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and their implications Jan Erik Linderberg, Yngve Ohrn

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Jan Erik Linderberg, Yngve Ohrn. Propagators and their implications. Molecular , Taylor & Francis, 2010, pp.1. ￿10.1080/00268976.2010.513342￿. ￿hal-00623307￿

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Propagators and their implications

Journal: Molecular Physics

Manuscript ID: TMPH-2010-0130

Special Issue Paper - Electrons, Molecules, Solids and Biosystems: Manuscript Type: Fifty Years of the Theory Project

Date Submitted by the 12-Apr-2010 Author:

Complete List of Authors: Linderberg, Jan; Aarhus University, Ohrn, Yngve; University of Florida, chemistry

Keywords: , density, response

Note: The following files were submitted by the author for peer review, but cannot be converted to PDF. You must view these files (e.g. movies) online.

Propagators.odt

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1 2 3 4 Propagators and their implications 5 6 Yngve Öhrn and Jan Linderberg 7 8 9 Quantum Theory Project, Departments of Chemistry and Physics, University of Florida, 10 Gainesville, Florida U. S. A. 11 12 YÖ: [email protected], JL: [email protected] 13 14 (Received 2For0 July 20 05Peer; final versio n Reviewreceived 17 August 20 0Only6) 15 16 17 Nearly fifty years of collaboration on the development of formulations of the many-electron 18 theory for molecular, atomic and condensed matter in terms of propagators is surveyed. 19 20 Keywords: propagator; density; response; 21 22 23 Initiation 24 25 theoretical methods were familiar to Per-Olov Löwdin from his early work with 26 27 relativity and electromagnetic theory. He was alerted to its impact in many electron 28 29 theory through the works by Gell-Mann and Brueckner and by Hubbard in the mid 50's. 30 31 32 Thus he wanted to establish the connection to the configuration space methods that he 33 34 was so familiar with and to this end he assigned one of us (JL) to give a seminar to the 35 36 Group at Uppsala on second in the fall of 1957. The 37 38 39 sources were the classical papers by Born and Jordan, the detailed treatment by Fock and 40 41 the treatises by Dirac and Corson. Löwdin followed up with formal field theory but it was 42 43 44 when Stig Lundqvist returned from a sabbatical with Brueckner that the concepts and 45 46 techniques became more familiar topics at the Group. The lecture notes taken by Lars 47 48 Hedin was the basis for the Technical Report from the Group [1] and Lundqvist's lectures 49 50 51 at the first Winter Institute at Gainesville in December 1960 and Sanibel Island in 52 53 January 1961. The authors spent a brief period together at Gainesville in early 1961 when 54 55 YÖ began a two-year stay at the University of Florida. 56 57 58 04/12/10 1 59 60 URL: http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics Page 2 of 31

1 2 3 Löwdin suggested in September 1961 that time was ripe for an attack on the 4 5 6 dispersion energy contribution to the cohesive energy of molecular solids, in particular 7 8 rare gas crystals, and assigned the task to JL who had just returned from a year with the 9 10 11 Quantum Theory Project. The treatment of fluctuating dipole fields as the source of long 12 13 range interactions was most readily tackled in terms of the concepts of propagators and 14 For Peer Review Only 15 associated elements of the field theoretical tool box. Decent results were found with 16 17 18 moderately elaborate calculations [2]. 19 20 The present authors were reunited at Uppsala in 1963 and were inspired by John 21 22 Hubbard's paper on electron correlation in narrow energy bands [3] to apply his Green 23 24 25 function decoupling procedure to the generalized Hückel model introduced by Pariser 26 27 and Parr and by Pople [4]. Our results were exciting and were gracefully communicated 28 29 by Coulson [5]. Additional results seemed to corroborate that we were onto to something 30 31 32 useful [6]. 33 34 Methods from field theory were not generally embraced in the molecular 35 36 37 electronic structure community. Slater was particularly skeptical towards the techniques 38 39 of that had become “stylish” [7] and Löwdin was concerned about 40 41 the N-representability issue. There were efforts under way at Cambridge to further the use 42 43 44 of second quantization, somewhat ironically published in a tribute to Slater [8]. 45 46 McLachlan spearheaded the applications in molecular orbital theory [9] and his paper 47 48 with Ball [10] remains a basic reference for time-dependent studies of electronic 49 50 51 properties. We were not alone and continued our exploration of the options offered by 52 53 second quantization, propagators and novel approximation methods. 54 55 56 57 58 04/12/10 2 59 60 URL: http://mc.manuscriptcentral.com/tandf/tmph Page 3 of 31 Molecular Physics

1 2 3 Second quantization algebra 4 5 Molecular electronic structure theory is almost exclusively based on a representation in 6 7 terms of an orbital basis set, be it either a formal or a precisely defined one. The Hückel 8 9 10 and Pariser-Parr-Pople models are examples of the former while the calculational 11 12 schemes emanating from Boys's, Roothaan's and Hall's early development represent the 13 14 latter. BothFor situations oPeerperate with a fiReviewnite basis of spin orbi taOnlyls U:{ u1(x), u2(x), …uM(x)} 15 16 17 and the associated power set P(U), which has a one-to-one correspondence to the set of 18 19 all Slater that can be formed from U. The empty set is included i P(U) and 20 21 represents the state without electrons, often called the vacuum. There are 2 M elements in 22 23 24 the power set and they are either denoted by the list of the particular spin orbitals for a 25 26 given element or by the occupation number of the individual basis orbitals, a M-digit 27 28 29 binary number where each position corresponds to a spin orbital in U. Elements of the 30 31 power set P(U) span a subspace of the full that can accommodate an arbitrary 32 33 number of electrons. 34 35 36 Transformations or mappings between elements of the power set are composed of 37 † 38 elementary creation (addition, ar ) or annihilation (removal, ar) operators for individual 39 40 spin orbitals. A consistent algebra requires the anticommutation relations 41 42 † † † † † † 43 ar as + as ar = 0; ar as + as ar = 0; ar as + as ar = δrs + Srs. 44 45 Overlap integrals S occur for most primitive bases and may be removed by linear 46 47 transformations. General many electron operators occur as linear combinations of 48 49 50 products, the hamiltonian, as an example, is 51 52 † † † H = Σrs ar hrs as + ½Σrr'ss'(rs|r's') ar ar' as'as. 53 54 55 Mulliken's notation is used for the Coulomb interaction between spin orbital densities 56 57 58 04/12/10 3 59 60 URL: http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics Page 4 of 31

1 2 3 u *(x)u (x) and u *(x')u (x'). The formalism exposes the structural similarity between 4 r s r' s' 5 6 methods where the integrals, hrs and (rs|r's'), are derived from specified spin orbitals and 7 8 proper integrations and methods where they are inferred from other sources. The 9 10 11 evaluation of a matrix representation of the hamiltonian from the elements of the power 12 13 set requires the standard rules as put forth by Löwdin and others [11]. 14 For Peer Review Only 15 The second quantization algebra offers an approach to avoid the explicit 16 17 18 construction of a matrix representative of the hamiltonian and still be able to infer 19 20 essential properties. It holds, in an orthonormal basis, that 21 22 † † † [ar ,H] = Σs hrs as + Σr'ss'(rs|r's')ar' as'as; [ [ar ,H], as ]+ = hrs + Σr's'{(rs|r's')-(rs'|r's)}ar' as' 23 24 † 25 and a mean field theory obtains when the product ar' as' is replaced by its 26 27 expectation value, the reduced density matrix γs'r' , in the second form. 28 29 30 Exponential operators provide compact representations of unitary transformations 31 32 of the basis and unusual symmetries, e.g. particle-hole relations in alternant hydrocarbon 33 34 molecules in the Pariser-Parr-Pople model, can be exposed. The developments of the 35 36 37 coupled-cluster variants in current usage would probably have been more awkward in 38 39 direct configuration space formulations. 40 41 The essential separation of the Pariser-Parr-Pople model appears rather 42 ΣΠ 43 44 directly in the second quantization formulation [12] and details the role of the Σ- 45 46 framework as a dielectric medium which screens the interaction within the Π-system. A 47 48 49 link is also established to the form of a Heisenberg spin-hamiltonian in a weak coupling 50 51 limit. 52 53 54 55 56 57 58 04/12/10 4 59 60 URL: http://mc.manuscriptcentral.com/tandf/tmph Page 5 of 31 Molecular Physics

1 2 3 Self-consistent fields 4 5 It was shown above that a mean field model for an electronic system results from a 6 7 replacement, in the operator equation of motion, of an operator product with its mean 8

9 † 10 value: γsr =〈 ar as 〉. A second step is then to relate the expectation value to the field. 11 12 Hartree introduced the criterium of self-consistency and this remains in electronic 13 14 structure thForeory as th e PeerHartree-Fock scReviewheme when proper ex cOnlyhange is accounted for. The 15 16 17 Fock matrix with elements frs = hrs + Σr's'{(rs|r's')-(rs'|r's)}γs'r' is hermitean and its 18 19 eigenvalues determine the canonical spin orbitals. These are used with the build up 20 21 22 principle to characterize the desired state. Neither spatial nor spin is explicitly 23 24 invoked and the general solution is rarely desirable. It is self-consistent to assume spin 25 26 invariance in a system with an even number of electrons and it is also consistent to 27 28 29 assume spherical symmetry for an atomic or ionic system with “magic” number of 30 31 electrons. Other cases will result in non-consistent situations. 32 33 Slater chose to evaluate the reduced density matrix elements from an ensemble 34 35 36 average based on an open shell configuration with a fixed number of electrons and 37 38 obtained a Fock matrix with spin orbitals appropriate for a central field. The open shell 39 40 41 spin orbitals were useful in further applications in the solid state but could not be related 42 43 to ionization processes. Applications to transition metal atoms uncovered another 44 45 awkwardness. A starting central field would indicate that the 4s-shell was filled and the 46 47 48 least bound spin orbitals were the 3d's. The field from this occupation would lead to the 49 50 opposite and a consistent solution was not obtained. A remedial approach introduced 51 52 fractional occupations and acceptable solutions. Slater's derivation was not based on an 53 54 55 ensemble average but came directly from an expression that is valid only for integer 56 57 58 04/12/10 5 59 60 URL: http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics Page 6 of 31

1 2 3 occupations. A proper ensemble was readily described in second quantization terms. Thus 4 5 † 6 it holds that averages 〈 ar as 〉 = qrδrs imply the ensemble operator 7 8 † † Γ = Πr [(1 – qr)arar + qrar ar]; Tr Γ = 1. 9 10 11 The standard Hartree- has occupation numbers qr either zero or one. The 12 13 ensemble gives an average energy as a function of the occupation numbers, 14 For Peer Review Only 15 E =Tr H = h q +½ [ (rr|ss) – (rs|sr) ] q q 16 Γ Σr rr r Σrs r s, 17 18 and optimization of the spin orbitals results in Fock equations with fractional occupation 19 20 numbers, is so desired. Derivatives of the energy, εr = ∂E/∂qr , are the orbital energies and 21 22 23 eigenvalues of the Fock matrix. The ensemble operator is not an eigenoperator of the 24 † 25 number operator Nop = Σr ar ar and the dispersion comes to 26 27 〈 2 〉 28 [ Nop - < Nop >] = Πr ( 1 – qr ) qr 29 30 showing that the ensemble has contributing of varying electron number, a grand 31 32 canonical ensemble. Such a construct serves to define a suitable set of basis spin orbitals 33 34 35 for situations when spatial and spin symmetry features are desirable. Kaijser 36 37 demonstrated the usefulness of such a basis for the determination of transition moments 38 39 and life times for the manifold of low lying states of the Ti II ion [13]. Viinikka [14] also 40 41 42 applied the grand canonical ensemble construct to define a basis that performed well in 43 44 the study of the multiplet structure of core hole states in transition metal ions. Recent 45 46 developments in basis set design have found the ensemble average with fractional 47 48 49 occupation numbers to have advantages [15]. 50 51 The grand canonical as well as the canonical ensemble construct were used by 52 53 54 Poul Jørgensen [16], one of many postdoctoral associates that have been exchanged 55 56 57 58 04/12/10 6 59 60 URL: http://mc.manuscriptcentral.com/tandf/tmph Page 7 of 31 Molecular Physics

1 2 3 between the JL and YÖ groups over the years. He compared the detailsof the theoretical 4 5 6 basis with that of the Slater construction of the Fock operator. He also applied [17] the 7 8 grand canonical ensemble construction for a time-dependent Hartree-Fock treatment of 9 10 11 the triplet-triplet absorption spectra of alternant hydrocarbons. 12 13 Detailed calculations on transition metals [18] showed that there was no 14 For Peer Review Only 15 indication that the orbital energy of the 4s-level should be below the 3d-level. It was 16 17 18 interesting to note that a simple function could represent the dependence on the 19 3/2 20 occupation number of the orbital energy: ε(q) = ε(0)[ 1 – q/qc ] . The maximum 21 22 occupation, q , gives a slightly negative ion. This form can be compared to Jørgensen's 23 c 24 25 differential ionization concept [19], Mulliken's electronegativity [20], and the concept of 26 27 absolute hardness [21]. 28 29 30 Calculations with fractional occupation numbers were also used to approximate 31 32 excitation energies. Slater [22] showed that for an energy expression that is a continuous 33 34 function of occupation numbers one may attempt to calculate energy differences from 35 36 37 Taylor series expansions: 38 39 E(…qr – 1…qs + 1…) - E(…qr …qs …) = ∂E/∂qs - ∂E/∂qr + … 40 41 while observing that the second order terms are eliminated when the derivatives are 42 43 44 evaluated at the midpoint, ( qr – ½, qs + ½), termed the transition state. Applications 45 46 confirmed that this concept offered a viable alternative to the determination of separate 47 48 self-consistent solutions to the two states. 49 50 51 52 53 Electron propagator approximations 54 Hubbard's model for narrow energy bands in solids [3] led him to a decoupling of the 55 56 57 58 04/12/10 7 59 60 URL: http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics Page 8 of 31

1 2 3 equations of motion for the electron propagator or Green function G (E) = <> . 4 rs r s E 5 6 His approximation was reformulated by means of a matrix representation in an extended 7 8 operator basis [23]. Later efforts were developed in operator spaces spanned by general 9 10 † † † 11 operator products. Useful forms are A = Σ ar (cr + crstas at + …) , they define two 12 † † 13 sequilinear forms, M =〈 [ A, A ]+ 〉 and K =〈 [ [ A, H ], A ]+ 〉, which may be 14 For Peer Review Only 15 brought to diagonal form. The metric M is hermitian and non-negative while the effective 16 17 18 hamiltonian K is hermitian. Linear dependencies may occur from an initial choice of 19 20 operators and reflect properties of the state or ensemble used for the averages. A 21 22 canonical set of operators { A † :〈 [ A , A † ] 〉= ,〈 [ [ A , H ], A † ] 〉= } is 23 j j k + δjk j k + εjδjk 24 25 defined and one infers that a useful approximation for the propagators follows: 26 27 † « Aj ; Ak »E = δjk / ( E – εj ). 28 29 30 Further analysis of consistency requires that the expectation values that are used to define 31 32 M and K are derived from the propagators, i. e. by a contour integral in the complex E- 33 34 plane: 35 36 † -1 † 37 〈 Ak Aj 〉 = (2πi) ∫C dE « Aj ; Ak »E 38 39 Fulfillment of these conditions can rarely be realized beyond the Hartree-Fock level and 40 41 42 it is necessary to exercise caution. 43 44 Numerous calculations have demonstrated that a moderate extension of the 45 46 operator manifold provides excellent interpretations of photoelectron spectra, including 47 48 49 shake-up and shake-off processes, albeit not fully self-consistent. 50 51 Applications of electron propagator theory became possible with the seminal 52 53 work of George Purvis, who joined the group of YÖ at QTP in 1970. He developed a 54 55 56 57 58 04/12/10 8 59 60 URL: http://mc.manuscriptcentral.com/tandf/tmph Page 9 of 31 Molecular Physics

1 2 3 program suite called GREENFUNC, which survived in bits and pieces for over a decade 4 5 6 in electronic structure software around the world. Early publications [24, 25, 26] set the 7 8 standard and were followed by many others from QTP. This development in the 9 10 11 propagator effort is covered in another publication in this issue of Molecular Physics. 12 13 14 PolarizatioForn propag atPeeror approxim atReviewions Only 15 Lindhard [27] was concerned with the stoppage of particles in matter and derived the 16 17 18 response of an electron gas to an external electric field by means of a self-consistent 19 20 determination of the induced effects in the gas. His wave length and frequency dependent 21 22 dielectric function exhibited the collective excitations as well as single particle 23 24 25 features. This was the first detailed application of the linearized time-dependent Hartree 26 27 method and remains a landmark in electronic structure theory. Dirac had delineated the 28 29 30 equation of motion for the reduced density matrix in the time domain [28] and 31 32 applications to non-homogeneous systems [10, 29] appeared as complements to the 33 34 elaborate diagrammatic and formal perturbation theory expansions that were the rule in 35 36 37 many-electron theory during quite a few years. 38 39 Knowledge of response functions such as generalized polarizabilities and 40 41 dielectric functions offers a route to correlation features through the fluctuation- 42 43 44 dissipation theorem. Thus Nozières and Pines [30] expressed the Coulomb interaction 45 46 energy in the electron gas through integrals over the dielectric function. Dispersion 47 48 energy contributions to the cohesion in crystals could similarly be obtained by 49 50 51 accommodating periodicity and exchange [2]. 52 53 Much emphasis has been centered on the possibility of the direct determination of 54 55 56 57 58 04/12/10 9 59 60 URL: http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics Page 10 of 31

1 2 3 excitation energies and transition moments from the poles and residues of the polarization 4 5 6 propagator. The basic approximation has several flavors: linearized time-dependent 7 8 Hartree-Fock, Random Phase Approximation, Equation of Motion method to mention the 9 10 11 most common ones. Their formal equivalence derives from an operator algebra adapted 12 13 to be similar to the one for the electron propagator. A number conserving operator is 14 For Peer Review Only 15 † † † considered of the form A = Σ ar [ crs + crtus at au + …] as and the sesquilinear forms 16 17 † † 18 Λ =〈 [ A, A ] 〉 and Ω =〈 [ [ A, H ], A ] 〉are such that Ω is a non-negative 19 20 hermitian form when the ground state/ensemble is stable towards distortions and an 21 22 23 energy minimum. The proper diagonal expressions for Λ and Ω offer pairs of adjoint 24 25 operators, A† and A, that can be normalized so that 26 27 1 =〈 [ [ A, H ], A† ] 〉=〈 [ [ A†, H ], A ] 〉; λ =〈 [ A, A† ] 〉> 0. 28 29 † 30 Eigenvalues λ are proportional to wave lengths of excitations created by A and the 31 32 † 2 associated propagator is « A; A »E = λ /( E λ – 1 ). The λ spectrum is bounded from 33 34 35 above by the inverse of the first excitation energy of the system but approximations in the 36 37 matrix evaluations causes the bound to be approximate as well. 38 39 An attractive feature of the random phase type approximations is that transition 40 41 42 moments and oscillator strengths can be calculated in either the dipole length or the 43 † 44 dipole velocity form if the spin orbital basis is such that the length operator D = Σ drsar as 45 46 † 47 and the velocity operator V = Σ vrsar as are related by the commutation relation 48 49 iV = [ D, H ]. 50 51 This holds for a complete basis and may be enforced in the Pariser-Parr-Pople model [31] 52 53 54 and, occasionally, in general cases. Limited basis sets may destroy fundamental 55 56 57 58 04/12/10 10 59 60 URL: http://mc.manuscriptcentral.com/tandf/tmph Page 11 of 31 Molecular Physics

1 2 3 commutation rules, e. g. [ D , D ] ≠ 0, which has consequences for the choice of 4 x y 5 6 reference system orientation. Transition moments involving magnetic fields need 7 8 operator representations that are consistent with [32]. 9 10 11 Consistency implies that average values are determined from the propagators, 12 13 vide infra, and can be achieved in simple situations [33]. The related quest for a explicit 14 For Peer Review Only 15 ground state representative resulted in the conclusion that the standard random phase 16 17 18 approximation is consistent with an antisymmetrized geminal power state [34]. Further 19 20 discussion of this form is presented in an another contribution to this issue. 21 22 23 Reflexions by old men 24 25 Momentous occurrences in the April quarter of 1968, among them the murders of Martin 26 27 Luther King Jr. and Robert F. Kennedy, interspersed a very active scientific collaboration 28 29 30 of the present authors at the Quantum Theory Project. We started a project that was to go 31 32 on for a few years and resulted in “the little yellow book” [35] and were much concerned 33 34 with the matters of fractional occupation numbers in variants of self-consistent theory. 35 36 37 Our paper on the underpinnings of the Pariser-Parr-Pople model [12] exposed the 38 39 problem of total energy calculations from different but equivalent expressions. These 40 41 difficulties arise from truncations in operator spaces and the lack of a proper variational 42 43 44 functional beyond the Hartree-Fock approximation. Nozières's textbook [36] deals with 45 46 higher order approximations and the ensuing Ward indentities. No entirely satisfactory 47 48 model has as yet been devised and the applications of propagators to total energy 49 50 51 questions cannot maintain a variational bound to the results. This is similar to the variants 52 53 of the coupled-cluster methods but these are being developed to offer reliable estimates to 54 55 56 57 58 04/12/10 11 59 60 URL: http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics Page 12 of 31

1 2 3 total energies as examplified in adjoining contributions to this issue. 4 5 6 Density functional theory relies on the concept of a universal relation between the 7 8 electron density and the total electronic energy. The precise functional has as yet eluded 9 10 11 the practioners but quite satisfactory approaches are in use. This will also be dealt with by 12 13 others in this issue. Time-dependent extensions lead to equations that are formally 14 For Peer Review Only 15 equivalent with the ones discussed above under the polarization propagator heading. 16 17 18 Several recent applications seem to deviate from the proper path by mixing elements such 19 20 as exchange with the pure density functional formalism. The present authors perceive that 21 22 electronic structure theory has reached a plateau where the next mountain range requires 23 24 25 new ideas, concepts, and mathematics. The discretization of three-space by Gaussian 26 27 basis sets was initiated by Boys and Preuss more than 50 years ago and the effect has 28 29 been tremendous. We ask ourselves, however, about the essential information content in 30 31 32 huge integral arrays and associated numerical processes that demand computational 33 34 efforts that increase as some large power of the number of atoms in our systems. Most of 35 36 37 the information is provided by very modest considerations and it is the small, albeit 38 39 important, deviation that has necessitated the very large scale machinery. Löwdin 40 41 maintained that theoretical advances could be as spectacular as the hardware 42 43 44 improvement, it remains to be seen. 45 46 Among the formal tools that might have potential for substantial changes in the 47 48 approach to electronic structure theory is the so called adiabatic connection. It is based on 49 50 51 the equality of the derivative of the expectation value of the hamiltonian with respect to a 52 53 parameter and the expectation of the derivative of the operator itself: 54 55 56 57 58 04/12/10 12 59 60 URL: http://mc.manuscriptcentral.com/tandf/tmph Page 13 of 31 Molecular Physics

1 2 3 E( ) = 〈 H( ) 〉; dE( )/d = 〈 dH( )/d 〉. 4 ξ ξ ξ ξ ξ ξ 5 6 This holds for eigenstates of the hamiltonian and for variationally stable states and is 7 8 known as the Hellmann-Feynman theorem [37]. The adiabatic concept allows the 9 10 11 integration to give 12 13 E(ξ1) – E(ξ0) = ∫d ξ〈 dH(ξ)/d ξ 〉 14 For Peer Review Only 15 and this is useful when a suitable expression is available for the integrand. This has been 16 17 18 realized in Hückel theory where formal charges and bond orders serve as derivatives [38]. 19 20 Changes in atomic parameters or bonds could then be estimated and indications be drawn 21 22 about reactivities, induced spin densities and similar properties. 23 24 25 Both Hellmann and Feynman were concerned with the forces between atoms and 26 27 explored the energy variation with changes in nuclear positions within the Born- 28 29 30 Oppenheimer picture. Their results are accordingly relevant for considerations based on 31 32 the virial theorem. Slater [39] deduced that the kinetic energy in the ground state of a 33 34 diatomic molecule at internuclear distance R equals T = - E(R) – R∂E(R)/∂R while the 35 36 37 potential energy is V =2E(R) + R∂E(R)/∂R. The first relation expresses the total energy in 38 39 terms of an integral over the expectation value os a one-electron operator that can be 40 41 evaluated from the electron propagator. Succes of such a procedure depends on the 42 43 44 consistency of the approximation of the propagator. 45 46 So is also the case when the fluctuation-dissipation theorem is applied to the 47 48 evaluation of the electron interaction energy in perturbation theory. The premise is that 49 50 † † † † † 51 expectation values 〈ar at auas〉=δst〈 ar au〉+Σn 〈0⎢ ar as⎢n〉〈n⎢ at au⎢0〉are 52 53 obtainable from the corresponding response function or propagator and that these are 54 55 56 57 58 04/12/10 13 59 60 URL: http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics Page 14 of 31

1 2 3 calculated as functions of a formal interaction strength parameter. The integration from 4 5 6 zero to unit strength offers an estimate of the total energy contribution. A simple 7 8 application to the linear chain gives a modest improvement over the basic 9 10 11 mean field approximation [40]. 12 13 Propagators have been part of these authors lives for the better part of the fifty 14 For Peer Review Only 15 years since the Quantum Theory Project was initiated. Their emphasis on processes rather 16 17 18 than stationary states appeals to the physico-chemical realization that knowledge is 19 20 obtained by probing the response of the material world. They do connect the pure state 21 22 formulation with ensemble structures and density matrices. 23 24 25 Jan Linderberg (*October 27, 1934) completed the requirements for the degree of Doctor of Philosophy at 26 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 27 the University of Florida and at the Department of Chemistry at the University of Utah. Some 100 scientific 28 papers and a couple of books have emanated through his efforts. 29 Yngve Öhrn (*June 11, 1934) completed the requirements for the degree of Doctor of Philosophy at 30 Uppsala University in May 1966. He was appointed to the faculty of the Department of Chemistry at the 31 University of Florida in 1966, served as chair 1977-83, director of the Quantum Theory Project 1983-98, 32 and entered emeritus status 2008. Editor-in-chief of International Journal of Quantum Chemistry. Several 33 books and some 100 scientific papers have resulted from his efforts. 34 35 36 References 37 [1] HEDIN, L. T. AND LUNDQVIST, S. O., 1960 Introduction to the Field Theoretical 38 Approach to the Many-Electron Problem (Uppsala: Quantum Chemistry Group 39 TIII, unpublished). 40 [2] LINDERBERG, J., 1964, Arkiv Fysik 26, 323, LINDERBERG, J. AND BYSTRAND, F., 1964 Arkiv 41 42 Fysik 26, 383. 43 [3] HUBBARD, J., 1963, Proc. Roy. Soc. (London) A276, 238. 44 [4] PARISER, R., AND PARR, R. G., 1953, J. Chem. Phys. 21, 466, POPLE, J., 1953, Trans. 45 Faraday Soc.49, 1375. 46 [5] LINDERBERG, J., AND ÖHRN, Y., 1965, Proc. Roy. Soc. (London) A285, 445. 47 48 [6] ÖHRN, Y., AND LINDERBERG, J., 1965, Phys. Rev. 139, A1063. 49 [7] SLATER, J. C., 1968, Am. J. Phys. 36, 69. 50 [8] LONGUET-HIGGINS, H. C., 1966, Quantum Theory of Atoms, Molecules,and the Solid 51 State (New York: Academic Press, Editor Per-Olov Löwdin) p. 105. 52 [9] MCLACHLAN, A. D., 1961, Mol. Phys. 4, 49. 53 [10] MCLACHLAN, A. D., AND BALL, M. A., 1964, Rev. Mod. Phys. 36, 844. 54 55 [11] LÖWDIN, P.-O., 1955, Phys. Rev. 97, 1474. 56 57 58 04/12/10 14 59 60 URL: http://mc.manuscriptcentral.com/tandf/tmph Page 15 of 31 Molecular Physics

1 2 3 [12] LINDERBERG, J., AND ÖHRN, Y., 1968, J. Chem. Phys. 49, 716. 4 5 [13] KAIJSER, P., AND LINDERBERG, J., 1973, J. Phys. B: Atom. Molec. Phys. 6, 1975. 6 [14] VIINIKKA, E.-K., AND ÖHRN, Y., 1975, Phys. Rev. B 11, 4168. 7 [15] JANSIK, B., HØST, S., JOHANSSON, M. P., OLSEN, J., AND JØRGENSEN, J., 2009, J. Chem. 8 Theory Comput. 5, 1027. 9 [16] JØRGENSEN, P., AND ÖHRN, Y., 1973, Phys. Rev. A 8, 112. 10 11 [17] JØRGENSEN, P., AND ÖHRN, Y., 1973, Chem. Phys. Letters 18, 261. 12 [18] ABDULNUR, S. F., LINDERBERG, J., ÖHRN, Y. AND THULSTRUP, P. W., 1972, Phys. Rev. A6, 13 889. 14 [19] JØRGENForSEN, C. K. , 1Peer962, Orbitals iReviewn Atoms and Molecul esOnly (New York: Academic 15 Press) p.85. 16 [20] MULLIKEN, R. S., 1934, J. Chem. Phys. 2, 782. 17 18 [21] PARR, R. G., AND PEARSON, R. G., 1983, J. Am. Chem. Soc. 105, 7512. 19 [22] SLATER, J. C., 1972, Adv. Quant. Chem. 6, 1, particularly p.30ff. 20 [23] LINDERBERG, J., AND ÖHRN, Y., 1967, Chem. Phys. Lett. 1, 295. cf ROTH, L. M., 1968, 21 Phys. Rev. Lett. 20, 1431. 22 [24] PURVIS, G. D., AND ÖHRN, Y., 1974, J. Chem. Phys. 60, 4063. 23 24 [25] PURVIS, G. D., AND ÖHRN, Y., 1975, J. Chem. Phys. 62, 396. 25 [26] TYNER REDMON, L., PURVIS, G. D., AND ÖHRN, Y., 1975, J. Chem. Phys. 63, 5011. 26 [27] LINDHARD, J., 1954, On the Properties of a Gas of Charged Particles, Det Kongelige 27 Danske Videnskabernes Selskab, Matematisk-fysiske Meddelelser 28, nr. 8, 28 particularly the Appendix. 29 [28] DIRAC, P. A. M., 1930, Proc. Cambridge Phil. Soc. 26, 376. 30 31 [29] EHRENREICH, H., AND COHEN, M. H., 1959, Phys. Rev. 115, 786, GOLDSTONE, J, AND 32 GOTTFRIED, K., 1959, Nuovo Cimento 13, 849. 33 [30] NOZIÈRES, P. AND PINES, D., 1958, Nuovo Cimento 9, 470. 34 [31] LINDERBERG, J., 1967, Chem. Phys. Lett. 1, 39. 35 [32] SEAMANS, L., AND LINDERBERG, J., 1972, Mol. Phys. 24, 1393. 36 37 [33] LINDERBERG, J., AND RATNER, M., 1970, Chem. Phys. Lett. 6, 37. 38 [34] LINDERBERG, J., AND ÖHRN, Y., 1977, Int. J. Quant. Chem. 12, 161, ÖHRN, Y., AND 39 LINDERBERG, J., 1979, Int. J. Quant. Chem. 15, 343. 40 [35] LINDERBERG, J., AND ÖHRN, Y., 1973, Propagators in Quantum Chemistry (London: 41 Academic Press) 42 [36] NOZIÈRES, P. 1963, Le problème a N corps (Paris: Dunod ) p. 234. 43 44 [37] HELLMANN, H., 1937, Einführung in die Quantenchemie, (Leipzig: Franz Deuticke) p. 45 285, FEYNMAN, R. P., 1939, Phys. Rev. 56, 340. 46 [38] COULSON, C. A., O'LEARY, B., AND MALLION, R. B. 1978, Hückel Theory for Organic 47 Chemists (London: Academic Press) offers a presentation of Coulson's pioneering 48 work and references to his original papers. 49 50 [39] SLATER, J. C., 1933, J. Chem. Phys. 1, 687. 51 [40] LINDERBERG, J., 1980, Physica Scripta 21, 373. 52 53 54 55 56 57 58 04/12/10 15 59 60 URL: http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics Page 16 of 31

1 2 3 4 5 6 7 8 9 10 11 12 13 14 For Peer Review Only 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 04/12/10 16 59 60 URL: http://mc.manuscriptcentral.com/tandf/tmph Page 17 of 31 Molecular Physics

1 2 3 4 Propagators and their implications 5 6 Yngve Öhrn and Jan Linderberg 7 8 9 Quantum Theory Project, Departments of Chemistry and Physics, University of Florida, 10 Gainesville, Florida U. S. A. 11 12 YÖ: [email protected] , JL: [email protected] 13 14 (Received 20For July 2005; Peer final version Reviewreceived 17 August 2006) Only 15 16 17 Nearly fifty years of collaboration on the development of formulations of the many-electron 18 theory for molecular, atomic and condensed matter in terms of propagators is surveyed. 19 20 Keywords: propagator; density; response; 21 22 23 Initiation 24 25 Field theoretical methods were familiar to Per-Olov Löwdin from his early work with 26 27 relativity and electromagnetic theory. He was alerted to its impact in many electron 28 29 theory through the works by Gell-Mann and Brueckner and by Hubbard in the mid 50's. 30 31 32 Thus he wanted to establish the connection to the configuration space methods that he 33 34 was so familiar with and to this end he assigned one of us (JL) to give a seminar to the 35 36 Quantum Chemistry Group at Uppsala on second quantization in the fall of 1957. The 37 38 39 sources were the classical papers by Born and Jordan, the detailed treatment by Fock and 40 41 the treatises by Dirac and Corson. Löwdin followed up with formal field theory but it was 42 43 44 when Stig Lundqvist returned from a sabbatical with Brueckner that the concepts and 45 46 techniques became more familiar topics at the Group. The lecture notes taken by Lars 47 48 Hedin was the basis for the Technical Report from the Group [1] and Lundqvist's lectures 49 50 51 at the first Winter Institute at Gainesville in December 1960 and Sanibel Island in 52 53 January 1961. The authors spent a brief period together at Gainesville in early 1961 when 54 55 YÖ began a two-year stay at the University of Florida. 56 57 58 August 2, 2010 1 59 60 URL: http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics Page 18 of 31

1 2 3 Löwdin suggested in September 1961 that time was ripe for an attack on the 4 5 6 dispersion energy contribution to the cohesive energy of molecular solids, in particular 7 8 rare gas crystals, and assigned the task to JL who had just returned from a year with the 9 10 11 Quantum Theory Project. The treatment of fluctuating dipole fields as the source of long 12 13 range interactions was most readily tackled in terms of the concepts of propagators and 14 For Peer Review Only 15 associated elements of the field theoretical tool box. Decent results were found with 16 17 18 moderately elaborate calculations [2]. 19 20 The present authors were reunited at Uppsala in 1963 and were inspired by John 21 22 Hubbard's paper on electron correlation in narrow energy bands [3] to apply his Green 23 24 25 function decoupling procedure to the generalized Hückel model introduced by Pariser 26 27 and Parr and by Pople [4]. Our results were exciting and were gracefully communicated 28 29 by Coulson [5]. Additional results seemed to corroborate that we were on to something 30 31 32 useful [6]. 33 34 Methods from field theory were not generally embraced in the molecular 35 36 37 electronic structure community. Slater was particularly skeptical towards the techniques 38 39 of second quantization that had become “stylish” [7] and Löwdin was concerned about 40 41 the N-representability issue. There were efforts under way at Cambridge to further the use 42 43 44 of second quantization, somewhat ironically published in a tribute to Slater [8]. 45 46 McLachlan spearheaded the applications in molecular orbital theory [9] and his paper 47 48 with Ball [10] remains a basic reference for time-dependent studies of electronic 49 50 51 properties. We were not alone and continued our exploration of the options offered by 52 53 second quantization, propagators and novel approximation methods. 54 55 56 57 58 August 2, 2010 2 59 60 URL: http://mc.manuscriptcentral.com/tandf/tmph Page 19 of 31 Molecular Physics

1 2 3 Second quantization algebra 4 5 Molecular electronic structure theory is almost exclusively based on a representation in 6 7 terms of an orbital basis set, be it either a formal or a precisely defined one. The Hückel 8 9 10 and Pariser-Parr-Pople models are examples of the former while the calculational 11 12 schemes emanating from Boys's, Roothaan's and Hall's early development represent the 13 14 latter. BothFor situations operatePeer with a finiteReview basis of spin orbitals Only U:{ u1(x), u2(x), 15 16 17 …uM(x)} and the associated power set P(U) , which has a one-to-one correspondence to 18 19 the set of all Slater determinants that can be formed from U. The empty set is included i 20 21 P(U) and represents the state without electrons, often called the vacuum. There are 2M 22 23 24 elements in the power set and they are either denoted by the list of the particular spin 25 26 orbitals for a given element or by the occupation number of the individual basis orbitals, 27 28 29 a M-digit binary number where each position corresponds to a spin orbital in U. Elements 30 31 of the power set P(U) span a subspace of the full Fock space that can accommodate an 32 33 arbitrary number of electrons. 34 35 36 Transformations or mappings between elements of the power set are composed of 37 † 38 elementary creation (addition, ar ) or annihilation (removal, ar) operators for individual 39 40 spin orbitals. A consistent algebra requires the anticommutation relations 41 42 † † † † † † 43 ar as + as ar = 0; a r as + as ar = 0; a r as + as ar = δrs + S rs . 44 45 Overlap integrals S occur for most primitive bases and may be removed by linear 46 47 48 transformations. General many electron operators occur as linear combinations of 49 50 products, the hamiltonian, as an example, is 51 52 † † † H = Σrs a r h rs a s + ½ Σrr'ss' (rs|r's') a r ar' as' as. 53 54 55 Mulliken's notation is used for the Coulomb interaction between spin orbital densities 56 57 58 August 2, 2010 3 59 60 URL: http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics Page 20 of 31

1 2 3 u *(x)u (x) and u *(x' )u (x' ). The formalism exposes the structural similarity between 4 r s r' s' 5 6 methods where the integrals, hrs and (rs |r's' ), are derived from specified spin orbitals and 7 8 proper integrations and methods where they are inferred from other sources. The 9 10 11 evaluation of a matrix representation of the hamiltonian from the elements of the power 12 13 set requires the standard rules as put forth by Löwdin and others [11]. 14 For Peer Review Only 15 The second quantization algebra offers an approach to avoid the explicit 16 17 18 construction of a matrix representative of the hamiltonian and still be able to infer 19 20 essential properties. It holds, in an orthonormal basis, that 21 22 [a ,H ] = Σ h a + Σ (rs |r's' )a †a a ; [ [ a ,H ], a †] = h + Σ {( rs|r's' )-(rs'|r's )} a †a 23 r s rs s r'ss' r' s' s r s + rs r's' r' s'

24 † 25 and a mean field theory obtains when the operator product ar' as' is replaced by its 26 27 expectation value, the reduced density matrix γs'r' , in the second form. 28 29 30 Exponential operators provide compact representations of unitary transformations 31 32 of the basis and unusual symmetries, e.g. particle-hole relations in alternant hydrocarbon 33 34 molecules in the Pariser-Parr-Pople model, can be exposed. The developments of the 35 36 37 coupled-cluster variants in current usage would probably have been more awkward in 38 39 direct configuration space formulations. 40 41 42 The essential Σ−Π separation of the Pariser-Parr-Pople model appears rather 43 44 directly in the second quantization formulation [12] and details the role of the Σ- 45 46 47 framework as a dielectric medium which screens the interaction within the Π-system. A 48 49 link is also established to the form of a Heisenberg spin-hamiltonian in a weak coupling 50 51 limit. 52 53 54 55 Self-consistent fields 56 It was shown above that a mean field model for an electronic system results from a 57 58 August 2, 2010 4 59 60 URL: http://mc.manuscriptcentral.com/tandf/tmph Page 21 of 31 Molecular Physics

1 2 3 replacement, in the operator equation of motion, of an operator product with its mean 4 5 † 6 value: γsr = 〈 ar as 〉. A second step is then to relate the expectation value to the field. 7 8 Hartree introduced the criterion of self-consistency and this remains in electronic 9 10 11 structure theory as the Hartree-Fock scheme when proper exchange is accounted for. The 12 13 Fock matrix with elements frs = h rs + Σr's' {( rs|r's' )-(rs'|r's )} γs'r' is hermitean and its 14 For Peer Review Only 15 16 eigenvalues determine the canonical spin orbitals. These are used with the build up 17 18 principle to characterize the desired state. Neither spatial nor spin symmetry is explicitly 19 20 invoked and the general solution is rarely desirable. It is self-consistent to assume spin 21 22 23 invariance in a system with an even number of electrons and it is also consistent to 24 25 assume spherical symmetry for an atomic or ionic system with “magic” number of 26 27 28 electrons. Other cases will result in non-consistent situations. 29 30 Slater chose to evaluate the reduced density matrix elements from an ensemble 31 32 average based on an open shell configuration with a fixed number of electrons and 33 34 35 obtained a Fock matrix with spin orbitals appropriate for a central field. The open shell 36 37 spin orbitals were useful in further applications in the solid state but could not be related 38 39 to ionization processes. Applications to transition metal atoms uncovered another 40 41 42 awkwardness. A starting central field would indicate that the 4s -shell was filled and the 43 44 least bound spin orbitals were the 3d 's. The field from this occupation would lead to the 45 46 opposite and a consistent solution was not obtained. A remedial approach introduced 47 48 49 fractional occupations and acceptable solutions. Slater's derivation was not based on an 50 51 ensemble average but came directly from an expression that is valid only for integer 52 53 54 occupations. A proper ensemble was readily described in second quantization terms. 55 † 56 Thus it holds that averages 〈 ar as 〉 = q rδrs imply the ensemble operator 57 58 August 2, 2010 5 59 60 URL: http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics Page 22 of 31

1 2 3 † † 4 Γ = Πr [(1 – qr)arar + q rar ar]; Tr Γ = 1. 5 6 The standard Hartree-Fock state has occupation numbers qr either zero or one. The 7 8 ensemble gives an average energy as a function of the occupation numbers, 9 10 11 E =Tr ΓH = Σr hrr qr +½ Σrs [ (rr|ss ) – (rs|sr ) ] q r qs, 12 13 and optimization of the spin orbitals results in Fock equations with fractional occupation 14 For Peer Review Only 15 numbers, if so desired. Derivatives of the energy, ε = ∂E/ ∂q , are the orbital energies 16 r r 17 18 and eigenvalues of the Fock matrix. The ensemble operator is not an eigenoperator of the 19 20 number operator N = Σ a †a and the dispersion comes to 21 op r r r 22 2 23 〈 [ Nop - < N op > ] 〉 = Πr ( 1 – qr ) qr 24 25 showing that the ensemble has contributions of varying electron number, a grand 26 27 28 canonical ensemble. Such a construct serves to define a suitable set of basis spin orbitals 29 30 for situations when spatial and spin symmetry features are desirable. Kaijser 31 32 demonstrated the usefulness of such a basis for the determination of transition moments 33 34 35 and life times for the manifold of low lying states of the Ti II ion [13]. Viinikka [14] also 36 37 applied the grand canonical ensemble construct to define a basis that performed well in 38 39 40 the study of the multiplet structure of core hole states in transition metal ions. Recent 41 42 developments in basis set design have found the ensemble average with fractional 43 44 occupation numbers to have advantages [15]. 45 46 47 The grand canonical as well as the canonical ensemble construct were used by 48 49 Poul Jørgensen [16], one of many postdoctoral associates that have been exchanged 50 51 between the JL and YÖ groups over the years. He compared the details of the theoretical 52 53 54 basis with that of the Slater construction of the Fock operator. He also applied [17] the 55 56 grand canonical ensemble construction for a time-dependent Hartree-Fock treatment of 57 58 August 2, 2010 6 59 60 URL: http://mc.manuscriptcentral.com/tandf/tmph Page 23 of 31 Molecular Physics

1 2 3 the triplet-triplet absorption spectra of alternant hydrocarbons. 4 5 6 Detailed calculations on transition metals [18] showed that there was no 7 8 indication that the orbital energy of the 4s -level should be below the 3d-level. It was 9 10 11 interesting to note that a simple function could represent the dependence on the 12 3/2 13 occupation number of the orbital energy: ε(q) = ε(0)[ 1 – q/q c ] . The maximum 14 For Peer Review Only 15 occupation, q , gives a slightly negative ion. This form can be compared to Jørgensen's 16 c 17 18 differential ionization concept [19], Mulliken's electronegativity [20], and the concept of 19 20 absolute hardness [21]. 21 22 23 Calculations with fractional occupation numbers were also used to approximate 24 25 excitation energies. Slater [22] showed that for an energy expression that is a continuous 26 27 function of occupation numbers one may attempt to calculate energy differences from 28 29 30 Taylor series expansions: 31 32 E(…qr – 1…qs + 1…) - E(…qr …qs …) = ∂E/ ∂qs - ∂E/ ∂qr + … 33 34 while observing that the second order terms are eliminated when the derivatives are 35 36 37 evaluated at the midpoint, ( qr – ½, qs + ½), termed the transition state. Applications 38 39 confirmed that this concept offered a viable alternative to the determination of separate 40 41 self-consistent solutions to the two states. 42 43 44 45 46 Electron propagator approximations 47 Hubbard's model for narrow energy bands in solids [3] led him to a decoupling of the 48 49 † 50 equations of motion for the electron propagator or Green function Grs (E) = << ar;a s >> E. 51 52 His approximation was reformulated by means of a matrix representation in an extended 53 54 55 operator basis [23]. Later efforts were developed in operator spaces spanned by general 56 57 58 August 2, 2010 7 59 60 URL: http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics Page 24 of 31

1 2 3 † † † 4 operator products. Useful forms are A = Σ a r (cr + c rst as at + … ) , they define two 5 † † 6 sequilinear forms, M = 〈 [ A, A ]+ 〉 and K = 〈 [ [ A, H ], A ]+ 〉, which may be 7 8 9 brought to diagonal form. The metric M is hermitian and non-negative while the effective 10 11 hamiltonian K is hermitian. Linear dependencies may occur from an initial choice of 12 13 operators and reflect properties of the state or ensemble used for the averages. A 14 For Peer Review Only 15 † † † 16 canonical set of operators { Aj :〈 [ Aj, A k ]+ 〉= δjk , 〈 [ [ Aj, H ], A k ]+ 〉= εjδjk } is 17 18 defined and one infers that a useful approximation for the propagators follows: 19 20 † 21 « Aj ; A k »E = δjk / ( E – εj ). 22 23 Further analysis of consistency requires that the expectation values that are used to define 24 25 26 M and K are derived from the propagators, i. e. by a contour integral in the complex E- 27 28 plane: 29 30 〈 A †A 〉 = (2 πi)-1 ∫ dE « A ; A † » 31 k j C j k E 32 33 Fulfillment of these conditions can rarely be realized beyond the Hartree-Fock level and 34 35 it is necessary to exercise caution. 36 37 38 Numerous calculations have demonstrated that a moderate extension of the 39 40 operator manifold provides excellent interpretations of photoelectron spectra, including 41 42 shake-up and shake-off processes, albeit not fully self-consistent. 43 44 45 Applications of electron propagator theory became possible with the seminal 46 47 work of George Purvis, who joined the group of YÖ at QTP in 1970. He developed a 48 49 program suite called GREENFUNC, which survived in bits and pieces for over a decade 50 51 52 in electronic structure software around the world. Early publications [24, 25, 26] set the 53 54 standard and were followed by many others from QTP. This development in the 55 56 propagator effort is covered in another publication in this issue of Molecular Physics. 57 58 August 2, 2010 8 59 60 URL: http://mc.manuscriptcentral.com/tandf/tmph Page 25 of 31 Molecular Physics

1 2 3 Polarization propagator approximations 4 5 Lindhard [27] was concerned with the stoppage of particles in matter and derived the 6 7 response of an electron gas to an external electric field by means of a self-consistent 8 9 10 determination of the induced effects in the gas. His wavelength and frequency dependent 11 12 dielectric function exhibited the collective plasmon excitations as well as single particle 13 14 features. ThisFor was the firstPeer detailed application Review of the linearized Only time-dependent Hartree 15 16 17 method and remains a landmark in electronic structure theory. Dirac had delineated the 18 19 equation of motion for the reduced density matrix in the time domain [28] and 20 21 applications to non-homogeneous systems [10, 29] appeared as complements to the 22 23 24 elaborate diagrammatic and formal perturbation theory expansions that were the rule in 25 26 many-electron theory during quite a few years. 27 28 29 Knowledge of response functions such as generalized polarizabilities and 30 31 dielectric functions offers a route to correlation features through the fluctuation- 32 33 dissipation theorem. Thus Nozières and Pines [30] expressed the Coulomb interaction 34 35 36 energy in the electron gas through integrals over the dielectric function. Dispersion 37 38 energy contributions to the cohesion in crystals could similarly be obtained by 39 40 accommodating periodicity and exchange [2]. 41 42 43 Much emphasis has been centered on the possibility of the direct determination of 44 45 excitation energies and transition moments from the poles and residues of the polarization 46 47 propagator. The basic approximation has several flavors: linearized time-dependent 48 49 50 Hartree-Fock, Random Phase Approximation, Equation of Motion method to mention the 51 52 most common ones. Their formal equivalence derives from an operator algebra adapted 53 54 55 to be similar to the one for the electron propagator. A number conserving operator is 56 57 58 August 2, 2010 9 59 60 URL: http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics Page 26 of 31

1 2 3 † † † 4 considered of the form A = Σ ar [ crs + c rtus a t a u + … ] as and the sesquilinear forms 5 6 Λ =〈 [ A, A † ] 〉 and = 〈 [ [ A, H ], A † ] 〉 are such that is a non-negative 7 8 9 hermitian form when the ground state/ensemble is stable towards distortions and an 10 11 energy minimum. The proper diagonal expressions for Λ and offer pairs of adjoint 12 13 operators, A† and A, that can be normalized so that 14 For Peer Review Only 15 † † † 16 1 = 〈 [ [ A, H ], A ] 〉 = 〈 [ [ A , H ], A ] 〉; λ =〈 [ A, A ] 〉 > 0. 17 18 Eigenvalues λ are proportional to wave lengths of excitations created by A† and the 19 20 † 2 21 associated propagator is « A; A »E = λ /( E λ – 1 ). The λ spectrum is bounded from 22 23 above by the inverse of the first excitation energy of the system but approximations in the 24 25 26 matrix evaluations causes the bound to be approximate as well. 27 28 An attractive feature of the random phase type approximations is that transition 29 30 31 moments and oscillator strengths can be calculated in either the dipole length or the 32 † 33 dipole velocity form if the spin orbital basis is such that the length operator D = Σ drs ar as 34 35 and the velocity operator V = Σ v a †a are related by the commutation relation 36 rs r s 37 38 iV = [ D, H ]. 39 40 This holds for a complete basis and may be enforced in the Pariser-Parr-Pople model [31] 41 42 43 and, occasionally, in general cases. Limited basis sets may destroy fundamental 44 45 commutation rules, e. g. [ Dx , D y ] ≠ 0, which has consequences for the choice of 46 47 reference system orientation. Transition moments involving magnetic fields need 48 49 50 operator representations that are consistent with translational symmetry [32]. 51 52 Consistency implies that average values are determined from the propagators, 53 54 vide infra, and can be achieved in simple situations [33]. The related quest for an explicit 55 56 57 58 August 2, 2010 10 59 60 URL: http://mc.manuscriptcentral.com/tandf/tmph Page 27 of 31 Molecular Physics

1 2 3 ground state representative resulted in the conclusion that the standard random phase 4 5 6 approximation is consistent with an antisymmetrized geminal power state [34]. Further 7 8 discussion of this form is presented in another contribution to this issue. 9 10 11 12 Reflexions by old men 13 Momentous occurrences in the April quarter of 1968, among them the murders of Martin 14 For Peer Review Only 15 Luther King Jr. and Robert F. Kennedy, interspersed a very active scientific collaboration 16 17 18 of the present authors at the Quantum Theory Project. We started a project that was to go 19 20 on for a few years and resulted in “the little yellow book” [35] and were much concerned 21 22 with the matters of fractional occupation numbers in variants of self-consistent theory. 23 24 25 Our paper on the underpinnings of the Pariser-Parr-Pople model [12] exposed the 26 27 problem of total energy calculations from different but equivalent expressions. These 28 29 30 difficulties arise from truncations in operator spaces and the lack of a proper variational 31 32 functional beyond the Hartree-Fock approximation. Nozières's textbook [36] deals with 33 34 higher order approximations and the ensuing Ward identities. No entirely satisfactory 35 36 37 model has as yet been devised and the applications of propagators to total energy 38 39 questions cannot maintain a variational bound to the results. This is similar to the variants 40 41 of the coupled-cluster methods but these are being developed to offer reliable estimates to 42 43 44 total energies as exemplified in adjoining contributions to this issue. 45 46 Density functional theory relies on the concept of a universal relation between the 47 48 electron density and the total electronic energy. The precise functional has as yet eluded 49 50 51 the practioners but quite satisfactory approaches are in use. This will also be dealt with by 52 53 others in this issue. Time-dependent extensions lead to equations that are formally 54 55 56 equivalent with the ones discussed above under the polarization propagator heading. 57 58 August 2, 2010 11 59 60 URL: http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics Page 28 of 31

1 2 3 Several recent applications seem to deviate from the proper path by mixing elements such 4 5 6 as exchange with the pure density functional formalism. The present authors perceive that 7 8 electronic structure theory has reached a plateau where the next mountain range requires 9 10 11 new ideas, concepts, and mathematics. The discretization of three-space by Gaussian 12 13 basis sets was initiated by Boys and Preuss more than 50 years ago and the effect has 14 For Peer Review Only 15 been tremendous. We ask ourselves, however, about the essential information content in 16 17 18 huge integral arrays and associated numerical processes that demand computational 19 20 efforts that increase as some large power of the number of atoms in our systems. Most of 21 22 the information is provided by very modest considerations and it is the small, albeit 23 24 25 important, deviation that has necessitated the very large-scale machinery. Löwdin 26 27 maintained that theoretical advances could be as spectacular as the hardware 28 29 improvement, it remains to be seen. 30 31 32 Among the formal tools that might have potential for substantial changes in the 33 34 approach to electronic structure theory is the so-called adiabatic connection. It is based on 35 36 37 the equality of the derivative of the expectation value of the hamiltonian with respect to a 38 39 parameter and the expectation of the derivative of the operator itself: 40 41 E( ξ) = 〈 H( ξ) 〉; dE( ξ)/d ξ = 〈 dH( ξ)/d ξ 〉. 42 43 44 This holds for eigenstates of the hamiltonian and for variationally stable states and is 45 46 known as the Hellmann-Feynman theorem [37]. The adiabatic concept allows the 47 48 49 integration to give 50 51 E( ξ1) – E( ξ0) = ∫d ξ 〈 dH( ξ)/d ξ 〉 52 53 and is useful when a suitable expression is available for the integrand. This has been 54 55 56 realized in Hückel theory where formal charges and bond orders serve as derivatives [38]. 57 58 August 2, 2010 12 59 60 URL: http://mc.manuscriptcentral.com/tandf/tmph Page 29 of 31 Molecular Physics

1 2 3 Changes in atomic parameters or bonds could then be estimated and indications be drawn 4 5 6 about reactivities, induced spin densities and similar properties. 7 8 Both Hellmann and Feynman were concerned with the forces between atoms and 9 10 11 explored the energy variation with changes in nuclear positions within the Born- 12 13 Oppenheimer picture. Their results are accordingly relevant for considerations based on 14 For Peer Review Only 15 the virial theorem. Slater [39] deduced that the kinetic energy in the ground state of a 16 17 18 diatomic molecule at an internuclear distance R equals T = - E (R) – R∂E(R)/∂R while the 19 20 potential energy is V =2E (R) + R∂E(R)/∂R. The first relation expresses the total energy in 21 22 terms of an integral over the expectation value of a one-electron operator that can be 23 24 25 evaluated from the electron propagator. Success of such a procedure depends on the 26 27 consistency of the approximation of the propagator. 28 29 So is also the case when the fluctuation-dissipation theorem is applied to the 30 31 32 evaluation of the electron interaction energy in perturbation theory. The premise is that 33 34 † † † † † expectation values 〈ar at auas〉 = δst 〈 a r au〉 + Σn 〈0′ar as′n〉〈n′at au′0〉 are 35 36 37 obtainable from the corresponding response function or propagator and that these are 38 39 calculated as functions of a formal interaction strength parameter. The integration from 40 41 42 zero to unit strength offers an estimate of the total energy contribution. A simple 43 44 application to the linear chain Hubbard model gives a modest improvement over the basic 45 46 mean field approximation [40]. 47 48 49 Propagators have been part of these authors’ lives for the better part of the fifty 50 51 years since the Quantum Theory Project was initiated. Their emphasis on processes rather 52 53 than stationary states appeals to the physico-chemical realization that knowledge is 54 55 56 obtained by probing the response of the material world. They do connect the pure state 57 58 August 2, 2010 13 59 60 URL: http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics Page 30 of 31

1 2 3 formulation with ensemble structures and density matrices. 4 5 6 Jan Linderberg (*October 27, 1934) completed the requirements for the degree of Doctor of Philosophy at 7 Uppsala University in May 1964. He was appointed to the chair of theoretical chemistry at Aarhus 8 University in 1968 and retired in 2003. He holds courtesy appointments at the Department of Chemistry at 9 the University of Florida and at the Department of Chemistry at the University of Utah. Some 100 scientific 10 papers and a couple of books have emanated through his efforts. 11 Yngve Öhrn (*June 11, 1934) completed the requirements for the degree of Doctor of Philosophy at 12 Uppsala University in May 1966. He was appointed to the faculty of the Department of Chemistry at the 13 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 14 books and someFor 100 scientific Peer papers have resultedReview from his efforts. Only 15 16 17 References 18 [1] HEDIN , L. T. AND LUNDQVIST , S. O., 1960 Introduction to the Field Theoretical 19 Approach to the Many-Electron Problem (Uppsala: Quantum Chemistry Group 20 TIII, unpublished). 21 22 [2] L INDERBERG , J., 1964, Arkiv Fysik 26 , 323, L INDERBERG , J. AND BYSTRAND , F., 1964 23 Arkiv Fysik 26 , 383. 24 [3] HUBBARD , J., 1963, Proc. Roy. Soc. (London) A276, 238. 25 [4] PARISER , R., AND PARR , R. G., 1953, J. Chem. Phys. 21, 466, POPLE , J., 1953, Trans. 26 Faraday Soc. 49, 1375. 27 [5] LINDERBERG , J., AND ÖHRN , Y., 1965, Proc. Roy. Soc. (London) A285, 445. 28 29 [6] ÖHRN , Y., AND LINDERBERG , J., 1965, Phys. Rev. 139, A1063. 30 [7] SLATER , J. C., 1968, Am. J. Phys. 36, 69. 31 [8] LONGUET -HIGGINS , H. C., 1966, Quantum Theory of Atoms, Molecules,and the Solid 32 State (New York: Academic Press, Editor Per-Olov Löwdin) p. 105. 33 [9] MCLACHLAN , A. D., 1961, Mol. Phys. 4, 49. 34 35 [10] MCLACHLAN , A. D., AND BALL , M. A., 1964, Rev. Mod. Phys. 36, 844. 36 [11] LÖWDIN , P.-O., 1955, Phys. Rev. 97 , 1474. 37 [12] LINDERBERG , J., AND ÖHRN , Y., 1968, J. Chem. Phys. 49, 716. 38 [13] KAIJSER , P., AND LINDERBERG , J., 1973, J. Phys. B: Atom. Molec. Phys. 6, 1975. 39 [14] V IINIKKA , E.-K., AND ÖHRN , Y., 1975, Phys. Rev. B 11 , 4168. 40 [15] JANSIK , B., HØST , S., JOHANSSON , M. P., OLSEN , J., AND JØRGENSEN , J., 2009, J. 41 42 Chem. Theory Comput. 5, 1027. 43 [16] J ØRGENSEN , P., AND ÖHRN , Y., 1973, Phys. Rev. A 8, 112. 44 [17] J ØRGENSEN , P., AND ÖHRN , Y., 1973, Chem. Phys. Letters 18 , 261. 45 [18] ABDULNUR , S. F., LINDERBERG , J., ÖHRN , Y. AND THULSTRUP , P. W., 1972, Phys. 46 Rev. A6 , 889. 47 48 [19] JØRGENSEN , C. K., 1962, Orbitals in Atoms and Molecules (New York: Academic 49 Press) p.85. 50 [20] M ULLIKEN , R. S., 1934, J. Chem. Phys. 2, 782. 51 [21] PARR , R. G., AND PEARSON , R. G., 1983, J. Am. Chem. Soc. 105, 7512. 52 [22] SLATER , J. C., 1972, Adv. Quant. Chem. 6, 1, particularly p.30ff. 53 [23] LINDERBERG , J., AND ÖHRN , Y., 1967, Chem. Phys. Lett. 1, 295. cf ROTH , L. M., 54 55 1968, Phys. Rev. Lett. 20 , 1431. 56 [24] PURVIS , G. D., AND ÖHRN , Y., 1974, J. Chem. Phys. 60, 4063. 57 58 August 2, 2010 14 59 60 URL: http://mc.manuscriptcentral.com/tandf/tmph Page 31 of 31 Molecular Physics

1 2 3 [25] PURVIS , G. D., AND ÖHRN , Y., 1975, J. Chem. Phys. 62, 396. 4 5 [26] TYNER REDMON , L., PURVIS , G. D., AND ÖHRN , Y., 1975, J. Chem. Phys. 63, 5011. 6 [27] LINDHARD , J., 1954, On the Properties of a Gas of Charged Particles, Det 7 Kongelige Danske Videnskabernes Selskab, Matematisk-fysiske Meddelelser 28, 8 nr. 8, particularly the Appendix. 9 [28] DIRAC , P. A. M., 1930, Proc. Cambridge Phil. Soc. 26 , 376. 10 11 [29] EHRENREICH , H., AND COHEN , M. H., 1959, Phys. Rev. 115, 786, GOLDSTONE , J, AND 12 GOTTFRIED , K., 1959, Nuovo Cimento 13 , 849. 13 [30] NOZIÈRES , P. AND PINES , D., 1958, Nuovo Cimento 9, 470. 14 [31] LINDERBERGFor, J., 1967,Peer Chem. Phys. Review Lett. 1, 39. Only 15 [32] S EAMANS , L., AND LINDERBERG , J., 1972, Mol. Phys. 24 , 1393. 16 [33] LINDERBERG , J., AND RATNER , M., 1970, Chem. Phys. Lett. 6, 37. 17 18 [34] LINDERBERG , J., AND ÖHRN , Y., 1977, Int. J. Quant. Chem. 12 , 161, ÖHRN , Y., AND 19 LINDERBERG , J., 1979, Int. J. Quant. Chem. 15 , 343. 20 [35] LINDERBERG , J., AND ÖHRN , Y., 1973, Propagators in Quantum Chemistry (London: 21 Academic Press) 22 [36] NOZIÈRES , P. 1963, Le problème a N corps (Paris: Dunod ) p. 234. 23 24 [37] HELLMANN , H., 1937, Einführung in die Quantenchemie, (Leipzig: Franz Deuticke) 25 p. 285, FEYNMAN , R. P., 1939, Phys. Rev. 56, 340. 26 [38] COULSON , C. A., O'L EARY , B., AND MALLION , R. B. 1978, Hückel Theory for 27 Organic Chemists (London: Academic Press) offers a presentation of Coulson's 28 pioneering work and references to his original papers. 29 [39] SLATER , J. C., 1933, J. Chem. Phys. 1, 687. 30 31 [40] LINDERBERG , J., 1980, Physica Scripta 21 , 373. 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 August 2, 2010 15 59 60 URL: http://mc.manuscriptcentral.com/tandf/tmph