Modifying the HF procedure to include screening effects

# Modifying the HF procedure to include screening effects

## Abstract

A self-consistent formulation is proposed to generalize the HF scheme with the incorporation of screening effects. For this purpose in a first step, an energy functional is defined by the mean value for the full Hamiltonian, not in a Slater determinant state, but in the result of the adiabatic connection of Coulomb plus the nuclear (jellium charge) in the Slater determinant. Afterwards, the energy functional defining the screening approximation is defined in a diagrammatic way, by imposing a special ”screening” restriction on the contractions retained in the Wick expansion. The generalized self-consisting set of equations for the one particle orbitals are written by imposing the extremum conditions. The scheme is applied to the homogeneous electron gas. After simplifying the discussion by assuming the screening as static and that the mean distance between electrons is close to the Bohr radius, the equations for the electron spectrum and the static screening properties are solved by iterations. The self-consistent results for the self-energies dispersion does not show the vanishing density of states at the Fermi level predicted by the HF self-energy spectrum. In this extreme non retarded approximation, both, the direct and the exchange potentials are strongly screened, and the energy is higher that the one given by the usual HF scheme. However, the inclusion of the retardation in the exact solution and the sum rules associated to the dielectric response of the problem, can lead to energy lowering. These effects will be considered in the extension of the work.

###### pacs:
71.10.Fd,71.15.Mb,71.27.+a,71.30.+h,74.20.-z,74.25.Ha,74.25.Jb, 74.72.-h

## I Introduction

The development of band structure calculation procedures is a theme to which an intense research activity has been devoted in modern Solid State Physics. This area of research has a long history due to existence of important unsolved and relevant questions concerning the structure of solids (1); (2); (3); (4); (5); (6); (7); (8); (9); (10); (11); (12); (13); (14); (15); (16); (17); (18); (19); (20); (21); (22); (23); (24); (25); (26); (27); (43); (29); (30). One central open problem is the connection between the so called first principles schemes with the Mott phenomenological approach. This important approach, furnishing the descriptions for a wide class of band structures of solids, is not naturally explained by the assumed fundamental first principles approaches (30); (3); (12); (7); (5). A class of materials which had been in the central place in the existing debate between these two conceptions for band structure calculations are the transition metal oxides (TMO) (10); (12); (7); (16); (25); (17); (20); (21). A particular compound which is closely related with the TMO is the superconductor material in which the first band structure calculations predicted metal and paramagnetic characters, which are largely at variance with the known isolator and antiferromagnetic nature of the material (29). Actually, these two qualities are fully believed to be purely strong correlation properties which are not derivable from HF independent particle descriptions (16); (25); (34). However, in Refs. (35); (36) a single band Hartree-Fock (HF) study was able to produce these assumed strong correlation properties of , as independent particle ones, arising from a combination of crystal symmetry breaking in and an entanglement of the spin and spacial structure of the single particle states. Therefore, this result opens the interesting possibility of clarifying the connection between the Mott and Slater viewpoints in the band theory of the TMO (35); (36). It should be underlined that a former HF study of in Ref. (37), also indicated an insulator an AF character of the material. However, the extremely large gap (  eV) which was predicted by that work had a radical discrepancy with the experimentally estimated value of eV. However, it could happens that the results of Ref. (37) could show a close link with the conclusions of Refs. (35); (36). Results in the literature supporting this possibility are the ones presented in Ref. (38), where the same recourse employed in Refs. (35); (36) of allowing a crystal symmetry breaking by doubling the unit cell of the lattice was employed. In that work, the spin was treated in a phenomenological way by including a spin dependent functional in the variational evaluation done. These two procedures allowed the authors to predict an insulator and AF structure for the material with a gap of 2 , well matching the experimental estimate.

The results in Ref. (35); (36) were obtained  by starting form a Tight Binding (T.B.) model with a square lattice size equal to the distance between Cu atoms in the CuO planes, by assuming a simple crystalline extended quadratic (near the Cu atoms) T.B. potential. The strength of the potential was fixed to reduce the width of the states to be smaller than the chosen lattice constant to obey the T.B. condition.  Afterwards the metal paramagnetic HF bands obtained when considering the Coulomb interaction in the model,  was fixed to approximately reproduce the single half filled band crossing the Fermi level, obtained in the evaluation of the band structure of in Ref. (29).  A curious point was that the value of the dielectric constant that was required to make equal the two bandwidths  was nearly . Therefore, it come to the mind that the introduction of a dielectric constant in the HF equations of Ref. (37), might be produced a band gap of nearly  eV for being very close to the measured one of eV.  This indicates that if a HF method could be extended to predict the screening properties, it could has the chance of  explaining the strongly correlation properties of and with luck, of its close related parents, the transition metal oxides.  The possibility exist that such an approach could furnish a simple way of catching the strong correlation properties of these materials giving in this way a clear explanation for the Mott properties, by showing its connection with the crystal symmetry breaking and spin-space entangled nature of single particle states (35); (36).

Therefore, in this paper we intend to start generalizing the HF scheme to introduce screening effects. A first possibility is considered here by constructing an energy functional to be optimized by a set of single particle states. The functional for the HF scheme is simply taken as the mean value of the exact Hamiltonian in the Slater determinant of formed by the single particle functions over which is optimized.

Since we are interested in introducing Coulomb interaction effects, a first general possibility consists in modifying the variational Slater determinant of the single particle functions by connecting in it the Coulomb interaction between the electrons and with nuclear ”jellium” potential by employing the Gell-Mann-Low procedure (40); (41); (42); (43).̇ This definition will have variational character, that is, will be equal to a mean value of the total Hamiltonian in one state, and thus the result after finding the extremes over the one particle states, should be greater than the ground state energy of the system.  However, the complications in solving the extremum equations following from introducing the Coulomb and nuclear interaction in an exact form will be enormous.  However, we are only interested in catching improvement on the HF scheme produced by more simple screening processes. Then, it becomes possible to approximate the above mentioned state (the one resulting of the connection of the electronic interactions) by restricting the sum over all the Feynman graphs to a set reflecting screening properties. Then, the simpler state will be chosen by only retaining in the sum given by the Wick expansion, only the graphs for which: a) for any line arriving to a point of a vertex and coming from a another different point, always exists another different line that joins the same two points.  This definition basically retains the diagrams in which are essentially the ”tree” diagrams (having now closed loops) in which arbitrary number of insertions of the polarization loop are included. The functional defined as the mean value of the Hamiltonian is also an upper bound for the ground state energy. However, this functional expression although simpler, yet will  lead to slightly cumbersome procedure. Thus, we defer its study to eventual future discussion.  In order to further simplify the form of the functional, but yet retaining screening effects, we even more simplify the functional by imposing the same screening approximation to the whole sum of Feynman diagrams which defines the functional in which the Coulomb and nuclear interactions are adiabatically connected.  This definition leads to scheme which is formulated in very close terms with the usual discussions of the screening approximation in Many Body theory (39).

Up to now, we had not been able to show that this approximation also has a variational character. However, at least it follows that the Feynman diagrams  defining the scheme represent allowed physical process, which suggests that the procedure can improve the HF results.  If the variational character turns to be valid, the result for energy will also furnish an upper bound for the ground state energy. This issue will be considered elsewhere.

After defining , the generalized HF equations are derived for the general case of inhomogeneous electron systems. They include exchange and direct like terms, but also show additional contributions being related with effect of the Coulomb interaction and nuclear potential on the kinetic energy of the electron which vanish for homogeneous systems. The equations are also written for the case homogeneous electron gas in which the full translation symmetry simplifies them. In this case, as in the usual HF scheme, only the kinetic and exchange terms are non vanishing. In this first exploration, it is assumed that the screening is static and the density is chosen to be one electron per sphere of radial dimension equal to the Bohr radius. Under these conditions, the self-consistent equations for the electron spectrum and the static screening properties are solved by iterations. The result for the dispersion does not present the vanishing density of states at the Fermi energy appearing in the fist in the HF self-energy spectrum (44); (45); (46); (47); (48).

The work proceeds as follows. In Section II, the generalization of the HF functional and equations of motions for the single particle orbitals are presented. Section III then considers the application to the homogenous electron gas. In the conclusions the results are reviewed and possible extensions of the work are commented.

## Ii Generalization of the HF scheme to include screening effects

### ii.1 Hartree Fock approximation

As it is known the HF approximation is obtained after minimizing the mean value of the full Hamiltonian of the system in a Slater determinant state  formed with the elements of a basis of wavefunctions .  For future purposes, let us define a free Hamiltonian in terms for these functions in the form

 H0=∑kϵk a+kak, (1)

where and are the creation and annihilation operators of the functions and the energy eigenstates are to be defined by the optimization process of the energy functional

 E[φ] = ⟨Φ0| H |Φ0⟩−∑klλkl(∫dx φ∗k(x)φl(x)−δkl), (2) ⟨x1,x2,...,xNp|Φ0⟩ =1Np!Det [φk(xl)] (3)

where is the number of electrons.  The bold letters will represent the combination of the position and spin coordinates as and the integrals will mean

 ∫dx=∑s=±1∫d →x.

In addition, in order to simplify the expressions when time dependence will be considered we will define also space-time coordinates

The system to be considered will be a set of electrons which move in the presence of a nuclear potential and interact among themselves with the Coulomb two body potential .  After adding the kinetic energy of the electrons, the total Hamiltonian is given in the Schrodinger coordinate representation as

 H =T+Vnuc+v, (4) T ≡−Np∑i=1ℏ22m∂2i, \ \ \ ∂i=∂∂ →xi, (5) Vnuc ≡−Np∑i=1∫d →z ρnuc(→z)e2|→z−→xi|, (6) v ≡Np∑i,j=1i≠j12e2|→xi−→xj|, (7)

and, in what follows, in order to simplify the notation, all the operators acting in coordinate space appearing, will be supposed to have a spin structure, which when not written,  are assumed to be the identity matrix in the representation of the functions on which the operators act.  Imposing the extremum conditions on over the variations of the functions, the equations for determining the single particle states follow.  The fact that, on the extremum values of the single particle functions, it is satisfied the implies

 E≥Eg, (8)

where is the ground state energy, which for a Hamiltonian system gives the lowest mean value possible in any state.

Let us now intend to construct a generalization of the method including screening effects.  For this purpose, initially, let us define the functional to be optimized in an alternative form assuring that the result includes Coulomb interaction effects. A possible form of this kind is the following one

 E = ⟨Φ0|(Uc α(0,−∞))+H Ucα(0,−∞) |Φ0⟩⟨Φ0|(Uc α(0,−∞))+Ucα(0,−∞) |Φ0⟩ =⟨Φ0|Uc α(∞,0) H Ucα(0,−∞) |Φ0⟩⟨Φ0|Uc α(∞,0)Ucα(0,−∞) |Φ0⟩, (9)

in which is the evolution operator connecting the Coulomb interaction among the electrons plus the nuclear potential in an initial previously defined Slater determinant (the ground state of the free Hamiltonian at a large time in the past

 Ucα(0,−∞) =∞∑n=0(−iℏ)nn!∫0−∞dt1∫0−∞dt2...∫0−∞dtn T [Hcα(t1),Hcα(t2),Hcα(t3)...,Hcα(tn)], =T [exp(−iℏ∫0−∞dt Hcα(t))], (10) Hcα(t) =exp(iℏH0t) Hcexp(−iℏH0t) e(−α | t |), (11) Hc =Vnuc+v, (12)

where is the sum of the Coulomb and nuclear electron interactions in a time independent Schrodinger representation, is the Gell-Mann-Low small parameter which implements the adiabatic connection of the interaction when taken in the limit The dependent exponential factor appearing in the definition of the interaction representation operators above, tends to one when vanishes (40); (41); (43) .  The symbol appearing is the time ordering operation which orders the operators from right to left for increasing  time arguments (40); (41); (43).

For the Hermitian conjugate operator it follows

 (Ucα(0,−∞))+ =∞∑n=0(iℏ)nn!∫0−∞dt1∫0−∞dt2...∫0−∞dtn T∗ [Hcα(t1),Hcα(t2),Hcα(t3)...,Hcα(tn)], =.T∗ [exp(iℏ∫0−∞dt Hcα(t))] =T [exp(iℏ∫0∞dt∗Hcα(t∗))]=T [exp(−iℏ∫∞0dt∗Hcα(t∗))] =Ucα(∞,0). (13)

Noticing that is equal to its interaction representation at

 H(0)=T(0)+Vnuc(0)+v(0), (14)

it follows that in formula (9), the operators are temporally ordered. Therefore, the Wick theorem can be applied.  The sum of the expressions associated to all the Feynman diagrams giving the denominator, coincides with the sum of all the possible terms associated to disconnected (unlinked) graphs of the numerator, multiplying the contribution of a given arbitrary connected graph.  Thus, this common factor cancels (40); (41); (43).  The expression for in this general form is given by the sum of the contributions being associated to all the diagrams being connected through their vertices and lines to the vertices defined by the operator In this case, since is again a mean value of in a given state, it follows that should be greater than the ground state energy

In spite of this property, the complexity of the sum over all the connected part will surely make the optimization problem an impossible to be solved one.  Thus, it is needed to further simplifying the functional incorporating

the screening effects.

### ii.2 Screening approximation

As it was just mentioned, the general form of the variational problem defined by (9) is too complicated for to be usable in concrete evaluations. Let us consider in this section two simplifications. They will be called as ”screening” ones, and will be implemented through an approximation of the Wick theorem. In the Wick expansion, the time ordered products of creation and annihilation operators are expressed as the sum of all normally ordered products including an arbitrary number of contractions between any pair of operators entering in the product. The terms in this expansion are related by a one-to-one mapping with all the diagrams generated by the Feynman rules of the theory (40); (41); (43).

The first ”screening” approximation to be considered will be one in which a variational state is defined after applying the Wick expansion to the operator entering in the definition of the state In the Feynman diagram expansion of the operator only the graphs will retained in which,  for any particular line joining two different ending points of any vertices, another different line always exists connecting the same two points (41).  We will describe this reduction of the diagrams associated to a time ordered operators by writing . Then, the variational state will have the form

 U(s)α|Φ0⟩ =[Ucα(0,−∞)]WS|Φ0⟩ =∞∑n=0(−iℏ)nn!∫0−∞dt1∫0−∞dt2...∫0−∞dtn [T [Hcα(t1),Hcα(t2),Hcα(t3)...,Hcα(tn)]]WS|Φ0⟩ (15)

and its Hermitian conjugate state satisfies

 ⟨Φ0|U(s)α =[⟨Φ0|Ucα(∞,0) ]WS =∞∑n=0(−iℏ)nn!∫∞0dt1∫∞0dt2...∫∞0dtn ⟨Φ0|[T [Hcα(t1),Hcα(t2),Hcα(t3)...,Hcα(tn)]]WS =⟨Φ0|[T [exp(−iℏ∫∞0dt Hcα(t))]]WS. (16)

It is clear that since the functional is defined as a mean value in the given normalized state, the minimization of process will be variational, that is Since the criterion for eliminating a contribution is only determined by the contractions between two linked vertices, the application to the screening approximation to a product of two to disconnected graphs and in which all the operators are contracted, gives the product of the two independently approximated graphs. That is, the same property as for the Wick expansion

 [G1G2]WS=[G1]WS[G2]WS. (17)

The approximation done in defining the state eliminates all the graphs in the Wick expansion of the state But when the scalar product defining  is taken, a new application of the Wick expansion to the full operator leads to diagrams not satisfying the defined screening approximation. That is, diagrams exist for which not all the internal fermion lines appears in polarization loops and tadpoles.  This fact is not an obstacle of principle, but it leads to an structure of the connected to sum of Feynman diagrams in which somewhat complicated three and four loops diagrams appear.  Then, in this work will not use the approximation in this form.  Rather, in order to retaining contributions consistently being given by diagrams showing the imposed ”screening” graphical restriction, the functional will be defined in the form

 E=⟨Φ0|[U(c)+αHU(c)α]WS|Φ0⟩⟨Φ0|[U(c)+αU(c)α]WS|Φ0⟩, (18)

in which, the modified Wick expansion is applied to the whole operators and .

After expanding the exponential defining the operators, the numerator has the expression

 ⟨Φ0|[U(c)+αHU(c)α]WS|Φ0⟩ =∞∑n2=0(−iℏ)n2n2!∫∞0dt1∫∞0dt2...∫∞0dtn2 ∞∑n1=0(−iℏ)n1n1!∫0−∞dt′1∫0−∞dt′2...∫0−∞dt′n1 ⟨Φ0|[T [Hcα(t1),Hcα(t2),Hcα(t3)...,Hcα(tn2)]× H× T [Hcα(t′1),Hcα(t′2),Hcα(t′3)...,Hcα(t′n1)]]WS|Φ0⟩, (19)

and the denominator has a similar one, obtained by omitting the operator.

The figure (1) is a pictorial representation of the terms associated to the product of the power in the series expansion of and the power of the expansion of the operator in formula (19) for the functional. Each square represents one interaction operator evaluated a given time, which grows from right to left in the picture.  The illustrated term represents a particular contribution to the approximated Wick expansion adopted. In it, the white squares represent vertices which have contractions between the operators defining them. Such contractions are in such a form that the graph associated to these vertices and contractions are not connected to the vertices in the operator The white channels joining such vertices represent the set of contractions in the illustrated contribution. Alternatively, the black squares represent all the vertices of the interactions being connected among them and with vertices in by contractions. The square labeled with the letter  represents any of the vertices in the total Hamiltonian. The number of black squares come from the expansion of and the number from the expansion of  The black channels in this case symbolize the set of contractions between operators associated to these vertices.  The fact that all the terms in the Wick expansion are normal orderings including contractions, allows to reorder in the time direction all the square boxes in an arbitrary way. This follows because each vertex has an even number of these operators and the normal ordering are even under a permutations of groups of even numbers of such operators.  Therefore, the same contribution to the result of the expansion will be obtained from the terms appearing the expansion of the exponential operator each of which has a coefficient. In the same way, the  number of identical contributions in the expansion of changes the original coefficient by   Thus, each connected contribution will be multiplied by the sum of the same contributions defining the denominator in which is not appearing. Therefore, the functional is defined by the sum of all the contributions associated to the Feynman diagrams only including ”screening” contributions being connected to the operator .  It is not ruled out this second type of screening approximation also gives a  value of being an upper bound for the ground state energy. However, we had yet found a proof of the property, and it will be expected to be considered elsewhere.  In spite of this fact, since the HF approximation furnishes good results for the energy of many systems of interest, leads to the expectation that the inclusion of the screening effect can provide improvements of the HF results for the total energy as well as for the excitation spectrum at least in some situations of physical interest.

### ii.3 Analytic expression of the functional

Let us write in more detail the Hamiltonian of the system as written in second quantization, and the rules for the Feynman diagrams in the considered general many electron problem. The definition and notation follow the ones in Ref. (42). The Hamiltonian in second quantization has the form

 H(t) =T(t)+Vnuc(t)+v(t), (20) T(t) ≡−∫d xℏ22mΨ+(x)∂2iΨ(x), \ \ \ ∂i=∂∂→xi, (21) Vnuc(t) ≡−∫d →z ρnuc(→z)e2|→z−→x|∫d x Ψ+(x)Ψ(x), (22) v(t) ≡∫d x∫d x\'{}12Ψ+(→x,s,t)Ψ+(→x´,s´,t ´)e2δ(t−t´)|→x−→x´|Ψ+(→x´,s´,t\'{}) Ψ+(→x,s,t), (23) Ψ(x) =∑kak(t) φk(x)=∑kakexp(−ihϵkt)φk(x), (24) Ψ+(x) =∑ka+k(t) φ∗k(x)=∑ka+kexp(ihϵkt)φ∗k(x), (25)

The above Hamiltonian leads to a diagram expansion as follows. The directed continuous lines represent the free fermion propagator (contractions) associated to the Hamiltonian

 G(x,x′) =∫dw2π∑ki \ φk(x)φ∗k(x′)exp(−i w(t−t′−δ))(w−1ℏϵk+i α sgn(ϵk−ϵf)) =∑kφk(x)φ∗k(x′)(θ(t−t′−δ)θ(ϵk−ϵf)−θ(t′−t+δ)θ(ϵf−ϵk), δ →0+,

where, as defined before, is the small parameter introduced for connecting the interaction and is small time splitting which is required to well define the time ordering for equal time operators in each interaction Hamiltonian according to the convention in Ref. (42). In particular it assures the appearance of the usual terms in the discussion in the limit of vanishing polarization effects. The wavy lines with two ending points represent the vertices of instantaneous Coulomb interaction potential

 (−iℏ)vc(x,x´)=(−iℏ)e2δ(t−t´)|→x−→x´|, (26)

in which a factor multiplying the Coulomb potential and a delta function reflect its instantaneous character. A circle with a central letter as a label and with a point attached to a Coulomb wavy line ending at point   (at which two fermion lines attach, one coming and another outgoing) represents the one body nuclear potential in (22)

 (−iℏ)∫d →z ρnuc(→z)e2|→z−→x|. (27)

The kinetic energy vertex (21)

 −ℏ22m∂2i, (28)

is represented by a circle with a central letter joint to a point on which one fermion line arrives (in the sense of its attached arrow) and another line departs. The Laplacian operator acts on the coordinate argument of the fermion propagator at which the fermion directed line arrives.  Finally a  black circle will represents the sum of the (negative) electron charge density and (positive) charge density of nuclear particles.

The generic types of connected contributions satisfying the screening approximation are illustrated in figure 2.  The first series corresponds to the addition of an infinite number of self-energy insertions of the fermion polarization loops in the Coulomb interaction line of the standard exchange term. The grey circles with attached wavy lines represents screened potential resulting from the described insertions. The second series corresponds to the sum of an arbitrary number of such insertions of the polarization loop in the Coulomb interaction line of the standard direct term in the HF scheme.  As defined before, this total density is the electron particle density minus the density of the nuclear charged particles.  This direct term includes the geometric series of polarization insertions in the Coulomb interaction between: the electron charges among themselves, the nuclear ”jellium” charges among them and the interaction between these two kind of charges. The last four  terms combined correspond to  screening contributions to the mean value of the kinetic energy, which are mainly determined by the total charge density. Each contribution having a given number of fermion polarization loops in its diagram has a sign .  The geometric series over all the insertions in the graphs of figure 2 reproduces the usual definition of the screened Coulomb potential (39); (42).

The polarization loops in the graphic have the expressions

 P(x,x′) =−iℏG(x,x′)G(x′,x) =∫φl(x)φ∗l(x′)φ∗k(x′)φ∗k(x)exp(−i w(t−t′))× 2ℏfk−fl(w+1ℏ(ϵk−ϵl)+i α(fk−fl) , (29) fk =θ(ϵf−ϵk), (30)

and the screened Coulomb potential arising form the summation of the geometric series of polarization insertions is given as

 vs(x,x′) =vc(x,x′)−∫dx1dx2vc(x,x1)P(x1,x2)vc(x2,x′)+ ∫dx1dx2dx3dx4vc(x,x1)P(x1,x2)vc(x2,x3)P(x3,x4)vc(x4,x′)+... (−1)n∫dx1dx2dx3dx4...dx2n−1dx2nvc(x,x1)P(x1,x2)vc(x2,x3)P(x3,x4)vc(x4,x′)... vc(x,x2n−1)P(x2n−1,x2n)vc(x2n,x′) =(vc11+Pvc)(x,x′) =(11+vcPvc)(x,x′) =∫dx1 ε(x,x1)vc(x1,x′), (31)

in which the usual definition ´ of the dielectric function in the screened approximation appears (39).

The screening effect on the kinetic energy can be expressed in terms of a correction to the electron propagator including all the self-energy insertions of the total potential generated by the total charge density , as follows. This screening modified Green function

 Gs(x,x′) =G(x,x′)+∫dx1G(x,x1)(−iℏ)Vt(x1)G(x1,x′)+ ∫dx1∫dx2G(x,x1)(−iℏ)Vt(x1)G(x1,x2)(−iℏ)Vt(x2)G(x2,x′)+... ∫dx1∫dx2G(x,x1)(−iℏ)Vt(x1)G(x1,x2)...(−iℏ)Vt(xn)G(xn,x′)... =(G11−(−iℏ)VtG)(x,x′) =(11−GVt(−iℏ)G)(x,x′), (32) Vt(x) =∫dx´vs(x,x´)ρt(x´), (33) ρt(x) =G(x,x+)+ρn(x), (34) x+ =(→x,t+δ), \ \ δ=0+.

where the addition of a small infinitesimal positive number to the time component of a four-vector has been indicated as    Finally the analytic expression for the functional can be written in the form

 E =−12∫dxdx′G(x,x′)G(x′,x)vs(x,x′)+ 12∫dxdx′ρt(x)vs(x,x′)ρt(x′))+ ∫dxdx′(−ℏ22m∇2xG(x,x′)|x′−>x+)Gs(x′,x), (35)

in which all the participating elements are functionals of the  basis functions through the propagator as follows

 vs(x,x′) =(vc11+Pvc)(x,x′) (36) Gs(x,x\'{}) =(G11−(−iℏ)VtG)(x,x′) (37) P(x,x´) =−iℏG(x,x′)G(x′,x) (38) Vt(x) =−iℏ∫dxvs(x,x´)ρt(x´) (39) ρt(x) =G(x,x+)+ρn(x) (40) G(x,x′) =∫dw2π∑ki φk(x)φ∗k(x′)exp(−i w(t−t′−δ))(w−1ℏϵk+i α sgn(ϵk−ϵf)) (41) sgn(a) =θ[a]−θ[−a]. (42)

Let us write the variational equations for the single particle wavefunctions in next subsection.

### ii.4 The equations for the single particle states

As usual, the set of equations for determining the basis functions   will be written as the vanishing derivatives of a Lagrange functional  with respect to all the functions, by also imposing on them the orthonormality constraints.  Since is a functional of the only through its dependence on the fermion Green functions, the chain rule for the functional derivatives is helpful. Then, the following expression for the derivative of with respect to any of the conjugate one particle wavefunctions can be employed

 δδφ∗k(x′′)G(x,x′) =φk(x) δ(3)(x′′−x′)fk(t−t′) fk(t−t′)= (θ(t−t′−δ)θ(ϵk−ϵf)−θ(t′−t+δ)θ(ϵf−ϵk)× (43) .exp(−iϵkℏ(t−t′−δ). (44)

That is, the functional derivative of  with respect to is proportional to the wavefunction at the same quantum number, as multiplied by an energy dependent factor of unit absolute value and by a Dirac Delta function. The Delta function when integrated over an internal variable will define as the external point of the diagram. The point will be the one at the other extreme of  fermion line where the arrow arrives to the point to in which the wavefunction  is evaluated.  Employing the Lagrange multipliers method to  find the extremal of by also imposing the orthonormality  constraints on the functions  the  effective  functional   to be used for writing the equations for the extremals, will be

 ELag=E−∑l≠kλkl (∫dx φ∗k(x)φl(x)−δkl). (45)

where are the Lagrange multipliers of the imposed orthonormality constraints.   Note that, the not yet specified energies defining the free Hamiltonian are not yet  related with the Lagrange multipliers .    Then, the above rule for the derivative of the propagators, allows to write the extremum condition for in the form

 δδφ∗k(x)ELag =0 =∫dx′HFock(x,x′)φk(x′)−∑l≠kλkl φl(x), (46) δδλ∗kl(x)ELag =∫dx φ∗k(x)φl(x)−δkl=0, (47)

in which  the modified Fock kernel including screening effects is written in the form

 HFock(x,x′) =T(x,x′)+Vkex(x,x′)+Vkdir(x,x′)+ (48) Vks(x,x′)+(1)VkT−v(x,x′)+(2)VkT−v(x,x′). (49)

The various kernels appearing in the above expression have the explicit forms

 T(x,x′) =−ℏ22mθ(ϵf−ϵk)∇2xδ(x−x′), (50) Vkex(x,x′) =1T∫dt dt′fk(t−t′)G(x,t,x′,t′)vs(x,t,x4,t4)ε(x4,t4,x′,t′), (51) Vkdir(x,x′) =−θ(ϵf−ϵk)vt(x)δ(x−x′) (52) Vks(x,x′) =ih∫dt dt′fk(t−t′)vt(x)G(x,x′)vt(x′)ρt(x′), (53) vt(x′) =∫vs(x,x′′)ρt(x′′)dx′′ (54)
 VkT(x,x′) =−δ(x−x′)∫dx′vs(x,x2)G(x2,