Maria Hellgren

Local vertex corrections from exchange-correlation kernels with a discontinuity

The fundamental gap of an interacting many-electron system is given by the sum of the singleparticle Kohn-Sham gap and the derivative discontinuity. The latter can be generated by advanced approximations to the exchange-correlation (XC) energy and is the key quantity to capture strong correlation with density functional theory (DFT). In this work we derive an expression for the derivative discontinuity in terms of the XC kernel of time-dependent density functional theory and demonstrate the crucial role of a discontinuity in the XC kernel itself. By relating approximate XC kernels to approximate local vertex corrections we then generate beyond-GW self-energies that include a discontinuity in the local vertex function. The quantitative importance of this result is illustrated with a numerical study of the local exchange vertex on model systems.

I. INTRODUCTION

Kohn-Sham (KS) density functional theory (DFT) is the most widely used many-electron approach for numerical simulations in physics, chemistry and materials science. 1,2 In both its static and time-dependent (TD) version, the interacting many-electron problem is reformulated into the simpler problem of non-interacting electrons moving in a self-consistent, single-particle KS potential. This construction is formally exact, but is expected to become more constrained when the electrons are strongly correlated. On the other hand, exact studies on simple systems have shown that the KS potential captures effects of strong correlation by exhibiting rather intuitive peak and step features. [START_REF] Almbladh | NATO Advanced Study Institute Series B: Physics[END_REF][START_REF] Perdew | NATO Advanced Study Institute Series B: Physics[END_REF][START_REF] Gritsenko | [END_REF][6][7][8] Standard local and semi-local approximations to the exchange-correlation (XC) energy, such as LDA and GGA functionals, miss these features and, as a consequence, tend to spread out or delocalize the electrons in the system.

In a seminal work of Perdew et al. 9 it was shown that a key feature of the exact XC energy functional is a derivative discontinuity (DD) at integer particle numbers. The DD corrects the largely underestimated singleparticle KS gap, and prevents delocalization through the formation of discontinuous XC steps in the KS potential. To capture strong correlation with (TD)DFT it is thus necessary to construct functionals with the DD. It should be noted that the concept of strong correlation can be ambiguous since it sometimes refers to effects beyond semi-local approximations in DFT and sometimes to multi-reference effects beyond Hartree-Fock (HF). In the first case the DD can often be captured by local exact-exchange, 10,11 hybrid 12 or corrective DFT+U functionals. [START_REF] Himmetoglu | Cococcioni[END_REF] In the latter, truly correlated case, the DD constitute the entire gap, and it still remains a challenge to find approximate functionals.

Approximations based on many-body perturbation theory (MBPT) have shown to successfully capture many effects of correlation such as screening, van der Waals forces and derivative discontinuities derived from nonlocal exchange. One example is the random phase ap-proximation (RPA), derived from the GW self-energy within MBPT. The RPA and the GW approximation (GWA) share many virtues but both fail in describing localized states in strongly correlated systems such as Mott insulators. [14][15][16][17] The GWA can be improved by including so-called vertex corrections. So far a HF-like vertex has been employed to construct various total energy expressions based on partial re-summations of particlehole diagrams. [18][19][20][21][22] Similarly, approximations based on the local exact-exchange vertex was studied in Refs. 23-26. Local vertex corrections have also been studied at the simpler level of LDA. 27 Improved local vertex corrections can be derived from improved XC kernels within TDDFT. 28,29 The XC kernel is defined as the functional derivative of the XC potential with respect to the density and is the crucial quantity to calculate excitation energies within linear response TDDFT. 30 Recently, it was shown that also the XC kernel exhibits discontinuities at integer particle number, important for charge-transfer excitations 11,31 and chemical reactivity indices. 32 In this work we will investigate the effects of the discontinuity of the XC kernel for constructing local vertex corrections to the many-body self-energy. It has, e.g., previously been shown that affinities need a threepoint vertex to be accurately described. 33 Here we show that this problem can reformulated into the problem of a missing dynamical discontinuity of the two-point XC kernel. From the ACFD (adiabatic connection fluctuation dissipation) expression to the total energy we further derive an exact expression to the fundamental gap in terms of the XC kernel and its corresponding discontinuity. Finally, we calculate the discontinuity of the exactexchange kernel and use it to determine correlated gaps beyond the GWA.

II. DERIVATIVE DISCONTINUITY IN (TD)DFT

In this section we review some of the mathematical formulas underlying the concept of the DD in static and time-dependent DFT.

The DD refers to a discontinuous change in the derivative of the XC energy, E xc , as the density passes integer particle numbers. As a consequence, the corresponding XC potential, v xc , jumps with a space-independent constant. The DD has been proven to be a feature of the exact theory, 9,[34][35][36] and to play a key role when applying (TD)DFT to different physical problems such as, e.g., charge-transfer processes, Kondo or Coulomb blockade phenomena. 7,31,[37][38][39][40][41] The most direct effect of the discontinuity is, however, seen when trying to extract the fundamental gap from DFT. The ionization energy I and the affinity A of a system with N 0 particles can be calculated as the left and right derivatives of the total energy with respect to particle number N , i.e,

-I = ∂E ∂N - , -A = ∂E ∂N + (1)
where the subscript ± refers to the value of the quantity at N ± 0 = N 0 + 0 ± . The ionization energy of the KS system is equal to the ionization energy of the true interacting system 42 but the KS affinity A s has to be corrected with the discontinuity ∆ xc of the XC potential. The gap of the interacting system is thus equal to the KS gap plus the discontinuity, i.e.,

I -A = I -A s + ∆ xc . (2)
It has been demonstrated that the DD is comparable in size to the KS gap for both solids 43,44 and molecules. 45,46 Moreover, in strongly correlated Mott insulators the discontinuity accounts for the entire gap. 47,48 Following Ref. 31 we will now derive a formal expression for the quantity ∆ xc which is useful for extracting the discontinuity of approximate functionals. Using the chain rule we can write the partial derivative of E xc with respect to N as

∂E xc ∂N = dr v xc (r) ∂n(r) ∂N . (3)
In the limit N - 0 = N 0 + 0 -this expression can be rewritten as

0 = ∂E xc ∂N - -dr v - xc (r)f -(r), (4)
where f (r) = ∂n(r)/∂N is the so-called Fukui function. 49,50 Let us now look at the right derivative of E xc , which will be different from the left derivative if a DD is present. From Eq. (3) we see that this difference can appear either in the XC potential v + xc (r) = v - xc (r) + ∆ xc or in the Fukui function. In fact, it is easy to see that in the case of local or semilocal functionals such as LDA and PBE the discontinuous behaviour is restricted to the Fukui function, i.e., ∆ xc = 0. Meta-GGAs have shown to exhibit a discontinuity in the XC potential, albeit very small. 51 On the other hand, orbital functionals based on MBPT are known to accurately capture the discontinuity of the XC potential.

Let us now write v xc (r) = v - xc (r) + ∆ xc (r) and cast Eq. (3) into

dr ∆ xc (r)f (r) = ∂E xc ∂N -dr v - xc (r)f (r). (5)
In the limit N → N + 0 , ∆ xc (r) → ∆ xc and we find an expression for the discontinuity of v xc

∆ xc = ∂E xc ∂N + -dr v - xc (r)f + (r). (6)
A systematic scheme for generating orbital functionals based on MBPT was presented in Ref. 52. The idea is to start from variational energy expressions of the Green's function G and then restrict the variational freedom to Green's functions G s coming from a local KS potential. One can, for example, start from the Klein functional 53

E[G] = -iΦ[G] + E H + iTr [GG -1 s -1 + ln(-G -1 )], (7)
which contains another functional Φ having the property that the self-energy Σ is generated as Σ = δΦ/δG. When restricting to KS Green's functions it is easy to see that the XC energy corresponds to

E xc = -iΦ[G s ].
Furthermore, the equation for the XC potential is nothing but the linearized Sham-Schlüter (LSS) equation [54][55][56][57][58]

d2 χ s (1, 2)v xc (2) = d2d3 Σ xc (2, 3)Λ(3, 2; 1). ( 8
)
Here we have simplified the notation by introducing 1 = r 1 , t 1 . Even with the exact self-energy this scheme will never be exact but it has shown to produce useful approximations such as the exact-exchange (EXX) approximation (based on the HF self-energy) and the random phase approximation (RPA) (based on the GW selfenergy). To determine the discontinuities of the Klein XC potentials, Eq. ( 7) and Eq. ( 8) must be generalized to ensemble Green's functions. This was done in Refs. 14 and 31 showing that the discontinuities of the Klein functional are given by

∆ xc = drdr ϕ * L (r)Σ + xc (r, r , ε + L )ϕ L (r ) -dr v - xc (r)|ϕ L (r)| 2 . ( 9
)
where L signifies the LUMO (lowest unoccupied molecular orbital). The superscript on the self-energy can be dropped as it is, in general, invariant with respect to a constant shift of the potential. Combining Eq. ( 9) with Eq. ( 2) leads to

-A = ε - L + ∆ xc = ε - L + ϕ L |Σ xc (ε L ) -v - xc |ϕ L ( 10
)
where the KS LUMO is calculated from v - xc . This equation is very similar to the first order quasi-particle equation within MBPT.

The above mathematical analysis was restricted to the static case but also the time-dependent v xc exhibits jumps. In the linear response regime this leads to a frequency and space-dependent discontinuity in the XC kernel given by

f + xc (r, r , ω) = f - xc (r, r , ω) + g xc (r, ω) + g xc (r , ω). (11)
In order to determine the discontinuity g xc (r, ω) of the XC kernel, given an approximation to v xc , we use a similar procedure as in the static case but with a generalized ensemble that allow densities to change particle number in time. In Ref. 31 it was shown that such an ensemble was necessary for functional derivatives to be uniquely defined.

Assuming that v xc is defined on such a generalised domain of densities we can evaluate the functional derivative of v xc with respect to the time-dependent number of particles, i.e.,

δv xc (rt) δN (t ) n + 0 = dr f - xc (r, r , t -t )f + (r ) + dr g xc (r , t -t )f + (r ) + g xc (r, t -t ).( 12
)
Let us now specialize to functionals derived from the Klein action functional of MBPT. The TD XC potential v xc is then obtained from the LSS equation (Eq. ( 8) above). Combined with Eq. ( 12), we find the following equation to determine

g xc d2 χ s (1, 2)g xc (2, t) = d2d3 Σ xc (2,
-d2d3 χ s (1, 2)f - xc (2, 3) δn(3) δN (t) n + 0 -d2 v + xc (2) δχ s (1, 2) δN (t) n + 0 . ( 13
)
The discontinuity of the XC kernel was in Ref. 31 shown to be finite and carry a strong frequency dependence. Already the static discontinuity turned out to give a significant contribution when calculating reactivity indices such as the Fukui function. 32 In the next sections we will show that the dynamical discontinuity give a significant contribution when calculating beyond-GW gaps from local vertex corrections within MBPT.

III. LOCAL VERTEX CORRECTIONS FROM fxc

The most popular approximation to the self-energy is the GWA, in which the bare Coulomb interaction v of the HF approximation is replaced by a dynamically screened Coulomb interaction, W . [59][60][61] Any effect beyond the GWA is usually referred to as a vertex correction, denoted by Γ, and the exact self-energy can be written as

Σ = iGW Γ ( 14
)
where

W = v + vP W, P = -iGGΓ (15)
and P is the so-called irreducible polarisability. The vertex function Γ is defined as

Γ = - δG -1 δV = 1 + δΣ δV (16)
where V = v ext + v H is the sum of the external and Hartree potential. If the vertex is set to 1 we obviously obtain the GWA. From the definition of the vertex we see that the next level of approximation can be generated iteratively from the derivative of the GW self-energy. This self-energy is expected to be very accurate but too complex to be applied to a real systems.

With the idea of approximating the numerically cumbersome three-point vertex function it has been suggested to replace the self-energy in Eq. ( 16) by local XC potentials derived from TDDFT. This generates local vertex functions depending only on two space and time variables. 27,29 The LDA potential was used in Ref. 27 but produced only a very small change to the GW gaps. Moreover, in Ref. 33 it was found that whereas ionization energy were well captured by local vertex corrections affinities were not. In order to overcome these limitations we will now look at more advanced XC potentials derived from the Klein MBPT scheme in the previous section, a scheme which can also be used to derive local vertex functions. These approximations are, e.g., guaranteed to be conserving due to the Φ-derivability of the allowed selfenergies, 62,63 a potentially important feature.

In general, we can write any local vertex function as

Γ xc = 1 + δv xc δV = 1 1 -Γ 1 xc (17)
where

Γ 1 xc = δv xc δV s , V s = v ext + v Hxc . (18)
Using the chain rule, the local vertex function is easily related to the XC kernel of TDDFT

Γ 1 xc = δv xc δn δn δV s = f xc χ s . (19)
We can now insert the local vertex of Eq. ( 17) into Eq. ( 14) in order to generate an approximate beyond-GW self-energy

Σ = iGW Γ xc . (20)
There are now two important issues to consider. Firstly, when calculating the affinity the self-energy should be evaluated at N + 0 . This is similar to the case of approximating the nonlocal self energy with a local XC potential, where the latter is discontinuous at N 0 . Since the ensemble XC kernel also has discontinuities, from Eq. (19) we see that the local ensemble vertex function must have discontinuities

Γ + xc = Γ - xc + G xc . (21)
For example, looking at the first order term in the expansion of the vertex correction (Eq. ( 19)) we see immediately that

G 1 xc = g xc χ s (22)
with g xc as defined in Eq. ( 13) above. Secondly, we notice that the self-energies used in Eq. ( 9) should be Φ-derivable whereas self-energies from Eq. ( 20) are, in general, not. We can, however, derive a very similar expression to Eq. ( 9) starting from the exact ACFD-formula to the total energy

E = E s + i 2 dω 2π dλTr{v[χ λ (ω) -δn]}. (23)
Here, E s is the total energy in the Hartree approximation and χ λ is the scaled density correlation function. We also define Tr {AB} = drdr A(r, r )B(r , r) and δn = δ(r, r )n(r). With the definition of the exact local vertex function in Eq. ( 17) it is easy to rewrite Eq. ( 23) in terms of Eq. ( 20). 64 We find

E = E s - i 2 dω 2π dλ λ Tr{Σ GW Γxc λ G s }. (24)
We see that for the total energy the discontinuity of the local vertex function will always vanish. Let us now take the derivative of the XC part with respect to particle number. We find

∂E xc ∂N + = -i 1 0 dλ λ Tr ΣGW Γ + xc λ ∂G s ∂N + - λ 2 χ λ vχ λ ∂f λ xc ∂N + . (25)
This expression together with Eq. ( 6) yields an exact expression for the discontinuity of the XC potential and hence the fundamental gap. We see that the first term is very similar to Eq. ( 9) but with the exact nonlocal self-energy replaced by a symmetrized self-energy with a local vertex function

ΣGW Γ + xc = iG s Γ + xc W v -1 W Γ + xc . (26)
Within the RPA (Γ xc = 1) the λ-integral can be carried out analytically and this term becomes exactly Eq. ( 9) with the GW self-energy. It is clear that if Γ xc is discontinuous the self-energy in Eq. ( 26) will be discontinuous, with important consequences for calculating fundamental gaps. In the next section we will quantify its contribution.

IV. CORRELATED GAPS FROM THE EXX VERTEX

In this section we will study the EXX approximation to the XC kernel, which can be considered the most simple, yet consistent, vertex correction for going beyond the GWA. The EXX approximation has been shown to support a discontinuity in both the EXX potential 65 and the EXX kernel, 31 necessary for EXX density to equal the HF density to first order, at equilibrium and in linear response, respectively.

The fully nonlocal HF vertex is defined as

Γ HF = 1 + δΣ HF δV , (27)
a three point function in time and space. The EXX local vertex function can instead be written as

Γ x = 1 + δv x δV = 1 1 -Γ 1 x , ( 28
)
where v x is the local time-dependent EXX potential given by the LSS equation (Eq. ( 8)) within the HF approximation to the self-energy. The first order term in the expansion of the EXX vertex is given by

Γ 1 x = δv x δV s , V s = v ext + v Hx . (29)
Using the chain rule we can write the EXX kernel (f x ) as

f x = δv x δn = Γ 1 x χ -1 s . ( 30
)
This kernel has been studied in several previous works. It is known to produce excellent total energies when used in the ACFD total energy expression of Eq. ( 23). 23,24 It captures excitonic effects 66 and charge-transfer excitations within the single-pole approximation. 31 In this work we have calculated the EXX kernel and the corresponding EXX vertex of one-dimensional soft-Coulomb systems using a precise and efficient spline-basis set. 11,67,68 Below we will study the effect the exchange vertex when calculating the fundamental gap of molecular systems. We will also compare the fully nonlocal approximation (Eq. ( 27)) to the local approximation (Eq. ( 28)).

A. MP2

In order to test the theory we start by looking at the most simple approximation to the self-energy that can be written in terms of the exchange vertex. Expanding the self-energy to second order in the Coulomb interaction we obtain what is known as the the second order Born or MP (Møller-Plesset) approximation, 69 in which the sum of first and second order exchange can be written in terms of the first order HF vertex function (Γ 1 HF ), i.e.,

Σ MP2 = iΓ 1 HF vG s + ivχ s vG s . (31)
The last term is a second order term related to the screened interaction. In the appendix we evaluate these terms explicitly (Eq. ( 38) and Eq. ( 39)), and we see that they are smooth as functions of particle number N . They are also invariant with respect to a constant shift of the external potential.

We will now approximate this self-energy with the local EXX vertex defined above. We thus write

Σ MP2 L = iΓ 1 x vG s + ivχ s vG s . ( 32
)
Since the EXX vertex is discontinuous due to a discontinuity of the EXX kernel we have to evaluate the first term (Σ Γ 1 x ) differently depending on if we are calculating affinities or ionization energies. In the limit N = N - 0 , relevant for ionization energies, we write

Σ - Γ 1 x = if - x χ s vG s , (33)
and, in the limit N = N + 0 , relevant for affinities, we write

Σ + Γ 1 x = if - x χ s vG s + ig x χ s vG s . (34)
The discontinuity g x gives rise to a second non-vanishing term. Furthermore, if we take the Fourier transform of Eq. (34) we see that the frequency dependence in g x has to be taken into account

Σ + Γ 1 x (r, r , ω) = i dω 2π G s (r, r , ω -ω ) × dr 1 dr 2 v(r, r 2 )χ s (r 2 , r 1 , ω )f - x (r 1 , r , ω ) + dr 1 dr 2 v(r, r 2 )χ s (r 2 , r 1 , ω )g x (r 1 , ω ) . (35)
Let us now specialize to a two-electron system. The EXX kernel at N = N - 0 is then simply given by

f - x (r 1 , r , ω ) = - 1 2 v(r 1 , r ). (36)
If we insert this kernel into the first term of Eq. (35) we see that it is equal to minus one half of the second term of Eq. ( 31). The second term of Eq. ( 35), the term containing the discontinuity, can be evaluated with the help of Eq. ( 13), adapted to EXX. In the appendix we have explicitly evaluated this term for a two-electron system (see Eq. ( 40)).

In Fig. 1 we present the results for a soft-Coulomb atomic system containing two electrons. The black dot signifies the location of the atom. In the lower panel we plot the EXX KS potential (full black line) obtained by imposing Eq. ( 4), which ensures the correct asymptotic limit of the potential. The dashed black line is the same potential but shifted by the EXX discontinuity ∆ x (Eq. ( )). This is the result one would get by evaluating the potential at N = N + 0 , as indicated to left with the arrows pointing upwards. The location of the KS HOMO (highest occupied molecular orbital) (-I) and the KS LUMO (lowest unoccupied molecular orbital) (-A s ) are indicated with full horizontal red lines. The red dashed horizontal line indicates the true affinity (-A) after correction with the discontinuity (Eq. ( 10)), also indicated by an arrow pointing upwards to the left.

The upper panel shows the correction to the affinity due to MP2 correlation. The red dashed A(EXX) line is the EXX affinity (a duplicate of the -A line in the lower panel) and the blue dashed line is the correction due to the first term of Eq. (34). Keeping only this term corresponds to using Eq. ( 33) to calculate affinities. Including the discontinuity term (second term in Eq. ( 34), here called G x ) further shifts the affinity (blue full horizontal line). Actually, we find that G x corresponds to 75% of the total correction. The black full line in the same panel denoted A(MP2), is obtained by using the fully nonlocal vertex, i.e., by evaluating Eq. (31). We see that the black and blue full lines almost coincide. We can thus conclude that in order to reproduce the nonlocal MP2 approximation the discontinuity (i.e. the G x term) of the EXX kernel is crucial.

B. RPAx

We will now construct a more advanced self-energy that takes into account both screening and vertex corrections to all orders in the Coulomb interaction. In this way, we will consistently incorporate the EXX vertex in both the screened interaction (Eq. ( 15)) and in the selfenergy (Eq. ( 14)). We call this self-energy RPAx

Σ RPAx = iG s W x Γ x . (37)
This is the self-energy (although symmetrized) one obtains from the ACFD-formula including the EXX kernel (i.e. the RPAx energy) but by ignoring variations of f x with respect to the density. 23 It can also be seen as the local approximation to the self-energy that includes vertex corrections at the time-dependent HF level. 21 We applied this approximation to the stretched hydrogen molecule, for which we expect screening to be more important. In the lower panel of Fig. 2 we plot the KS potential at the RPA level (black full line). We also show the EXX potential in the background (grey thin line). The KS HOMO-LUMO gap is vanishing small but if we add the RPA discontinuity ∆ xc (obtained from Eq. ( 9) with the GW self-energy) the affinity shifts substantially. We also note that it is larger than the corresponding EXX correction, in agreement with the analysis in Ref. 16. Neither the HF nor the GWA is able to correctly describe the gap of stretched H 2 since it is a strongly correlated Mott-like system. 15 Including exchange effects in the vertex is not expected to qualitatively improve upon the GWA. Indeed, in the upper panel we present the results from the RPAx self-energy and we see that the correction is of the wrong sign. However, in this case, we see that the discontinuity is even more important to reproduce the nonlocality of the self-energy, accounting for almost the entire correction.

V. CONCLUSIONS

In this work we have derived approximate self-energies based on local vertex corrections derived from XC kernels within the Klein MBPT formulation of TDDFT. These vertex corrections capture a dynamical discontinuity of the XC kernel, needed to accurately describe electron affinities. A numerical study on model molecular systems shows that the discontinuity of the local vertex corresponds to the largest correction to the GW affinity.

From the ACFD formula to the total energy we further show that an exact expression for the discontinuity of the XC potential can be written in terms of the XC kernel and its derivative. Although this work focused on the DD as a correction to the gap, a very important consequence of the discontinuity is to localise electrons in strongly correlated systems. The ability of an XC potential derived from the ACFD expression to do so will strongly rely on the discontinuous nature of the XC kernel.

VI. APPENDIX

Here we give explicit expressions for the formulas in Sec. 4. The MP2 self-energy is composed of two terms. The second term in Eq. ( 31) is a first order screening correction so we denote it with a subscript W 1 . In terms of orbitals and eigenvalues it is given by Σ W1 (r, r , ω) = 2 kq ϕ k (r) dr 1 v(r, r 1 )f * q (r 1 ) ×ϕ * k (r ) dr 1 v(r , r 1 )f q (r 1 )

× n k ω -ε k + Z q -iη + 1 -n k ω -ε k -Z q + iη ,( 38
)
where q is the particle-hole index, f q is the 'bare' excitation function given by a product of occupied and unoccupied KS orbitals, and Z q is the corresponding eigenvalue difference. The first 'vertex' term is instead given by Σ Γ 1 HF (r, r , ω) = ksp ϕ * k (r) dr 1 v(r, r 1 )ϕ s (r 1 )ϕ p (r 1 )

× ϕ s (r ) dr 1 v(r , r 1 )ϕ k (r 1 )ϕ * p (r 1 ) × (-1)

× n k n s (1 -n p ) ω -ε s + ε p -ε k -iη + (1 -n k )n p (1 -n s ) ω -ε s + ε p -ε k + iη . ( 39
)
The sum of the terms (Eq. (38) and Eq. ( 39)) constitute the MP2 self-energy. The local MP2 self-energy also contains the term in Eq. ( 38) and in addition two local terms, one containing f x and one its discontinuity g x . For a two-electron system the f x -term is just minus one half of Eq. (38). The second term is generated from Eq. ( 13) yielding ig x χ s vG s = k,p =L ϕ k (r ) dr 1 v x (r 1 )ϕ p (r 1 )ϕ * L (r 1 )

×ϕ * k (r) dr 1 v(r, r 1 )ϕ L (r 1 )ϕ * p (r 1 )

× n k ω + ε L -ε p -ε k -iη + 1 -n k ω -ε L + ε p -ε k + iη + s,k,p =L
ϕ k (r )n s dr 1 dr 2 ϕ s (r 1 )ϕ * s (r 2 ) v(r 1 , r 2 )ϕ p (r 1 )ϕ * L (r 2 )

×ϕ * k (r) dr 1 v(r, r 1 )ϕ L (r 1 )ϕ * p (r 1 )

× n k ω + ε L -ε p -ε k -iη + 1 -n k ω -ε L + ε p -ε k + iη + kq ϕ k (r) dr 1 v(r, r 1 )f * q (r 1 )
×ϕ * k (r ) dr 1 dr 2 |ϕ L (r 2 )| 2 v(r 2 , r 1 )f q (r 1 ) ×ϕ * k (r ) dr 1 dr 2 ϕ * L (r 1 )ϕ L (r 2 )v(r 2 , r 1 )ϕ * s (r 1 )ϕ p (r 2 )

× n k ω -ε k + Z q -iη + 1 -n k ω -ε k -Z q + iη
× n k n s (1 -n p ) ω -ε k + ε p -ε s -iη + (1 -n k )n s (1 -n p ) ω -ε k -ε p -ε s + iη (40)

FIG. 1 .FIG. 2 .

 12 FIG.1. Lower panel: KS potential of a soft-Coulomb twoelectron atom in the EXX approximation (black full and dashed curves). Horizontal lines indicates the KS HOMO and LUMO energy levels (red full) and the corrected affinity (red dashed). Upper panel: MP2 corrected affinity (blue dashed and full lines are without and with discontinuity, respectively). Black full line is the nonlocal MP2 result.

+

  kq ϕ k (r) dr 1 v(r, r 1 )ϕ s (r 1 )ϕ * p (r 1 )


R S Mishra

Kaushalendra Kr Dubey
email: dubey.kaushalendra@gmail.com

Kaushalendra Kr

Dubey

Orc=organic Rankine

Cycle Scp=specific

IMAT3E'18 International Meeting on Advanced Technologies in Energy and Electrical Engineering Solar Integrated Energy Efficient Systems for Waste Heat Recovery of Power Generation Industry Using Different Eco-Friendly Energy Materials

Keywords: Global Warming Potential, Ozone Déplétion Potential, Solar Integrated Waste Heat Recovery System. IMAT3E'18

The two major issues are being disused in these days globally, CO2 concentration in atmosphere on one hand and energy crisis on other hand. All promising savings of reducing CO2 generation, fossil fuels depletion, control in energy prices increments, achievable by adaptation of eco-friendly energy efficient material based technology. The un-arrested heat discharge from captive power plants, cement production, steel industry, Oil-refineries, like energy intensive industries have tremendous potential for multiple energy generation (heating-power & cooling) for further industrial process with dumped heat recovery. This paper explores eco-friendly material R134a and activated carbon-methanol based organic Rankine model and solar integrated vapor adsorption cooling system for waste heat utilization of condenser unit of steam power plant. The results of proposed model analysis is to explain the environmental parameters of used material, effect of operating parameters and cooling-power output.

Introduction

The uncovered heat from power plants and heavy industry is dumped into atmosphere and it causes of severe impact on atmosphere in terms of green house effect. The US Dept of Energy reported that the cement/ captive power plants have, nearly 35% heat is lost, and this corresponds to around 70 to 75 MW of thermal energy. EPA estimated that the 65% of world CO2 emission is recorded from green house gas emission from fissile fuel burning during intensive industrial process [1,2]. This un-used or waste heat have tremendous potential to generate heating-power and cooling generation for industrial process by using eco-friendly material based novel thermodynamic system like, organic rankine system (ORC), kalina system for cooling-power generation, vapor absorption and adsorption refrigeration techniques. The employment of solar energy will enhance the system performance. All these systems work as waste heat recovery generator (WHRG). For a 6000 ton per day (tpd) capacity of cement plant can be reduced around 70,000 ton/annum of CO2 by installation of WHRG [1]. The proposed analysis is concern with the condenser heat recovery of reheating-rankine steam power plant by using eco-friendly material based ORC and vapor adsorption cooling system for simultaneously power and cooling generation. Two models have been analyzed in this paper, model 1 (fig- 1) integrated with solar ORC and model 2(fig- 2) equipped with double bed adsorption refrigeration system. Model2 has two different adsorber beds, bed1 is connected with condenser outlet of reheating-rankine steam plant and AC-Methanol (Activated Carbon)material pair is used and bed2 integrated with solar parabolic collector for condenser water

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International Meeting on Advanced Technologies in Energy and Electrical Engineering heating that heated water circulating to bed2 for generator purpose. The main objective of this analysis to estimate the multiple thermal effect (heating-power and cooling) from available un-covered heat of power plant by using different novel thermal system. The application of R134a and AC-Methanol type material helps to produce required cooling effect at low grade of thermal availability with environmental safety aspect.The numerous researches have been developed and analyzed eco-friendly material based novel thermodynamic model in environmental concern. The selection of a suitable working material depends on its great influence in the design of the process, application, heat source and the temperature level. The used material must have optimum thermodynamic properties at the lowest possible temperatures and pressures, and also satisfy several criteria such as economical, nontoxic, nonflammable, environmentally friendly. An extensive literature review of R134a was chosen as working fluid. This selection has been done on the basis that R134a: is a nontoxic and nonflammable fluid and its Ozone Depletion Potential (ODP) is zero. R134a has a high molecular mass (chemical formula: CF3CH2F, MM=102kg/kmol). It has a temperature and critical pressure of 101.1ºC and 40.6 bar, respectively, R134a based heat recovery ORC of condenser operates at a higher pressure than atmospheric, and therefore air in-leakages do not occur. [START_REF] Vélez | A technical, economical and market review of organic Rankine cycles for the conversion of low-grade heat for power generation[END_REF][START_REF] Saleh | Working fluids low temperature organic Rankine cycles[END_REF][START_REF] Quolin | Thermo-Economic optimization of waste heat recovery organic Rankine cycles[END_REF][START_REF] Tchange | Fluid selection for a lowtemperature solar organic Rankine cycle[END_REF] Several researchers have investigated the application and performance of ORC with R134a as a working fluid, the efficiency of the ORC using benzene, ammonia, R134a, R113, R11 and R12 was analyzed [START_REF] Hung | A review of organic Rankine cycles (ORC`s) for the recovery of low-grade waste heat[END_REF][START_REF] Chen | A review of thermodynamic cycles and working fluids for the conversion of low-grade heat[END_REF][START_REF]Class I Ozone Depleting Substances[END_REF] An analysis of a regenerative ORC based on the parametric optimization, using R12, R123, R134a, and R717 as working fluids superheated at constant pressure was carried out and. results revealed that selection of a regenerative ORC during overheating using R123 as working fluid appears to be a good system for converting low-grade waste heat to power [START_REF] Roy | Parametric optimization and performance analysis of a waste heat recovery system using organic Rankine cycle[END_REF]. In a low-temperature solar organic Rankine cycle system was designed and built with R134a as working fluid that works between 35.0ºC and 75.8ºC for reverse osmosis desalination in Greece, the results showed a system efficiency of about 7% and 4%, respectively. Other studies that have analyzed the use of R134a as working fluid in the ORC cycles for reverse osmosis desalination at an experimental level [START_REF] Manolakos | Design of an autonomous low-temperature solar Rankine cycle system for reverse osmosis desalination[END_REF][START_REF] Manolakos | Experimental evaluation of an autonomous low-temperature solar Rankine cycle system for reverse osmosis desalination[END_REF][START_REF] Manolakos | On site experimental evaluation of a low-temperature solar organic Rankine cycle system for RO desalination[END_REF], and also as a theoretical manner showed a simulation to estimate the increase in the efficiency and the energy available for desalination of an upper ORC coupled with a lower ORC with R134a, obtaining an efficiency for the latter of 4.2%. Other cycles with R134a for applications for geothermal sources are reported also and R134a integrated with internal combustion engine for heat recovery as bottoming cycle [START_REF] Manolakos | Identification of behaviour and evaluation of performance of small scale, low-temperature organic Rankine cycle system coupled with a RO desalination unit[END_REF][START_REF] Bruno | Modelling and optimisation of solar organic Rankine cycle engines for reverse osmosis desalination[END_REF][START_REF] Delgado-Torres | Analysis and optimization of the low-temperature solar organic Rankine cycle[END_REF][START_REF] Thanche | Exergy analysis of microorganic Rankine power cycles for a small scale solar driven reverse osmosis desalination system[END_REF][START_REF] Schuster | Simulation of an innovative stand-alone solar desalination system using an organic Rankine cycle[END_REF][START_REF] Karellas | Investigation of an autonomous hybrid solar thermal ORC-PV RO desalination system. The Chalki island case[END_REF][START_REF] Kosmadakis | Parametric theoretical study of a two-stage solar organic Rankine cycle for RO desalination[END_REF][START_REF] Franco | Optimal design of binary cycle power plants for water-dominated, medium-temperature geothermal fields[END_REF][START_REF] Aneke | Performance analysis of the Chena binary geothermal power plant[END_REF][START_REF] Astolfi | Technical and economical analysis of a solargeothermal hybrid plant based on an organic Rankine cycle[END_REF]. The overall screening of all heat recovery fluids R134a was found to be the most suitable in terms of most eco-friendly, non-toxic, high thermal performance for cooling and power generation with optimized system. Another category of energy recovery material for green refrigeration technology is vapor adsorption refrigeration. Activated carbon is a form of carbon that has been processed to make it extremely porous, and it has a large surface area available for adsorption. Methanol and ammonia are the most common refrigerants paired with activated carbon. Activated carbon-methanol is one of the most promising working pairs in practical systems because of its large adsorption quantity and low adsorption heat (about 1800 to 2000 kJ•kg-1 [START_REF] Wang | A review on adsorption working pairs for refrigeration[END_REF]. Low adsorption heat is beneficial to the system's performance because the majority of heat consumption in the desorption phase is the adsorption heat. Another advantage of activated carbon-methanol is low desorption temperature (about 100°C), which is within a suitable temperature range for using solar energy as a heat source. Activated carbon-ammonia has almost the same adsorption heat as the activated carbon-methanol working pair.The drawbacks of this working pair are the toxicity and pungent smell of ammonia. Silica gel is a granular, highly porous form of silica made synthetically from sodium silicate. For the silica gel-water working pair, the adsorption heat is about 2500 kJ/kg and the desorption temperature could be as low as 50°C [START_REF] Wang | A review on adsorption working pairs for refrigeration[END_REF][START_REF] Wang | State-of-the-Art Review on Crystallization Control Technologies for Water/LiBr Absorption Heat Pumps[END_REF].

Model-1 -R134a

Results and Discussion

Increasing the mass flow rate of air in HE2 increases the heat obtained thereby reducing the cooling effect and heating the air more. In operation cycle with solar heater the change in heat obtained and the rise in air temperature for a small increase in mass flow rate of air is much more than that in the operation cycle without solar heater. Fig 3 depicts the efficiency of scroll type expander is in the range of 14% to 25% of model1. The expander in case of cycle without solar integration will do more work. In case of solar integration, more heat for industrial process will be obtained. when solar heating is not implemented, the heat for process obtained from HE 1 will be more in both cycles than that obtained when solar heating is implemented. Moreover, the gain in the

IMAT3E'18

International Meeting on Advanced Technologies in Energy and Electrical Engineering heat obtained from HE 1, for process, per unit increase of mass flow rate of air passing through the HE 1, in case of operation cycle, is greater when solar heating is not implemented than when solar heating is implemented. All parametric results are concluded in Table 1&2 of both model 1&2.

Conclusions

The selection of a suitable working material depends on its process, application, heat source and the temperature level. The used material must have optimum thermodynamic properties at the lowest possible temperatures and pressures, and also satisfy the several criteria such as being economical, nontoxic, nonflammable, and environmentally friendly. All conventional refrigeration systems consume precious fuel or electricity to attain refrigeration. ORC and Adsorption systems are heat-operated units that need only heat in terms of un-covered heat from the energy intensive sectors, so they can utilize waste heat and generate multiple effects like heating, power and cooling. The major conclusion are drawn from both model analysis-1. Cycle with solar coupling provides more heat for process. This is nearly two times that of the heat provided for process by the cycle without solar coupling, but the total work achieved in case of solar heating is less than that achieved without solar heating. 2. More power from the expanders is achieved when solar heating is not implemented. This is due to the heat generated by the solar heater is converted into process heat. Therefore, solar integration is recommended, for the purpose of obtaining more process heat, in summer and for the purpose of power generation in winter. 3. Solar integrated thermal circuit in combined power generation system enhances the cooling effect due to more heat addition through solar heating in R134a ORC system. R134a will attain superheating, and produce cooling effect. 4. The parametric table-6 concludes that the performance indicators (thermal efficiency, heating and cooling effect) of proposed system is increased by 30-40% using solar system. 5. Parametric results, which are shown in table-9 of double bed, vapor adsorption refrigeration system for heat recovery of Reheating-Rankine condenser, the overall COP and SCP enhanced by 10-16% shown in figure-6. 6. Solar connected vapor adsorption bed gives more COP individually as compare to condenser connected vapor adsorption bed. The specific cooling power (SCP) of solar vapor adsorption bed is increasing but increasing of evaporation temperature will decrease the SCP of solar vapor adsorption bed due to conversion of solar gain into heating (Qheat-solar) by double rate. The variation of SCP between solar and without solar (bed-1 and bed-2) shown in figure-5. 7. The present combination of adsorbent-adsorbate material is suitable for low-grade heat recovery with solar thermal integration for continuous cooling effect generation and applicable for space cooling purpose. The used material R134a and methanol with activated carbon is most suitable pair under the category of eco-friendly material as per AHRAE guidelines of environmental safety protocol.All ecogriendly material have been shown in appendix tables. 8. The different materials have reviewed in present analysis. The two parameters are very important for environmental aspect, like GWP and ODP values. Minimum value of GWP &OPD shows non-toxic and eco-friendly nature of material for power and cooling generation system.The proposed material also applicable for low-grade heat recovery purpose due to its low boiling and freezing temperature.

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International Meeting on Advanced Technologies in Energy and Electrical

Engineering

Appendix-

Fig- 1 ,

 1 Fig-1, Schematic of R134a Based ORC heat recovery Unit Fig-2: Solar Integrated Vapor Adsorption System for Condenser Heat Recovery (Model-1) (Model-2)

  done Expander 1 -Start Up cycle With Solar Heating Work done Expander 2 -Start Up cycle With Solar Heating Work done Expander 1 -Operation cycle With Solar Heating Work done Expander 2 -Operatoon cycle With Solar Heating Work done Expander 1 -Start Up cycle Without Solar Heating Work done Expander 1 -Operation cycle Without Solar Heating

Figure 3 :

 3 Figure 3: Effect of expander efficiency on work done by expander Figure 4: Effect of mass flow rate of air flowing through HE 1, on Q process heat in operation cycle and startup cycle

  Fig 4 explains that when solar heating is not implemented, the heat for process obtained from HE 1 will be more in both cycles than that obtained when solar heating is implemented. Moreover, the gain in the

Fig- 5 COP

 5 Fig-5 COP Vs Source Temp Of VADRS Fig-6-Cpmbined COP Vs Source Temp of VADRS

Based Solar Integrated ORC Waste Heat Recovery System. Model-2-Activated Carbon based Solar Integrated Vapour Adsorption Cooling System For Condenser Waste Heat Recovery.

Table - 1-Parametric Results Of ORC Heat Recovery System

 -

	Parametric Results	Cycle without solar	Cycle with solar
	ηthermal	8-15%	12-27%
	ηexpander	10-25%	10-25%
	Process Heat (Qprocess heat)	900-1100 kW	1200-1500 kW
	Cooling Effect (QCooling)	14-19.25 kW	35-38.5 kW
	Expander work (Wexpander)	200-470 kW	260-650 kW
	Solar Heating (QSolar)	N/A	460 kW
	Solar Collector Efficiency	N/A	20-25%
	Water Temp through solar field	N/A	K

Table - 2, Parametric Results of combined double bed vapor adsorption System

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	Performance	Resultant	Performance	Resultant	Performance	Resultant Values
	Parameter	Values	Parameter	Values	Parameter
	Turbine work	294.7 &	Condenser heat	2133 kJ/kg COP of combined	0.46 to 0.51 (13-15%
	output (WHPT&	1072 kJ/kg	exhaust (QOUT)		VAdRS	increment in overall
	WLPT)				(COPCOMBINEDVADRS)	COP)
	Pump Work	16.23 kJ/kg	Plant Efficiency	0.388	Sp. Cooling Power	70-90 W/kg (10-16%
	(WPUMP)		(ηP)		(SCPCOMBINEDVADRS)	increment in overall
						SCP)
	Boiler Heat	3484 kJ/kg	Net Work	1350 kJ/kg COP gain by Solar	0.48-0.74 (most
	Generation		Output (WNET)		energy (COPSOLAR)	effective COP)
	(QIN)

18 International Meeting on Advanced Technologies in Energy and Electrical Engineering

Table - A Energy recovery and its sources [26]

 -

	Heat Categories	Heat Sources	Temperature in 0 C	Suggested recovery technology
		Solid waste	650-1000	Air preheating
		Nickel refining furnace	1370-1650	Steam generation for heating
		Copper reverberator furnace	760-815	Thermoelectric and thermal PV
	High Grade (>650 0 C)	Glass melting furnace Hydrogen plant	1000-1550 650-1000	Heat exchanger for preheating Thermal PV
		Steam boiler exhaust	230-480	Steam rankine cycle
		Gas turbine exhaust	370-540	Organic rankine cycle
	Medium Grade	Drying and baking ovens	230-600	Thermal PV
	(230-650 0 C)	Catalytic crackers	425-650	Thermal PV
		Reciprocating engine	315-600	Thermoelectric
		exhaust
		Welding and injection	32-88	Kalina cycle
	Low Grade	molding
	(>230 0 C)	Hot processed liquids and	32-233	Organic rankine cycle
		solids
		Drying, Baking and Curing	93-230	Absorption and adsorption
		ovens		cooling
		Bearing	32-88	Piezoelectric
	.

Table - B: Summary of Commercialized Working fluids [27-29]

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		Working		Physical Data		Safety	Environmental Data
		Fluids					Data
			Molecul	Normal	Critica	Critic	ASHRA	Atmosph	Ozone	Global
			ar Mass	Boiling	l Temp	al	E	eric Life	Depletion	Warming
	S.		(Kg/K	Point	(Tcriti)	Pressu	Group	Time	potential	Potential
	No		mol)	Temp	in 0 C	re		(ALT) in	(ODP)	(GWP) of 100
				(Tbp) in 0 C		(Pcriti)		years		years
						in
						MPa
	1	RC118	200.03	-6.0	115.2	2.778	A1	3200	0	10225
	2	R600a	58.12	-11.7	135	3.647	A3	0.017	0	˜˜20
	3	R114	170.92	3.6	145.7	3.289	A1	300	1.00	10040
	4	R600	58.12	-0.5	152	3.796	A3	0.018	0	˜˜20
	5	R601	72.15	36.1	196.5	3.364	NA	0.01	0	˜˜20
	6	R113	187.38	47.6	214.1	3.439	A1	85	1.000	6130
	7	Cyclohex	84.16	80.7	280.5	4.075	A3	NA	NA	NA
		ane`
	8	R290	44.10	-42.1	96.68	4.247	A3	0.041	0	~20
	9	R407C	86.20	-43.6	86.79	4.597	A1	NA	0	1800
	10 R32	52.02	-51.7	78.11	5.784	A2	4.9	0	675
	11 R500	99.30	-33.0	105.5	4.455	A1	NA	0.738	8100
	12 R152a	66.05	-24.0	113.3	4.520	A2	1.40	0	124
	13 R717	17.03	-33.3	132.3	11.333 B2	0.1	0	<1
		(Ammon
		ia)
	14 Ethanol	46.07	78.4	240.8	6.148	NA	NA	NA	NA

18 International Meeting on Advanced Technologies in Energy and Electrical Engineering

	15 Methanol 32.04	64.4	240.2	8.104	NA	NA	NA	NA
	16 R718	10.2	100	374	22.064 A1	NA	0	<1
	(Water)
	17 R134a	102.03	-26.1	101	4.059	A1	14.0	0	1430
	18 R12	120.91	-29.8	112	4.114	A1	100	1.000	10890
	19 R123	152.93	27.8	183.7	3.668	B1	1.3	0.02	77
	20 R141b	116.95	32.0	204.2	4.249	NA	9.3	0.120	725
	21 R245fa	134.05	15.3	154.1	3.64	B1	8.8	0	820
	22 R236fa	152.0	-1.5	124.0	3.20	-	209	0	6300
	23 R227ea	170.0	-17.5	102.0	2.95	-	36.5	0	2900
	24 R1234yf 114.02	-29.45	94.7	3.382	A1	NA	~0	4

Table - C-Energy Recovery Materials for Adsorption Refrigeration Techniques of Heat Recovery [24]

 -

	Material		Heat Source	Performance	Index	Application
				(COP)
			20 MJ/m2 per Day	0.12
			18 to 19.2 MJ/m2 per	0.12 to 0.14
			Day
	AC-Methanol		17 to 20 MJ/m2 per Day	0.13 to 0.15
			<120 degree C	0.18
	AC-NH3		105 degree C	0.10		Ice Making
	Silica Gel-Water		15.4 MJ/m2 per Day	0.16
	AC-Blanked	steel-	20 MJ/m2 per Day	0.18
	Methanol
	AC+CaCl2-NH3		115 degree C	0.39
			55 degree C	0.36
			65 degree C	0.28
			75-90 MJ/m2 per Day	0.35 to 0.60
	Silica Gel-Water		80 to95 MJ/m2 per Day	0.35 to 0.60		Chilled Water
			80 MJ/m2 per Day	0.33to 0.50
			232 degree C	0.42 to 1.19
	AC-NH3		100 degreeC	0.20
			204 degree C	0.60 to 1.60
			230 degreeC	0.41		Air Conditioning
	Zeolite-Water		310 degreeC	0.38
			230 to 300 degreeC	0.20 to 0.21