Faraday Symp. Chem. SOC.,1984 19 85-95 Generalizations of the Multiconfigurational Time-dependen t Hartree-Fock Approach BY DANNY L. YEAGER* Chemistry Department Texas A and M University College Station Texas 77843 U.S.A. AND JEPPE OLSEN AND POUL J0RGENSEN Chemistry Department Aarhus University 8000 Aarhus C Denmark Received 29th August 1984 We extend and generalize the time-dependent Hartree-Fock (TDHF) and multiconfigura- tional time-dependent Hartree-Fock (MCTDHF) approaches so that electronic transition energies from non-singlet reference states to states of pure spin symmetry other than the reference state can be described. Initial calculations are presented for the Be atom using the (2~3s) 3Sstate and the (2s2) l,!3 state as reference states.The accuracy of the calculated excitation egergies is examined as a function of extending the active space of the MCSCF reference state. Close agreement between the excitation spectra as obtained from either of the reference states is obtained when inner-outer correlation effects are considered in the MCSCF reference state. Over the past decade direct calculations of excitation energies transition moments and second-order molecular properties have proved to be very Various formulations have been given for the direct calculation of excitation processes. Examples are the equation-of-motion (EOM) method3 and the two-particle Green’s function or polarization propagator appr~ach.~ The simplest direct approach is the random phase approximation (RPA) or time- dependent Hartree-Fock (TDHF) approach where a single configurational state is used as reference state.l? 5+ For closed-shell systems with little correlation TDHF singlet excitation energies are generally determined to within an accuracy of 10% from experiment.Triplet excitation spectra may be structurally incorrect (triplet instabilities).For a reference state of spin different from zero calculations of excitation energies to states of the same spin multiplicity as the reference state are often very unreliable using TDHF.?? * A general description of how to evaluate excitation energies to states of spin multiplicity different from the reference state has not previously been given when the reference state has total spin different from zero. For closed-shell systems perturbational extensions of the TDHF approximation have been de~cribed.~? Substantial improvement has been obtained for both the singlet and triplet excitation energies in a second-order extension of the TDHF approximation.Second-order excitation energies appear to be accurate to ca. 0.5 eV for non-highly correlated systems. Third-order TDHF calculations have been describeds but the accuracy of such calculations is not yet well established.1° Open-shell perturbational extensions of the TDHF approximation have been described.* However severe problems need to be solved before accurate open-shell perturbation TDHF calculations can be carried out routinely. An alternative way of improving the TDHF approximation is to use the multi- configurational TDHF (MCTDHF) approximation where a multiconfiguration self- 85 GENERALIZATIONS OF THE HARTREE-FOCK APPROACH consistent field (MCSCF) state is used as reference state?* For both closed- and open-shell singlet reference states singlet and triplet excitation energies have been accurately evaluated with relatively short reference-state configuration lists (10-1 02 configurati~ns).~~ For a reference state of spin different from zero excitation energies to states of the same spin as the reference state can be described with equal accuracy.In this paper we demonstrate how extensions of the TDHF and MCTDHF approximation may be derived so that excitation energies to states of pure spin symmetry different from the reference state can be evaluated even when the reference state has a total spin that is different from zero.We do this by modifying the simple triplet particle-hole excitation operators such that the modified particle-hole operators when operating on the reference state of pure spin symmetry give states of pure spin symmetry. In electron propagator calculations on states of total spin different from zero annihilation and creation operators have previously been modified in a similarl43l5 manner to that used for the modification of the particle-hole excitation operators in this paper. This was done to assure that the ionization potentials and electron affinities of an electron propagator calculation are to states of pure spin symmetry. Initial calculation^^^^ l6on N2,0,and F with this approach [knownas the multiconfigurational electron propagator method (MCEP)] have yielded accurate ionization potentials for both principal and shake-up ionizations.In this paper we report results for beryllium and demonstrate for the (2~3s)3S+lS transitions that electronic transitions to pure spin states from a degenerate spin reference state may be accurately determined with the generalized multiconfigurational time-dependen t Hartree-Fock approach (GMCTDH F). THEORY SPECTRAL REPRESENTATION OF THE POLARIZATION PROPAGATOR The two-particle Green's function in the energy representation is defined using the Zubarev notation17 as ((A; B))E = (A+I (Ef+A)-' I B) (1) where A and B are number conserving operators and fand H are the superoperator identity and the superoperator Hamiltonian respectively.These are defined as !A = A (2) HA = [H,A]. (3) The Hamiltonian H is given as U au' where the creation and annihilation operators are indexed by a formally complete orthonormal basis of spatial orbitals and a spin index 0 which has the value of a or p spin. The superoperator binary product in eqn (1) is defined as (AI B) = <Wt" I [A+,4I Wt") (5) * It is apparently frequently forgotten that an AGP state may be thought of as a type of MCSCF state. AGP reference states usually do not obey the 'killer' condition and many calculations with AGP reference states arbitrarily zero the B matrix to force the 'killer' condition to be fulfilled. For theoretical and calculational analyses see for example ref.(1 2). D. L. YEAGER J. OLSEN AND P. JORGENSEN where Ity,"") refers to the exact N-electron reference state of total spin S and S component M. The spectral resolution of eqn (1) is (V,"" IA IV:'"') (VZ"' IB IV,"") -c E+ ~f-E:' nS'M' (V,"" 1 B IV:'"') (V:'"' IA I VfM> +c E+ E:'-~f nS'M' where the summations in the first and second terms contain a complete set of N-electronic eigenstates Eqn (6) shows that the poles of the two-particle Green's function occur at the excitation energies (E$-Ef)of the molecular system. The residues give the transition amplitudes (tyfMIA Iv/:'"') (~2'"' IB Iv/fM) which for example express the electric dipole transition probabilities if A and B refer to the electric dipole moment operator.APPROXIMATE CALCULATIONS The spectral representation in eqn (6)exhibits the physical content of the two-particle Green's function but it gives little insight into how approximate two-particle Green's function calculations can be carried out. To bring the Green's function to a form which is efficient for describing approximate Green's function calculations we take the inner projection1* of the superoperator resolvent in eqn (1) ((A; B))E =(B+Ih)(hIEf+ H Ih)-l (h1 A). (8) If the projection manifold h is complete eqn (8) and (1) are identical. Approximate two-particle Green's functions are determined by truncating the projection manifold and using an approximate state as the reference state. Below we describe in more detail how to carry out such approximate calculations.Excitation energies occur at the poles of the Green's function and may be determined according to eqn (8) as eigenvalues of the matrix equation (hIEf+ I? Ih) =0. (9) Eqn (9) is the same as the equations-of-motion method (EOM) for excitation energieslg <Vf" I[soh[K0R11 I V,"") =%(VfM I Wi,011IVf") (10) when the excitation operator 01is expanded in the projection manifold h and coAis the excitation energy. In practical calculations with either eqn (9) or (10) we will use the symmetric double commutator [A CI =:([A,[B'Cll+ "A BI Cl) (1 1) if it is not specified otherwise. EXCITATION ENERGIES FOR A NON-TOTALLY SYMMETRIC REFERENCE STATE The excitation energies E:' -E,S appearing in the spectral representation of the two-particle Green's function in eqn (7) are between states of pure spatial and spin symmetry.This was obtained because the complete set of states in eqn (6) was assumed to be eigenstates of the non-relativistic Hamiltonian in the Born-Oppenheimer approximation. In approximate two-particle Green's function calculations special GENERALIZATIONS OF THE HARTREE-FOCK APPROACH precautions must be taken to assure that the excitation energies have pure spatial and spin symmetry. From eqn (6) it is clear that the conditions necessary for obtaining excitation energies of pure spatial and spin symmetry consist of requiring (1) that the approxi- mate reference state has pure spatial and spin symmetry and (2) that the approximate projection manifold is constructed such that when operating on the approximate reference state states of pure spatial and spin symmetry are created.The first condition is trivial to satisfy. The second condition while easily fulfilled for a totally symmetric reference state is by no means trivially fulfilled for a non-totally symmetric reference state. To illustrate this point consider for example a reference state of total spin S and spin projection M I rySM). The singlet particle-hole tensor operator (for convenience the MCSCF orbitals are arranged in the order core valence and then the virtual orbitals) when operating on I rySM) will automatically create states of total spin S and spin projection M. However if one component of the triplet spin tensor particle-hole operators operates on I vSM) states of mixed total spin S+ 1 S or S-1 are formed.Excitation to states of mixed spin symmetry will therefore be obtained if the operators in eqn (13) are used in an approximate projection manifold. In the following we describe how to modify the operators in eqn (13) to ensure that excitation energies in an approximate Green's function calculation will be of pure spin symmetry. States of pure total spin S+ 1 S or S-1 can be formed by coupling the operators of eqn (13) with the various components of the reference state using appropriate vector coupling coefficients. For example a state of total spin S+ 1 and Sz component M may be formed as where the analytical expressions for the vector coupling coefficients given for example in Tinkham20 have been used.When the identities D. L. YEAGER J. OLSEN AND P. J0RGENSEN are inserted into eqn (14) we obtain (S-M+1) ‘2 +((S+M+ 1) ) apsa s+]VSM). I From eqn (1 7) it is obvious that modified (and unnormalized) particle-hole operators T,+,(1,O) may be defined as TFS(1,O) = -(S+ M+ 1) a& aspS-+(S+M + 1) (S-M + 1) (a:a asa -a& asp)+(S-M + 1) a asaS+. (18) These operators when acting on a state of totai spin S and S component M ( I SM)) give (unnormalized) states of total spin S+ 1 and S component M ( I S+ 1 M)). Modified particle-hole operators TTS(O,0) and T,+,(-1,O) which form states of total spin S and S-1 respectively when acting on 1 SM) may be derived in a similar manner as TFs(O,0) = a& aspS-+M(a&asa-a& asp)+a$ asaS+ (19) T:s(-1,O) = -(S-M)a&aspS--(S-M)(S+ M)(a& asa-a& asp)+(S+ M)a:p as&S+.(20) Use of the modified particle-hole operators of eqn (18)-(20) in an approximate projection manifold ensures that the excitation energies of an approximate Green’s function calculation are between states of pure spin symmetry provided the reference state has pure spin symmetry. MULTICONFIGURATIONALTIME-DEPENDENT HARTREE-FOCK (MCTDHF) We now discuss the multiconfigurational time-dependent Hartree-Fock (MCTDHF) approximation. The reference state is chosen as a multiconfigurational self-consistent field (MCSCF) state I OSM).21In most of our subsequent discussions we use a complete active space (CAS) MCSCF reference state. This choice is made to simplify evaluation of certain matrix elements by avoiding the calculation of three-body density matrices.To set up the projection manifold we also define the MCSCF orthogonal complement of states to JOSM) denoted (ITS’M’)}.For a CAS MCSCF state the orthogonal complement space { I TS’M’)} is here considered to be all linearly independent N-electron states of all possible symmetries of the CAS space orthogonal to J OSM). The projection manifold of the MCTDHF approximation consists of the state- transfer excitation R+and de-excitation R operators where ITSM) is a state in the orthogonal complement MCSCF space and the non-redundant set2’ of spatial totally symmetric particle-hole [Q+(OO) of eqn (12)] and hole-particle Q(0,O) excitation operators.The approximation projection manifold of the MCTDHF approximation may thus be written as h = {Q+<o,01 WO O) Q(0,O) R(O,0)). (22) 90 GENERALIZATIONS OF THE HARTREE-FOCK APPROACH Inserting this projection manifold into eqn (9) and using the MCSCF state I OSM) as reference state gives the MCTDHF approximation for excitation energies to states of the same spin multiplicity as the MCSCF reference state provided the superoperator Hamiltonian is defined to operate always first on the orbital excitation operators in evaluating coupling elements between orbital and state excitation operators.ll EXTENSIONS OF MULTICONFIGURATIONAL TIME-DEPENDENT HARTREE-FOCK When the reference state is totally symmetric in spin space (S = 0) an extension of the MCTDHF approximation may straightforwardly be defined for determining excitation energies to states of triplet spin symmetry.The projection manifold then may be chosen to consist of the simple particle-hole and hole-particle triplet excitation operators of eqn (1 3) and appropriate state-transfer excitation and de-excitation operators. For a reference state which is not totally symmetric in spin space care must be taken as described previously in the selection of the projection manifold to assure that the excitation energies are to states of pure spin symmetry. For a reference state of total spin S the projection manifold for excitation energies to states of total spin S+ 1 may be chosen to consist of where the T+ and T operators are defined in eqn (18).Excitation energies to states of total spin S-1 may similarly be obtained by using the projection manifold h(-1,O) = { T+(-1 O) R+(-1 O) T(-1,O) R(-1,O)). (24) A straightforward extension of the MCTDHF approximation for excitation energies to states of total spin S can also be defined by using the projection manifold The inclusion of the P(0,O)and T(0,O)operators relative to the MCTDHF manifold is justified because these excitation operators when operating on the reference state I OSM)create states that are single excitation in the spatial part of the wavefunction but with an additional spin flip. Such states are of approximately the same energy as the states formed by the simple orbital excitation Q+(O,0) and de-excitation Q(0,O) operators. Of course for a reference state which is totally symmetric in spin space (S= 0) all contributions from the l"t(0,O)and T(0,O)operators vanish.Furthermore care must be taken to assure that operators in eqn (25) are linearly independent. GENERALIZED MULTICONFIGURATION HARTREE-FOCK EIGENVALUE EQUATION Using the MCSCF reference state 1 OSM) and the projection manifolds of eqn (23F(25) allows us to define extensions of the MCTDHF approximation for determining excitation energies to states of total spin S+ 1 S and S-1 respectively. A generalized MCTDHF eigenvalue equation is obtained from eqn (9) As an example consider the case where excitation energies are determined to states 91 D. L. YEAGER J. OLSEN AND P. J0RGENSEN of total spin S+ 1. The matrices A(S+ 1,M) and S(S+ 1,M) of eqn (26) are then defined as A@+ 1 M) (OSM I[T(1,O) H W1,O)l I OSM)(OSM I “T(1,O) HI,R+(1 011 I OSM) =((OSMI[R(I,O),[H 1-~(1,0)111~S~)(~~~I[R(l,O),H,R+(1,0)11~SM) (27) S(S+ 1,M) (OSM I [T(l,O) P(1,O)l I OSM)(OSM IIT(L O) R+(L011 I OSM)).(28) =((OSMI [R(l,O) W1,O)I I OSM)(OSMI [R(1,O),R+(l,O)I I OSM) The matrices B(S+ 1,M) and A(S+ 1,M) are obtained from A(S+ 1,M) and S(S+ 1,M) by replacing T(1,O) and R(1,O) with T+(1,O) and R+(1 0) respectively. Note that in defining coupling elements between the orbital and state excitation operators the matrices A(S+ 1,M) and B(S+ 1,M) are defined such that the Hamil- tonian is commuted first with the orbital excitation operator and then the result is commuted with the state-transfer excitation operator.Matrix elements of A and B which refer exclusively to either the orbital or the state space are defined in terms of the symmetric double commutator. The matrix elements that are required to determine excitations to states of total spin S or S-1 are defined in a similar manner. In ref. (2)we give the explicit expressions for the matrix elements of the A B S and A matrices corresponding to excitations to states of total spin S-1 S or S+ 1 for a CAS MCSCF reference state. A CAS MCSCF state is considered because the evaluation of A and B simplifies in this case. Only two-electron density matrix elements are required in evaluating A and B for a CAS MCSCF reference state while for a general MCSCF reference state three-electron density matrix elements may also have to be constructed.Straightforward evaluation of the matrix elements of the A and B matrices requires knowledge of one- and two-electron density matrices with different M components of the reference state. Using the Wigner-Eckart theorem the expressions for the A and B matrix elements may be simplified to only require knowledge of one- and two-electron density matrix elements with the same M component of the reference state. To illustrate both these points we consider the evaluation of one of the right-hand side components of the A matrix (SMI[Q+(l -I>S+,[H,Q+ (1 -l)~+lllSM) =2/((S+ M+ 1)(S+ M+ 2)(S-M)(S-M-1)) x (SMI [Q+<l,-I) [H,Q+(l -I)]] I SM+2) +2/(2(S+M+l)(S-M))(SMI(Q+<l -1) x [H,Q+<1,0)l-[H,Q+(l -1)1 Q+(l 0))I SM+ 1).(29) Consider initially how to evaluate the first term of eqn (29).The operator inside the bracket is a tensor operator of rank 2 and component -2 p-2 = [Q(l -I) [H,Q+(l -I)]]-(30) Using the Wigner-Eckart theorem we obtain (SMI VySM+2) = (S(M+2)2-2(SM)(SII VIIS) (31) 92 GENERALIZATIONS OF THE HARTREE-FOCK APPROACH where (S(M+2)2 -2 ISM) is the vector coupling coefficient and (SI I v2 I I S) is the reduced matrix element. To evaluate the first term in eqn (29) we need thus to evaluate the reduced matrix element (S(I v2 I IS). Using eqn (30) we determine the tensor operator q as 1 1 v2 -[S+ ?el] =-[S+ [S+,P-,I] "~'6 2d6 The reduced matrix element may therefore be determined from and the first term in eqn (29) may be evaluated as (SMI [Q+(l -11 [H,Q+<l,-I)]] I SM+2) +2[Q'(l 01 [H,Q+(l,O)ll+ Q+(l -11 [H,Q+<l 1)11> ISM).(34) Eqn (34) requires only one- and two-electron density matrices referring to the state ISM) to be evaluated. For a general MCSCF state the second term in eqn (29) will contain three-electron density matrix elements. For a CAS MCSCF however where orbital excitation operators within the active space are redundant we have defined the ordering of orbitals so that QrsISM) = 0. [See the discussion following eqn (12).] We may therefore write the second term in eqn (29) as a double commutator d(2(S+ M+ 1)(S-M))(SM I {[Q+(l,-11 [H,Q+(l O)]] +[Q+(l,o) [H,Q+(l -1)11>I SM+ 1). (35) It requires then that only one- and two-electron density matrix elements be evaluated.Using that we may write eqn (35) as 2d((S+M+ 1) (S-M))(SM I P1I SM+ 1) = 22/((S+ M+ 1)(S-M))(S(M+ 1)2 -1ISM) (SII PII S) (37) and using eqn (31) the second term in eqn (29) may be easily evaluated with only knowledge of one- and two-electron density matrix elements involving the state I SM). RESULTS AND DISCUSSION We report now some initial calculations of the excitation energies of the Be atom using the TDHF MCTDHF and generalized MCTDHF schemes outlined above. First we use the (2s2)lS state as reference state and calculate excitation energies to states of lS and 3S symmetry. The calculated MCTDHF excitation energies are D. L. YEAGER J. OLSEN AND P. J0RGENSEN Table 1. Excitation energies to states of 'S and 3S symmetry in Be (29) 1s -+ (2sns) ' 3s (2~3s)3S+ (2sns) 'l3S state expa 2sb 2s2pb 2~2~~2~2~'~2s2s'2p2p'3db 2s3sb- 2~2~'3~3~'2p2p'~* (2s2) 1s (2~3s)'S (2~4s)lS (2~5s)'S (2~6s)'S 0 6.78 8.09 8.59 8.84 0 6.12 7.26 7.74 8.98 0 6.95 8.26 8.79 9.17 0 6.76 8.07 8.60 8.98 0 6.77 8.08 8.60 8.98 0 5.67 6.51 7.03 7.44 0 6.74 8.22 8.64 10.23 (2~3s)3S (2~4s)3S (2x5s) 3S (2~6s) 6.46 8.00 8.56 8.82 5.49 7.10 7.67 8.88 6.50 8.15 8.74 9.06 6.43 7.98 8.56 8.88 6.43 7.98 8.56 8.88 5.00 6.44 7.00 7.34 6.39 7.93 8.50 8.83 a S.Bashkin and J. 0.Stoner Jr Atomic Energy Levels and Grotrian Diagrams I Addenda (North-Holland Amsterdam 1978). Complete active space in specified orbitals. Entries in this column were obtained by adding the calculated (2~3s) 3S -,(2s2)lS excitation energy of 5.00 eV to the obtained excitation energies.Entries in this column were obtained by adding the calculated (2~3s)3S -+ (2s2)'S excitation energy of 6.39 eV to the obtained excitation energy. examined as a function of the quality of the CAS space in the MCSCF reference state. The excitation energies are then calculated using the (2x3s) 3S state as reference state and finally the quality of the excitation spectra using the (2s2)?S state and the (2~3s) 3S state as reference states are compared. The basis set we use is a 5s contraction22 of a H~zinaga~~ 10s Gaussian basis set five sets of uncontracted Gaussian p functions24 and a set of Gaussian dfunctions with exponent 0.65. Four additional diffuse s functions with exponents (0.020,0.008,0.003 0.001) four additional sets of diffuse Gaussian p functions with exponents (0.016 0.007,0.003,0.001) and two sets of diffuse Gaussian dfunctions with exponents (0.015 0.0032) were added.In table 1 the obtained excitation energies are reported. In the calculations of the third column the active space consists of the 2s orbital. The reference state therefore is the SCF (2s2)lSstate and the excitation energies become those of the TDHF approximation. The excitation energies of Be have previously been calculated in the TDHF approximation using a 50 Slater-type orbital basis. The excitation energies of both calculations are essentially identical indicating that the reported TDHF excitation energies must be close to the complete basis limit. In the fourth column the active space is extended with the 2p orbital.The MCSCF reference state therefore contains two configurations (2s2 and 2p2). MCTDHF excitation energies have previously been reported in ref. (13) using the same two- configuration reference state and with a 50 Slater-type orbital basis. The lowest two excitation energies of both singlet and triplet symmetry are identical to all reported values in the two calculations indicating that these excitation energies must be close to the complete basis limit for the two-configuration case. In column 5 we report the MCTDHF excitation energies that are obtained by considering the so-called inner-outer correlation effects in the MCSCF reference state. The results of column 5 show the importance of inner-outer correlation of the MCSCF reference state.In column 6 we have further included the angular correlation effects which originate from GENERALIZATIONS OF THE HARTREE-FOCK APPROACH including a 3d orbital in the MCSCF reference state. The MCTDHF excitation energies do not change by including the 3d orbital in the active space indicating that the MCTDHF excitation energies are stable toward extending the active space with orbitals of higher angular momentum. The TDHF excitation energies are generally of rather poor quality and differ from experiment by as much as 0.5-1.1 eV. A substantial improvement is obtained when the reference state is correlated with the 2p2configuration. The MCTDHF excitation energies (column 5) then differ from experiment by 0.1-0.2 eV.When the inner-outer correlation is included in the MCSCF reference state agreement with experiment to within 0.01-0.03 eV is obtained for the lowest three excitation energies of both singlet and triplet symmetry. The larger deviation in the excitation energies to states originating from the 2s6s electronic configuration is due to the fact that our basis only contains 9s Gaussian basis functions and therefore cannot accurately describe more than the few lowest excitation energies. In columns 7 and 8 we report the excitation energies obtained using the (2~3~)~s state as reference state. In column 7the 2s and 3s orbitals are active and the reference state becomes the (2~3s) 3SSCF state. The TDHF excitation energies of column 7 are very unreliable for excitations of triplet-triplet and triplet-singlet character.Note that for the SCF reference state (2~3s) 3S we have the option when evaluating excitation energies to states of ‘5 symmetry either to include the 2s+ 3s orbital excitation operator in the calculation or to include the state-transfer operators containing the 2s2,3s2and 2s3s singlet states. The results in column 7 use the state-transfer operators as only this choice gives the possibility of describing all the excitation energies in table 1 in a THDF approach. In column 8 we report the excitation energies which are obtained by including the 2s2s’3s3s’2p2p’orbitals in the active space. This active space describes both inner-outer correlation and some angular correlation in the MCSCF reference state.As is clear from the results in column 8 very good agreement is obtained with experi- ment for the lowest 3 excitation energies of both triplet-triplet and the triplet-singlet character. The triplet-triplet excitation energies are evaluated using the MCTDHF approach. Inclusion of the T operators in the projection manifold [see eqn (25)]gives no significant improvement and introduces in some cases linear dependencies in the projection manifold. The inclusion of the T operator has not been explored further for triplet-triplet excitations. The triplet-singlet excitation energies in column 8 have been evaluated using the projection manifold of eqn (24). The excitation spectrum in column 8 which uses the (2~3s) 3S state as reference state and the spectrum in column 5 which uses the (2s2)lSstate as reference state are in very close agreement and agree with experimental excitation energies.Excitation spectra may therefore equally reliably be evaluated using the ground or an excited state as reference state. It also appears that excitation energies involving a change in the total spin may be evaluated as reliably as excitation energies which conserve the total spin. Of course many more calculations have to be carried out before definite conclusions can be made about this point. D.L.Y. thanks the National Science Foundation (grant no. CHE-8023352) and the Robert A. Welch Foundation (grant no. A-770) for support and NATO (grant no. RG 193.80) for travel support. D. L. YEAGER J.OLSEN AND P. J0RGENSEN 95 J. Linderberg and Y. 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