 2010	9

Page 25 of 31 URL: http://mc.manuscriptcentral.com/tandf/tmph

  = Σ a r † [ c rs + c rtus a t † a u + …] a s and the sesquilinear formsΛ =〈 [ A, A † ] 〉 and Ω =〈 [ [ A, H ], A † ] 〉 are such that Ω is a non-negativehermitian form when the ground state/ensemble is stable towards distortions and an energy minimum. The proper diagonal expressions for Λ and Ω offer pairs of adjoint operators, A † and A, that can be normalized so that

	Molecular Physics
	considered of the form A †
	F o
	r
	P
	e e r
	R e v i e w
	O n
		l y
	August 2, 2010	10

URL: http://mc.manuscriptcentral.com/tandf/tmph

URL: http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics

formulation with ensemble structures and density matrices. Jan Linderberg (*October 27, 1934) completed the requirements for the degree of Doctor of Philosophy at Uppsala University in May 1964. He was appointed to the chair of theoretical chemistry at Aarhus University in 1968 and retired in 2003. He holds courtesy appointments at the Department of Chemistry at the University of Florida and at the Department of Chemistry at the University of Utah. Some 100 scientific papers and a couple of books have emanated through his efforts. Yngve Öhrn (*June 11, 1934) completed the requirements for the degree of Doctor of Philosophy at Uppsala University in May 1966. He was appointed to the faculty of the Department of Chemistry at the University of Florida in 1966, served as chair 1977-83, director of the Quantum Theory Project 1983-98, and entered emeritus status 2008. Editor-in-chief of International Journal of Quantum Chemistry. Several books and some 100 scientific papers have resulted from his efforts. formulation with ensemble structures and density matrices. Jan Linderberg (*October 27, 1934) completed the requirements for the degree of Doctor of Philosophy at Uppsala University in May 1964. He was appointed to the chair of theoretical chemistry at Aarhus University in 1968 and retired in 2003. He holds courtesy appointments at the Department of Chemistry at the University of Florida and at the Department of Chemistry at the University of Utah. Some 100 scientific papers and a couple of books have emanated through his efforts. Yngve Öhrn (*June 11, 1934) completed the requirements for the degree of Doctor of Philosophy at Uppsala University in May 1966. He was appointed to the faculty of the Department of Chemistry at the University of Florida in 1966, served as chair 1977-83, director of the Quantum Theory Project 1983-98, and entered emeritus status 2008. Editor-in-chief of International Journal of Quantum Chemistry. Several books and some 100 scientific papers have resulted from his efforts.


Nina-Katri J Gustafsson
email: nina-katri.gustafsson@sorad.su.se

-Katri

Mats Ramstedt

Changes in alcohol-related harm in Sweden after increasing alcohol import quotas and a Danish tax decrease -an interrupted time-series analysis for 2000-2007

Keywords: alcohol-related harm, register data, tax change, traveller allowances quotas

Changes in alcohol-related harm in Sweden after increasing alcohol import quotas and a Danish tax decrease -an interrupted time-series analysis for 2000-2007

Nina-Katri Jessica Gustafsson, Mats Ramstedt

When Denmark decreased their tax on spirits and Sweden shortly thereafter abolished its private import quotas on alcohol to unlimited levels if it was for private consumption, it was assumed that people in the southern parts of Sweden would increase their imports of alcohol. Because private imports already corresponded to about 50% of the total alcohol consumption among Swedes in the south, it was expected that per capita alcohol consumption in this region would increase, and thus also the alcohol-related harms. The present results in this study did not allow for any definite conclusion regarding whether alcohol-related harm increased in southern Sweden. Using an interrupted time-series analysis, an increase in alcohol poisonings was found in relation to the repeal of the alcohol import quotas, but not in relation to the Danish tax decrease. Similarly, no increase was found in violent assaults or drunk driving. It is concluded that the results raise important questions about the association between changes in availability and alcohol-related harms in Sweden today. Methods. Interrupted time-series analysis was performed with monthly data on cases of hospitalization due to acute alcohol poisoning, number of reported violent assaults and drunk driving for the years 2000-2007 in southern Sweden using the northern parts of Sweden as a control and additionally controlling for two earlier major changes in quotas. Results. The findings were not consistent with respect to whether alcohol-related harm increased in southern Sweden after the decrease in Danish spirits tax and the increase in Swedish alcohol import quotas. On the one hand, an increase in acute alcohol poisonings was found, in particular in the age group 50-69 years old: on the other hand, no increase was found in violent assaults and drunk driving. Conclusions. The present results raise important questions about the association between changes in availability and alcohol-related harms.

Introduction

Few countries have traditionally regarded alcohol as a public health issue to the same extent as the Nordic countries [START_REF] Room | The idea of alcohol policy. NAT (Nordisk alcohol-& narkotikatidskrift)[END_REF]. As a result, the alcohol policies of particularly Sweden, Finland and Norway have been very restrictive, the goal being to limit not only alcohol consumption among problem drinkers, but also the overall level of drinking. The most important tools for keeping alcohol consumption (total volume consumed) at a low level have been extensive

restrictions on physical availability and high alcohol prices [START_REF] Babor | Alcohol: No ordinary commodity: research and public policy[END_REF]. To make these tools efficient, Sweden has also applied strict restrictions on the amount of alcohol that can be brought back privately from abroad, but these restrictions have been repealed since Sweden joined the European Union (EU) in 1995 (Table 1).

[TABLE 1 ABOUT HERE] With higher import quotas, raising alcohol taxes was no longer regarded as efficient owing to the risk of substitution effects, i.e. that more alcohol would be purchased abroad instead.

Further, availability was increasing in practice, especially among residents in southern Sweden who live close to low-tax countries like Germany and Denmark, from which most of the privately imported alcohol comes [START_REF] Ramstedt | Tal om alkohol 2008: En statistisk årsrapport från Monitor-projektet[END_REF]. Earlier studies have also shown that especially southern Sweden has a high rate of private imports and that such imports account for almost half of all alcohol consumed in this part of Sweden [START_REF] Ramstedt | Tal om alkohol 2008: En statistisk årsrapport från Monitor-projektet[END_REF]. Therefore, it was assumed that the amount of alcohol consumed in this region would increase when Denmark, on October 1 st 2003, decreased its tax on spirits by 45% -which was equivalent to a 25% decrease in the price of the cheaper brands of spirits -and when Sweden shortly thereafter, on January 1 st 2004, increased its alcohol import quotas to the levels required by the EU (i.e. indicative levels of when import can be considered to be for one's own use). Thus, although the decrease in price did not actually take place in Sweden, the already high market share for privately A recent study based on large repeated cross-sectional surveys was able to establish that traveller's imports did increase substantially in southern Sweden in 2004 [START_REF] Ramstedt | Increasing travellers' allowances in Sweden -how did it affect travellers' imports and Systembolaget's sales? NAT (Nordisk alcohol-& narkotikatidskrift)[END_REF]. However, surveys measuring actual drinking did not find any increase in alcohol consumption or alcohol-related social problems. For instance, a comparison between 2003 and 2004 did not show any increase in volume of alcohol consumed in any region in Sweden, neither in the repeated cross-sectional samples for the total population nor in the longitudinal data [START_REF] Mäkelä | Changes in volume of drinking after changes in alcohol taxes and traveller's allowances: results from a panel study[END_REF]. A study using a longer follow-up period did not change this conclusion, and no change was found in southern Sweden when data up until 2006 were analysed [START_REF] Gustafsson | Alcohol consumption in southern Sweden after major decreases in Danish spirits tax and increases in traveller's quotas[END_REF]. In accordance with the results on consumption, alcohol-related social problems in southern Sweden did not increase as expected [START_REF] Bloomfield | Changes in alcohol-related problems after alcohol policy changes in Denmark, Finland and Sweden[END_REF]. Thus, the results from population surveys did not verify that alcohol consumption increased following these changes in alcohol policy.

As these unexpected results are based on survey data, with their limitations (i.e. in reaching heavy drinkers), further analyses on data less affected by such problems (i.e. various forms of register data) are warranted. In the present study, the number of hospital admissions for acute alcohol poisonings and the number of police-reported violent assaults and drunk-driving incidents were used to assess the impact of lower taxes in Denmark and higher Swedish import quotas. Analyses from other Nordic countries suggest that an analysis of this kind of register data can give new insights compared with studies based on survey data. In Finland, the raised quotas for private alcohol import were shown to have a great effect on heavier drinkers [START_REF] Mustonen | People are buying and importing more alcohol than ever before. Where is it all going?[END_REF][10][START_REF] Mäkelä | Halvat huvit -millaiset seuraukset? [Cheap pleasures? -what are the consequences?[END_REF], among whom there was a striking increase in alcohol-related deaths between 2003 and 2004. Herttua, Mäkelä and Martikainen [START_REF] Herttua | Changes in alcohol-related mortality and its socioeconomic differences after a large reduction in alcohol prices: A natural experiment based on register data[END_REF] studied the effect of the reduction in price following the tax decrease in Finland in 2004 and also found an increase in 5

F o r R e v i e w O n l y
alcohol-related mortality (mainly for chronic causes and particularly for liver diseases) after the change. In a study using interrupted time-series analysis, Koski et al. [START_REF] Koski | Alcohol tax cuts and increase in alcohol-positive sudden deaths -a time-series intervention analysis[END_REF] found an increase in alcohol-positive sudden deaths related to the tax cut on alcohol. Other sources of information in Finland have supported these findings as well, stating that substance abusers have been shown to be in poorer health after the changes [START_REF] Mustonen | People are buying and importing more alcohol than ever before. Where is it all going?[END_REF][START_REF] Murto | Alkoholihaittojen torjunta vaatii nopeita toimia [The prevention of alcohol-related harms demands fast interventions[END_REF] and that intoxicated persons have been taken into custody more often [START_REF] Mustonen | People are buying and importing more alcohol than ever before. Where is it all going?[END_REF][START_REF] Noponen | Poliisin suorittamat päihtyneiden säilöön ja säilöönottojen asiakaskunta Helsingissä[END_REF]. As for Denmark, time-series analysis performed on violent assaults and hospitalizations for acute alcohol intoxication January 2000 through December 2005 showed an increase in hospitalizations due to acute alcohol intoxication after the Danish tax reduction in 2003 among individuals 15 years and younger, but did not find such an increase in the total population or for violent assaults [START_REF] Bloomfield | Changes in alcohol-related harm after alcohol policy changes in Denmark[END_REF].

For Sweden, no study using register data has yet been carried out to examine the effects on alcohol-related harm of the changes in private import quotas or the reduction in the Danish alcohol tax in 2003. According to a study of the relatively modest increase in travellers' allowances of wine and beer in 1995, no increases in alcohol-related mortality, traffic accidents or reported criminal violence were found [START_REF] Norström | Cross-border trading of alcohol in southern Sweden -Substitution or Addition? In: Holder HD. Sweden and the European Union: Changes in national alcohol policy and their consequences[END_REF].

The aim of the present paper is thus to study whether the abolishment of alcohol import quotas in 2004 and the Danish reduction in spirits tax three months earlier had any effect on alcohol-related harm in southern Sweden, using register data on alcohol poisonings, drunk driving and assaults as indicators of alcohol-related harm (more details are presented in the method section). As the hospitalization data on alcohol poisonings are available by age, we will also test the possibility of age-specific effects in order to replicate a similar study based on Danish data [START_REF] Bloomfield | Changes in alcohol-related harm after alcohol policy changes in Denmark[END_REF]. Our starting point is that the effect may be stronger in age groups known to travel and import the most alcohol, which are mainly middle aged people [START_REF] Svensson | Travellers' alcohol imports to Sweden at the beginning of the 21 st Century. NAT (Nordisk alcohol-& narkotikatidskrift)[END_REF].

Furthermore, a recent analysis of the impact of the increasing allowances and tax reduction in

6 F o r R e v i e w O n l y
Finland suggested that the increase in alcohol-related mortality was basically seen in the age group 50-69 years [START_REF] Herttua | The Effects of the 2004 Reduction in the Price of Alcohol on Alcohol-Related Harm in Finland: A Natural Experiment Based on Register Data[END_REF].

7

F o r R e v i e w O n l y

Data and method

The selection of alcohol poisoning, drunk driving and assault as indicators of increasing harm in the analysis was justified by the following arguments. Naturally, all three indicators have a strong link to high alcohol consumption. Alcohol poisonings and drunk driving are by definition alcohol-related, and it is well established that a high proportion of perpetrators [20,[START_REF] Brå | Om utsatthet, trygghet och förtroende Rapport 2010:2[END_REF] as well as victims of violent assaults [START_REF] Brå | Om utsatthet, trygghet och förtroende Rapport 2010:2[END_REF][START_REF] Pernanen | Proportions of crimes associated with alcohol and other drugs in Canada[END_REF] are intoxicated. In addition, all three indicators have been found to have a positive association with sales at Systembolaget [the Swedish Alcohol Retail Monopoly stores] as well as with estimates of travellers' alcohol imports in southern Sweden [START_REF] Norström | Unregistered alcohol consumption and alcohol-related harm in Sweden, 2001-2005. NAD (Nordiska nämnden för alcohol-och drogforskning)[END_REF]. Furthermore, data are available on a monthly basis, thus providing a sufficient number of data points for a meaningful time-series analysis as well as allowing us to come as close as possible in time to the specific changes. It was also considered important to have indicators that respond rapidly to an increase in drinking, thus ensuring that the problem of lagged effects will play a minor role.

Alcohol poisonings were measured as hospitalizations in which alcohol poisoning was the main or secondary diagnosis. These data are recorded in the Swedish Hospital Discharge

Records and cover all public in-patient care in Sweden (for more information on this register, see http://www.sos.se/epc/english/ParEng.htm). The validity of the data is considered to be high, and the quality is continuously being evaluated. The selection of ICD codes (10 th revision) that represent a case of alcohol poisoning was guided by the criteria set up by the Swedish Board of Health and Welfare and includes "Acute intoxication" (F100), "Harmful use" (F101) and "Toxic effect of alcohol" (T51).

Statistics on alcohol-related crimes were collected from the Swedish National Council for do not include driving while under the influence of drugs. It should be noted that some of these crimes do not lead to a sentence; this share accounts for only a few percentage points each year. As noted in the descriptive figure for violence (Figure 2), there was a peak in June-July 2001 in southern Sweden that was caused by riots in Gothenburg (southern Sweden) in relation to an EU meeting at this time. The series was therefore corrected using the linear interpolation command in SPSS 16.0 for these months.

Monthly data covering the period January 2000 until December 2007 were analysed using interrupted time-series analysis, more specifically the technique developed by Box and Jenkins [START_REF] Box | Time-series analysis: Forecasting and control[END_REF], often referred to as ARIMA models, in this case ARIMA impact analyses [START_REF] Mcdowall | Interrupted time-series analysis, Sage University Paper Series on Quantitative Applications in the Social Sciences[END_REF].

Intervention components (dummy variables) were constructed to test whether there was any impact of the two interventions that could not be accounted for by normal fluctuations and long-term trends within the harm series. The first dummy took the value 0 before the tax change and 1 afterwards, and similarly the second dummy took the value 0 before the quota change and 1 afterwards. The final model controlled for changes in harm in northern Sweden as well as for two earlier changes in private import quotas of alcohol on January 1 st 2002 and 2003 (see Table 1) using dummies computed as for the other quota dummy.

In ARIMA modelling, data need to be stationary, which was assessed using the Dickey-Fuller test. Additionally long-term trends should be removed prior to the analyses. Most series showed seasonal peaking (every 12 th lag) with gradual attenuation, which indicates that the series requires seasonal differencing for stationarity [START_REF] Yaffee | Time Series Analysis and Forecasting With applications of SAS and SPSS[END_REF]. With a seasonal differencing of Y t , defined as Y t -Y t-12 , the relationship between the changes in the same month but during the following years are analysed rather than the relationship between the raw series Y t and Y t+1 , one month compared to the following month during the same year. Further, the method requires that the noise term (including other etiological factors) be allowed to have a temporal means that there is no structure of autocorrelation in the noise term. This was tested using the Box-Ljung Q-test [START_REF] Ljung | Measure of Lack of Fit in Time-Series Models[END_REF].

Other factors, in addition to the Danish tax cut and the increasing import quotas, are likely to affect both alcohol consumption and harm rates in southern Sweden, i.e. general trends in purchasing power and travelling habits in Sweden. There may also exist national trends in police enforcement and health policies that affected the selected harm indicators. Therefore, we used corresponding harm rates in northern Sweden as controls in our analysis, the assumption being that the long distance to the Danish and German borders would mean little effect of the reforms and that we would still capture the impact of other relevant factors. This idea was supported in separate analyses by site (without using a control site), where no effects of the interventions were found for northern Sweden (results not shown). It should be noted, however, that the findings for southern Sweden were little affected by the inclusion of northern Sweden as a control in the analysis and that the main results remained without this control variable (analysis not reported here).

The following specification was used in the final model:

Results

The graphs (Figure 1-3) reveal an increase in alcohol poisonings as well as in the number of reported violent crimes in both northern and southern Sweden, but no obvious increase in the number of drunk driving cases for the period January 2000 until December 2007. With respect to alcohol poisonings, the development during the pre-intervention period was similar in both regions, but between 2003 and 2006, the rates increased more in southern Sweden than in the northern regions. In 2007, however, the rates were once again more similar.

[FIGURE 1 ABOUT HERE]

[FIGURE 2 ABOUT HERE]

[FIGURE 3 ABOUT HERE]

To test whether the two interventions had any effect on these kinds of harm, a series of interrupted time-series analyses were performed. The findings from the final model controlling for harm in northern Sweden and earlier quota changes are presented in Table 2.

The only effect revealed in all these ARIMA models was the effect of the traveller's allowance change in 2004 on alcohol poisonings in southern Sweden. After applying the conversion formula to the estimate of 0.10 in the final model, we note that the intervention effect of the increased import quotas was followed by a 10.5% increase in the number of hospitalizations due to alcohol poisoning. However, the tax decrease in Denmark was not shown to have affected the different kinds of harm, and none of the interventions had any impact on violent crime or drunk driving. A further analysis showed that there was a strong correlation (Rxy=0.76) between the estimates of the quota change and the tax decrease in southern Sweden, which produced uncertain results when both variables were estimated in the same model. Thus, we cannot rule out that the tax decrease, too, was part of the effect found for the increase in private import quotas of alcohol.

An increase in alcohol poisonings due to the change in the traveller's allowance in 2004 was not observed among all age groups (Table 2), but was in fact only observed among those aged 50-69 years. The changes in quotas implied a 13% increase in alcohol poisonings in southern Sweden in this group.

[TABLE 2 ABOUT HERE]

All models were satisfactory with respect to the residual autocorrelation; thus the auto correlations were close to zero, indicating that the time-series are random (white noise). It should be noted that the semi-logarithmic model specification applied here assumes multiplicative effects of the two policy changes. As this may not necessarily be the case, the analyses were also performed using linear models, which, however, did not change the outcome.

13

F o r R e v i e w O n l y

Discussion

The Danish decrease in spirits tax in 2003 and the abolishment of Swedish alcohol import quotas on alcohol in 2004 suggested that alcohol consumption and alcohol-related harm would increase, especially in the southern parts of Sweden. This expectation has not been verified in previous studies using survey data, and the aim of the present study was to investigate the same issue by analysing various register data using an interrupted time-series approach. Hospitalization for acute alcohol poisoning and police-reported cases of drunk driving and violent assaults were used as indicators of alcohol-related harm.

Overall, the results of the presentthese analyses did not allow for any definite conclusion as to whether alcohol-related harm increased in southern Sweden. On the one hand, an increase in acute alcohol poisonings was found after the increase in import quotas, particularly among those aged 50-69 years, but on the other hand, no increase was found in violent assaults or in drunk driving. Thus, the findings with respect to assaults and drunk driving supported previous findings based on survey data and the conclusion that problems had not increased, whereas the increase in alcohol-related harm (alcohol poisonings) in southern Sweden suggests that some problems may have worsened, at least among middle-aged people in southern Sweden.

One interpretation of the results is simply that some problems increased whereas some did not and that these harm indicators measure specific problems rather than alcohol-related harm in general. Thus, drunk driving and alcohol-related violence were not affected by the increase in from the Swedish Council for Crime Prevention). Thus, the finding that travellers' imports of alcohol are most common among people over 50 years [START_REF] Svensson | Travellers' alcohol imports to Sweden at the beginning of the 21 st Century. NAT (Nordisk alcohol-& narkotikatidskrift)[END_REF] could explain some part of the difference, although further studies are needed to test this idea. It is also worth noting that there are some similarities with Finnish experiences related to their abolishment of import quotas and reduction of alcohol taxes in 2004. An analysis by Herttua [START_REF] Herttua | The Effects of the 2004 Reduction in the Price of Alcohol on Alcohol-Related Harm in Finland: A Natural Experiment Based on Register Data[END_REF] found no impact on assaults after the changes, but an increase in alcohol-related hospitalizations, which was largest in the age group 50-69 years. Thus the increases in both countries can be linked to older heavy drinkers, who are known to be less well represented in surveys. Our findings also have some similarities with a study from Denmark using the same time-series approach. This study examined changes in alcohol-related harm in Denmark between 2003 and 2005 after changes in alcohol policies were introduced and found no increase in violent assaults. On the other hand, an increase in alcohol poisonings was revealed but only among those aged 15 years or below [START_REF] Bloomfield | Changes in alcohol-related harm after alcohol policy changes in Denmark[END_REF]. Although the overall findings from these countries all indicate an increase in harms, the results vary by sub-group, suggesting that the effects of this kind of increase in availability may be influenced by cultural differences and that findings for one country should not automatically be generalized to other countries. In this specific case, it is also important to consider that the Danish and Finnish changes were domestic, whereas in Sweden a trip abroad was required to take advantage of the new possibilities to buy cheaper alcohol.

Naturally, one major limitation of this kind of study, which uses a quasi-experimental design, is that the intervention and control groups have not been randomly selected. Thus, there may have been large systematic initial differences between southern and northern Sweden, meaning that the latter was a poor control site.

  Changes in alcohol-related harm in Sweden after increasing alcohol import quotas and a Danish tax decrease -an interrupted time-series analysis for 2000-2007

  Denmark decreased its tax on spirits by 45% on October 1 st 2003. Shortly thereafter, on January 1 st 2004, Sweden increased its import quotas of privately imported alcohol, allowing travellers to bring in much larger amounts of alcohol from other EU countries. Whereas these changes were assumed to increase alcohol-related harm in Sweden, particularly among people living close to Denmark, analyses based on survey data collected before and after these changes have not supported this assumption. The present paper tests whether alcohol-related harm in southern Sweden was affected by these changes by analysing other indicators of alcohol-related harm, e.g. harm recorded in different kinds of registers.

  southern Sweden and the short distance to Denmark suggested that the changes were equivalent to a price reduction in southern Sweden.

  structure of the noise term (N) was modelled and estimated in terms of the autoregressive parameters (AR) or moving average parameters (MA). The model and noise specification are specified by the expression (p,d,q) (P,D,Q), the first parenthesis referring to the regular noise modelling and the second to the seasonality component. The order of the auto-regressive and moving average parameters is indicated by p and q, respectively, and the order of differencing is indicated by d. The corresponding symbols for the seasonal parameters are P, D and Q. The residuals of the estimated model should be white noise, which

  LnH t = b 1 T t + b 2 Q1 t + b 3 Q2 t + b 4 Q3 t + b 5 HNt +LnN t 10 symbolizes the harm indicator in the southern site at time t, i.e. by month. Ln signifies that the harm series are logaritmized. This means that the estimates of b 1 -b 5 can be expressed as change in % by applying the formula: 100*(exp(b)-1). T is the dummy variable for the Danish tax change in 2003, and Q1-3 represents the Swedish quota change in 2004 and the two previous quota changes in Sweden in 2002 and 2003. HN t is the harm indicator in northern Sweden. The estimates of N signify the noise term.

  figure for those hospitalized for alcohol poisoning is about 50 per cent (according to data

Figure 1 .Figure 2 .Figure 3 .

 123 Figure 1. Descriptive figure over the trends in number of alcohol poisonings in southern and northern

Table 1 .

 1 The higher level of drinking and harm in the southern site are two possible examples. However, in the present study, we had to identify a region affected to a lesser degree by the major interventions of interest: the Danish tax decrease and the abolishment of import quotas. Although certainly not a perfect control area, the northern parts of Sweden were the best choice available for fulfilling these criteria and, as previously mentioned, no effect on harm in northern Sweden could be established in relation to these reforms.In conclusion, because previous survey analyses have not revealed any increase in alcoholrelated problems in southern Sweden, the present findings highlight the importance of using various data sources when studying policy changes and their potential impacts on alcoholrelated harm. It also stresses the fact that various data sources are able to cover different segments of the population and that choosing one over the other may result in missing an existing effect. Additionally, the results presented here raise important questions about the association between changes in availability and alcohol-related harms in Sweden today. Changes in Swedish alcohol quotas from other EU countries since membership in the EU. Quotas refer to the amount (in litres) that can be brought into Sweden by travellers aged 20 years old and above without any Swedish alcohol tax being paid.

	F o r
	R
	e v
	i e
	w
	O n l
	y

Table 2 .

 2 Estimated effects of Danish tax-cut in October 2003 and increased alcohol quotas in January 2004 on alcohol poisonings, violent crimes and drunk driving. For alcohol poisonings, also by age. Semi-logarithmic ARIMA models estimated on monthly data January 2000 -December 2007 for southern Sweden, controlling for changes in the harm rates in northern Sweden and quota changes of 2002 and 2003.

	Output
		Estimate	SE	p-value	95% Conf. Interval	Box-Young	p-value
	Alcohol poisoning			(1,0,0) (1,1,0,12)		6.15	0.91
	Tax change 03	0.031	0.044	0.480	-0.055	0.118
	Quota change 04	0.100** 0.039	0.009	0.025	0.176
	Violent crime			(0,0,0) (1,1,0,12)		14.29	0.28
	Tax change 03	-0.037	0.031	0.237	-0.098	0.024
	Quota change 04	0.025	0.026	0.347	-0.027	0.076
	Drunk-driving			(0,0,0) (0,1,1,12)		10.80	0.55
	Tax change 03	-0.009	0.049	0.856	-0.106	0.088
	Quota change 04	0.031	0.049	0.529	-0.065	0.126
	Alcohol poisoning
	10-19			(0,0,0) (0,1,1,12)		7.92	0.79
	Tax change 03	-0.000	0.145	0.998	-0.285	0.284
	Quota change 04	0.138	0.144	0.338	-0.144	0.421
	20-29			(0,0,0)		14.08	0.30
	Tax change 03	-0.076	0.182	0.675	-0.434	0.281
	Quota change 04	0.121	0.177	0.493	-0.226	0.469
	30-49			(0,0,1) (1,1,0,12)		17.23	0.14
	Tax change 03	0.106	0.092	0.249	-0.074	0.285
	Quota change 04	0.050	0.085	0.556	-0.117	0.218
	50-69			(0,0,0) (1,1,0,12)		11.17	0.51

Acknowledgement

This work was supported by the US National Institute on Alcohol Abuse and Alcoholism (R01 AA014879), as part of the study 'Effects of major changes in alcohol availability'.