
Citation: Jean-Louis Bretonnet. Basics of the density functional theory[J]. AIMS Materials Science, 2017, 4(6): 1372-1405. doi: 10.3934/matersci.2017.6.1372
[1] | Paola Goatin, Chiara Daini, Maria Laura Delle Monache, Antonella Ferrara . Interacting moving bottlenecks in traffic flow. Networks and Heterogeneous Media, 2023, 18(2): 930-945. doi: 10.3934/nhm.2023040 |
[2] | Felisia Angela Chiarello, Paola Goatin . Non-local multi-class traffic flow models. Networks and Heterogeneous Media, 2019, 14(2): 371-387. doi: 10.3934/nhm.2019015 |
[3] | Jan Friedrich, Oliver Kolb, Simone Göttlich . A Godunov type scheme for a class of LWR traffic flow models with non-local flux. Networks and Heterogeneous Media, 2018, 13(4): 531-547. doi: 10.3934/nhm.2018024 |
[4] | Abraham Sylla . Influence of a slow moving vehicle on traffic: Well-posedness and approximation for a mildly nonlocal model. Networks and Heterogeneous Media, 2021, 16(2): 221-256. doi: 10.3934/nhm.2021005 |
[5] | Christophe Chalons, Paola Goatin, Nicolas Seguin . General constrained conservation laws. Application to pedestrian flow modeling. Networks and Heterogeneous Media, 2013, 8(2): 433-463. doi: 10.3934/nhm.2013.8.433 |
[6] | Caterina Balzotti, Simone Göttlich . A two-dimensional multi-class traffic flow model. Networks and Heterogeneous Media, 2021, 16(1): 69-90. doi: 10.3934/nhm.2020034 |
[7] | Dong Li, Tong Li . Shock formation in a traffic flow model with Arrhenius look-ahead dynamics. Networks and Heterogeneous Media, 2011, 6(4): 681-694. doi: 10.3934/nhm.2011.6.681 |
[8] | Raimund Bürger, Kenneth H. Karlsen, John D. Towers . On some difference schemes and entropy conditions for a class of multi-species kinematic flow models with discontinuous flux. Networks and Heterogeneous Media, 2010, 5(3): 461-485. doi: 10.3934/nhm.2010.5.461 |
[9] | Raimund Bürger, Christophe Chalons, Rafael Ordoñez, Luis Miguel Villada . A multiclass Lighthill-Whitham-Richards traffic model with a discontinuous velocity function. Networks and Heterogeneous Media, 2021, 16(2): 187-219. doi: 10.3934/nhm.2021004 |
[10] | Alexander Kurganov, Anthony Polizzi . Non-oscillatory central schemes for traffic flow models with Arrhenius look-ahead dynamics. Networks and Heterogeneous Media, 2009, 4(3): 431-451. doi: 10.3934/nhm.2009.4.431 |
Macroscopic traffic flow models based on fluid-dynamics equations have been introduced in the transport engineering literature since the mid-fifties of last century, with the celebrated Lighthill, Whitham [11] and Richards [13] (LWR) model. Since then, the engineering and applied mathematical literature on the subject has considerably grown, addressing the need for more sophisticated models better capturing traffic flow characteristics. Indeed, the LWR model is based on the assumption that the mean traffic speed is a function of the traffic density, which is not experimentally verified in congested regimes. To overcome this issue, the so-called "second order" models (e.g. Payne-Whitham [12,15] and Aw-Rascle-Zhang [3,16]) consist of a mass conservation equation for the density and an acceleration balance law for the speed, thus considering the two quantities as independent.
More recently, "non-local" versions of the LWR model have been proposed in [5,14], where the speed function depends on a weighted mean of the downstream vehicle density to better represent the reaction of drivers to downstream traffic conditions.
Another limitation of the standard LWR model is the first-in first-out rule, not allowing faster vehicles to overtake slower ones. To address this and other traffic heterogeneities, "multi-class" models consist of a system of conservation equations, one for each vehicle class, coupled in the speed terms, see [4] and references therein for more details.
In this paper, we consider the following class of non-local systems of
∂tρi(t,x)+∂x(ρi(t,x)vi((r∗ωi)(t,x)))=0,i=1,...,M, | (1) |
where
r(t,x):=M∑i=1ρi(t,x), | (2) |
vi(ξ):=vmaxiψ(ξ), | (3) |
(r∗ωi)(t,x):=∫x+ηixr(t,y)ωi(y−x)dy, | (4) |
and we assume:
We couple (1) with an initial datum
ρi(0,x)=ρ0i(x),i=1,…,M. | (5) |
Model (1) is obtained generalizing the
Due to the possible presence of jump discontinuities, solutions to (1), (5) are intended in the following weak sense.
Definition 1.1. A function
∫T0∫∞−∞(ρi∂tφ+ρivi(r∗ωi)∂xφ)(t,x)dxdt+∫∞−∞ρ0i(x)φ(0,x)dx=0 |
for all
The main result of this paper is the proof of existence of weak solutions to (1), (5), locally in time. We remark that, since the convolution kernels
Theorem 1.2. Let
In this work, we do not address the question of uniqueness of the solutions to (1). Indeed, even if discrete entropy inequalities can be derived as in [5,Proposition 3], in the case of systems this is in general not sufficient to single out a unique solution.
The paper is organized as follows. Section 2 is devoted to prove uniform
First of all, we extend
To this end, we approximate the initial datum
ρ0i,j=1Δx∫xj+1/2xj−1/2ρ0i(x)dx,j∈Z. |
Similarly, for the kernel, we set
ωki:=1Δx∫(k+1)ΔxkΔxω0i(x)dx,k∈N, |
so that
Vni,j:=vmaxiψ(Δx+∞∑k=0ωkirnj+k),i=1,…,M,j∈Z. | (6) |
We consider the following Godunov-type scheme adapted to (1), which was introduced in [8] in the scalar case:
ρn+1i,j=ρni,j−λ(ρni,jVni,j+1−ρni,j−1Vni,j) | (7) |
where we have set
We provide here the necessary estimates to prove the convergence of the sequence of approximate solutions constructed via the Godunov scheme (7).
Lemma 2.1. (Positivity) For any
λ≤1vmaxM‖ψ‖∞, | (8) |
the scheme (7) is positivity preserving on
Proof. Let us assume that
ρn+1i,j=ρni,j(1−λVni,j+1)+λρni,j−1Vni,j≥0 | (9) |
under assumption (8).
Corollary 1. (
‖ρni‖1=‖ρ0i‖1,i=1,…,M, | (10) |
where
Proof. Thanks to Lemma 2.1, for all
‖ρn+1i‖1=Δx∑jρn+1i,j=Δx∑j(ρni,j−λρni,jVni,j+1+λρni,j−1Vni,j)=Δx∑jρni,j, |
proving (10).
Lemma 2.2. (
T<(M‖ρ0‖∞vmaxM‖ψ′‖∞W0)−1. |
Proof. Let
ρn+1i,j=ρni,j(1−λVni,j+1)+λρni,j−1Vni,j≤ˉρ(1+λ(Vni,j−Vni,j+1)) | (11) |
and
|Vni,j−Vni,j+1|=vmaxi|ψ(Δx+∞∑k=0ωkirnj+k)−ψ(Δx+∞∑k=0ωkirnj+k+1)|≤vmaxi‖ψ′‖∞Δx|+∞∑k=0ωki(rnj+k+1−rnj+k)|=vmaxi‖ψ′‖∞Δx|−ω0irnj++∞∑k=1(ωk−1i−ωki)rnj+k|≤vmaxi‖ψ′‖∞ΔxM‖ρn‖∞ωi(0) | (12) |
where
‖ρn+1‖∞≤‖ρn‖∞(1+MKvmaxM‖ψ′‖∞W0Δt), |
which implies
‖ρn‖∞≤‖ρ0‖∞eCnΔt, |
with
t≤1MKvmaxM‖ψ′‖∞W0ln(K‖ρ0‖∞)≤1Me‖ρ0‖∞vmaxM‖ψ′‖∞W0, |
where the maximum is attained for
Iterating the procedure, at time
tm+1≤tm+mMem‖ρ0‖∞vmaxM‖ψ′‖∞W0. |
Therefore, the approximate solution remains bounded, uniformly in
T≤1M‖ρ0‖∞vmaxM‖ψ′‖∞W0+∞∑m=1mem≤1M‖ρ0‖∞vmaxM‖ψ′‖∞W0. |
Remark 1. Figure 1 shows that the simplex
S:={ρ∈RM:M∑i=1ρi≤1,ρi≥0fori=1,…,M} |
is not an invariant domain for (1), unlike the classical multi-population model [4]. Indeed, let us consider the system
∂tρi(t,x)+∂x(ρi(t,x)vi(r(t,x)))=0,i=1,...,M, | (13) |
where
Lemma 2.3. Under the CFL condition
λ≤1vmaxM(‖ψ‖∞+‖ψ′‖∞), |
for any initial datum
ρn+1j=ρnj−λ[F(ρnj,ρnj+1)−F(ρnj−1,ρnj)], | (14) |
with
ρnj∈S∀j∈Z,n∈N. | (15) |
Proof. Assuming that
ρn+1i,j=ρni,j−λ[vmaxiρni,jψ(rnj+1)−vmaxiρni,j−1ψ(rnj)]. |
Summing on the index
rn+1j=M∑i=1ρn+1i,j=M∑i=1ρni,j−λM∑i=1[vmaxiρni,jψ(rnj+1)−vmaxiρni,j−1ψ(rnj)]=rnj+λψ(rnj)M∑i=1vmaxiρni,j−1−λψ(rnj+1)M∑i=1vmaxiρni,j. |
Defining the following function of
Φ(ρn1,j,…,ρnM,j)=rnj+λψ(rnj)M∑i=1vmaxiρni,j−1−λψ(rnj+1)M∑i=1vmaxiρni,j, |
we observe that
Φ(0,…,0)=λψ(0)M∑ivmaxiρni,j−1≤λ‖ψ‖∞vmaxM≤1 |
if
Φ(ρn1,j,...,ρnM,j)=1−λψ(rnj+1)M∑i=1vmaxiρni,j≤1 |
for
∂Φ∂ρni,j(ρnj)=1+λψ′(rnj)M∑i=1vmaxiρni,j−1−λψ(rnj+1)vmaxi≥0 |
if
ρn+1i,j=ρni,j(1−λvmaxiψ(rnj+1))+λvmaxiρni,j−1ψ(rnj)≥0 |
if
Lemma 2.4. (Spatial
T≤mini=1,…,M 1H(TV(ρ0i)+1), | (16) |
where
Proof. Subtracting the identities
ρn+1i,j+1=ρni,j+1−λ(ρni,j+1Vni,j+2−ρni,jVni,j+1), | (17) |
ρn+1i,j=ρni,j−λ(ρni,jVni,j+1−ρni,j−1Vni,j), | (18) |
and setting
Δn+1i,j+1/2=Δni,j+1/2−λ(ρni,j+1Vni,j+2−2ρni,jVni,j+1+ρni,j−1Vni,j). |
Now, we can write
Δn+1i,j+1/2=(1−λVni,j+2)Δni,j+1 | (19) |
+λVni,jΔni,j−1/2−λρni,j(Vni,j+2−2Vni,j+1+Vni,j). | (20) |
Observe that assumption (8) guarantees the positivity of (19). The term (20) can be estimated as
Vni,j+2−2Vni,j+1+Vni,j==vmaxi(ψ(Δx+∞∑k=0ωkirnj+k+2)−2ψ(Δx+∞∑k=0ωkirnj+k+1)+ψ(Δx+∞∑k=0ωkirnj+k))=vmaxiψ′(ξj+1)Δx(+∞∑k=0ωkirnj+k+2−+∞∑k=0ωkirnj+k+1)+vmaxiψ′(ξj)Δx(+∞∑k=0ωkirnj+k−+∞∑k=0ωkirnj+k+1)=vmaxiψ′(ξj+1)Δx(+∞∑k=1(ωk−1i−ωki)rnj+k+1−ω0irnj+1)+vmaxiψ′(ξj)Δx(+∞∑k=1(ωki−ωk−1i)rnj+k+ω0irnj)=vmaxi(ψ′(ξj+1)−ψ′(ξj))Δx(+∞∑k=1(ωk−1i−ωki)rnj+k+1−ω0irnj+1)+vmaxiψ′(ξj)Δx(+∞∑k=1(ωk−1i−ωki)(rnj+k+1−rnj+k)+ω0i(rnj−rnj+1))=vmaxiψ″(˜ξj+1/2)(ξj+1−ξj)Δx(+∞∑k=1M∑β=1ωkiΔnβ,j+k+3/2)+vmaxiψ′(ξj)Δx(M∑β=1N−1∑k=1(ωk−1i−ωki)Δnβ,j+k+1/2−ω0iΔnβ,j+1/2), |
with
ξj+1−ξj=ϑΔx+∞∑k=0ωkiM∑β=1ρnβ,j+k+2+(1−ϑ)Δx+∞∑k=0ωkiM∑β=1ρnβ,j+k+1−μΔx+∞∑k=0ωkiM∑β=1ρnβ,j+k+1−(1−μ)Δx+∞∑k=0ωkiM∑β=1ρnβ,j+k=ϑΔx+∞∑k=1ωk−1iM∑β=1ρnβ,j+k+1+(1−ϑ)Δx+∞∑k=0ωkiM∑β=1ρnβ,j+k+1−μΔx+∞∑k=0ωkiM∑β=1ρnβ,j+k+1−(1−μ)Δx+∞∑k=−1ωk+1iM∑β=1ρnβ,j+k+1=Δx+∞∑k=1[ϑωk−1i+(1−ϑ)ωki−μωki−(1−μ)ωk+1i]M∑β=1ρnβ,j+k+1+(1−ϑ)Δxω0iM∑β=1ρnβ,j+1−μΔxω0iM∑β=1ρnβ,j+1−(1−μ)Δx(ω0iM∑β=1ρnβ,j+ω1iM∑β=1ρnβ,j+1). |
By monotonicity of
ϑωk−1i+(1−ϑ)ωki−μωki−(1−μ)ωk+1i≥0. |
Taking the absolute values we get
|ξj+1−ξj|≤Δx{+∞∑k=2[ϑωk−1i+(1−ϑ)ωki−μωki−(1−μ)ωk+1i]+4ω0i}M‖ρn‖∞≤Δx{+∞∑k=2[ωk−1i−ωk+1i]+4ω0i}M‖ρn‖∞≤Δx6W0M‖ρn‖∞. |
Let now
∑j|Δn+1i,j+1/2|≤∑j|Δni,j+1/2|(1−λ(Vni,j+2−Vni,j+1))+ΔtHK1, |
where
∑j|Δn+1i,j+1/2|≤∑j|Δni,j+1/2|(1+ΔtG)+ΔtHK1, |
with
∑j|Δni,j+1/2|≤eGnΔt∑j|Δ0i,j+1/2|+eHK1nΔt−1, |
that we can rewrite as
TV(ρΔxi)(nΔt,⋅)≤eGnΔtTV(ρ0i)+eHK1nΔt−1≤eHK1nΔt(TV(ρ0i)+1)−1, |
since
t≤1HK1ln(K1+1TV(ρ0i)+1), |
where the maximum is attained for some
ln(K1+1TV(ρ0i)+1)=K1K1+1. |
Therefore the total variation is uniformly bounded for
t≤1He(TV(ρ0i)+1). |
Iterating the procedure, at time
tm+1≤tm+mHem(TV(ρ0i)+1). | (21) |
Therefore, the approximate solution has bounded total variation for
T≤1H(TV(ρ0i)+1). |
Corollary 2. Let
Proof. If
TV(ρΔxi;[0,T]×R)=nT−1∑n=0∑j∈ZΔt|ρni,j+1−ρni,j|+(T−nTΔt)∑j∈Z|ρnTi,j+1−ρnTi,j|⏟≤Tsupt∈[0,T]TV(ρΔxi)(t,⋅)+nT−1∑n=0∑j∈ZΔx|ρn+1i,j−ρni,j|. |
We then need to bound the term
nT−1∑n=0∑j∈ZΔx|ρn+1i,j−ρni,j|. |
From the definition of the numerical scheme (7), we obtain
ρn+1i,j−ρni,j=λ(ρni,j−1Vni,j−ρni,jVni,j+1)=λ(ρni,j−1(Vni,j−Vni,j+1)+Vni,j+1(ρni,j−1−ρni,j)). |
Taking the absolute values and using (12) we obtain
|ρn+1i,j−ρni,j|≤λ(vmaxi‖ψ′‖∞M‖ρn‖∞ωi(0)Δx|ρni,j−1|+vmaxi‖ψ‖∞|ρni,j−1−ρni,j|). |
Summing on
∑j∈ZΔx|ρn+1i,j−ρni,j|=vmaxi‖ψ′‖∞M‖ρn‖∞ωi(0)Δt∑j∈ZΔx|ρni,j−1|+vmaxi‖ψ‖∞Δt∑j∈Z|ρni,j−1−ρni,j|, |
which yields
nT−1∑n=0∑j∈ZΔx|ρn+1i,j−ρni,j|≤vmaxM‖ψ‖∞Tsupt∈[0,T]TV(ρΔxi)(t,⋅)+vmaxM‖ψ′‖∞MW0Tsupt∈[0,T]‖ρΔxi(t,⋅)‖1‖ρΔxi(t,⋅)‖∞ |
that is bounded by Corollary 1, Lemma 2.2 and Lemma 2.4.
To complete the proof of the existence of solutions to the problem (1), (5), we follow a Lax-Wendroff type argument as in [5], see also [10], to show that the approximate solutions constructed by scheme (7) converge to a weak solution of (1). By Lemma 2.2, Lemma 2.4 and Corollary 2, we can apply Helly's theorem, stating that for
nT−1∑n=0∑jφ(tn,xj)(ρn+1i,j−ρni,j)=−λnT−1∑n=0∑jφ(tn,xj)(ρni,jVni,j+1−ρni,j−1Vni,j). |
Summing by parts we obtain
−∑jφ((nT−1)Δt,xj)ρnTi,j+∑jφ(0,xj)ρ0i,j+nT−1∑n=1∑j(φ(tn,xj)−φ(tn−1,xj))ρni,j+λnT−1∑n=0∑j(φ(tn,xj+1)−φ(tn,xj))Vni,j+1ρni,j=0. | (22) |
Multiplying by
−Δx∑jφ((nT−1)Δt,xj)ρnTi,j+Δx∑jφ(0,xj)ρ0i,j | (23) |
+ΔxΔtnT−1∑n=1∑j(φ(tn,xj)−φ(tn−1,xj))Δtρni,j | (24) |
+ΔxΔtnT−1∑n=0∑j(φ(tn,xj+1)−φ(tn,xj))ΔxVni,j+1ρni,j=0. | (25) |
By
∫R(ρ0i(x)φ(0,x)−ρi(T,x)φ(T,x))dx+∫T0∫Rρi(t,x)∂tφ(t,x)dxdt, | (26) |
as
ΔxΔtnT−1∑n=0∑jφ(tn,xj+1)−φ(tn,xj)ΔxVni,j+1ρni,j=ΔxΔtnT−1∑n=0∑jφ(tn,xj+1)−φ(tn,xj)Δx(ρni,jVni,j+1−ρni,jVni,j)+ΔxΔtnT−1∑n=0∑jφ(tn,xj+1)−φ(tn,xj)Δxρni,jVni,j. | (27) |
By (12) we get the estimate
ρni,jVni,j+1−ρni,jVni,j≤vmaxi‖ψ′‖∞ΔxM‖ρ‖2∞ωi(0). |
Set
ΔxΔtnT∑n=0∑jφ(tn,xj+1)−φ(tn,xj)Δx(ρni,jVni,j+1−ρni,jVni,j)≤ΔxΔt‖∂xφ‖∞nT∑n=0j1∑j=j0vmaxi‖ψ′‖∞M‖ρ‖2∞ωi(0)Δx≤‖∂xφ‖∞vmaxi‖ψ′‖∞M‖ρ‖2∞ωi(0)Δx2RT, |
which goes to zero as
Finally, again by the
ΔxΔtnT−1∑n=0∑j(φ(tn,xj+1)−φ(tn,xj))Δxρni,jVni,j−12→∫T0∫R∂xφ(t,x)ρi(t,x)vi(r∗ωi)dxdt. |
In this section we perform some numerical simulations to illustrate the behaviour of solutions to (1) for
In this example, we consider a stretch of road populated by cars and trucks. The space domain is given by the interval
{∂tρ1(t,x)+∂x(ρ1(t,x)vmax1ψ((r∗ω1)(t,x)))=0,∂tρ2(t,x)+∂x(ρ2(t,x)vmax2ψ((r∗ω2)(t,x)))=0, | (28) |
with
ω1(x)=2η1(1−xη1),η1=0.3,ω2(x)=2η2(1−xη2),η2=0.1,ψ(ξ)=max{1−ξ,0},ξ≥0,vmax1=0.8,vmax2=1.3. |
In this setting,
{ρ1(0,x)=0.5χ[−1.1,−1.6],ρ2(0,x)=0.5χ[−1.6,−1.9], |
in which a platoon of trucks precedes a group of cars. Due to their higher speed, cars overtake trucks, in accordance with what observed in the local case [4].
The aim of this test is to study the possible impact of the presence of Connected Autonomous Vehicles (CAVs) on road traffic performances. Let us consider a circular road modeled by the space interval
{∂tρ1(t,x)+∂x(ρ1(t,x)vmax1ψ((r∗ω1)(t,x)))=0,∂tρ2(t,x)+∂x(ρ2(t,x)vmax2ψ((r∗ω2)(t,x)))=0,ρ1(0,x)=β(0.5+0.3sin(5πx)),ρ2(0,x)=(1−β)(0.5+0.3sin(5πx)), | (29) |
with
ω1(x)=1η1,η1=1,ω2(x)=2η2(1−xη2),η2=0.01,ψ(ξ)=max{1−ξ,0},ξ≥0,vmax1=vmax2=1. |
Above
As a metric of traffic congestion, given a time horizon
J(β)=∫T0d|∂xr|dt, | (30) |
Ψ(β)=∫T0[ρ1(t,ˉx)vmax1ψ((r∗ω1)(t,ˉx))+ρ2(t,ˉx)vmax2ψ((r∗ω2)(t,ˉx))]dt, | (31) |
where
The authors are grateful to Luis M. Villada for suggesting the non-local multi-class traffic model studied in this paper.
We provide here alternative estimates for (1), based on approximate solutions constructed via the following adapted Lax-Friedrichs scheme:
ρn+1i,j=ρni,j−λ(Fni,j+1/2−Fni,j−1/2), | (32) |
with
Fni,j+1/2:=12ρni,jVni,j+12ρni,j+1Vni,j+1+α2(ρni,j−ρni,j+1), | (33) |
where
Lemma A.1. For any
λα<1, | (34) |
α≥vmaxM‖ψ‖∞, | (35) |
the scheme (33)-(32) is positivity preserving on
Lemma A.2. (
T<(M‖ρ0‖∞vmaxM‖ψ′‖∞W0)−1. | (36) |
Lemma A.3. (
Δt≤22α+Δx‖ψ′‖∞W0vmaxM‖ρ‖∞Δx, | (37) |
then the solution constructed by the algorithm (33)-(32) has uniformly bounded total variation for any
T≤mini=1,...,M1D(TV(ρ0i)+1), | (38) |
where
[1] |
Hartree DR (1928) The Wave Mechanics of an Atom with a Non-Coulomb Central Field. Part I. Theory and Methods. Mathematical Proceedings of the Cambridge Philosophical Society, 24: 89–110. doi: 10.1017/S0305004100011919
![]() |
[2] |
Fock V (1930) Näherungsmethode zur Lösung des quantenmechanischen Mehrkörperproblems. Z Physik 61: 126–148. doi: 10.1007/BF01340294
![]() |
[3] | Slater JC (1930) Note on Hartree's Method. Phys Rev 35: 210–211. |
[4] | Raimes S (1967) The Wave Mechanics of Electrons in Metals, Amsterdam: North-Holland. |
[5] |
Thomas LH (1927) The calculation of atomic fields. Mathematical Proceedings of the Cambridge Philosophical Society, 23: 542–548. doi: 10.1017/S0305004100011683
![]() |
[6] |
Fermi E (1928) Eine Statistische Methode zur Bestimmung einiger Eigenschaften des Atoms und ihre Anwendung auf die Theorie des periodischen Systems der Elemente. Z Physik 48: 73–79. doi: 10.1007/BF01351576
![]() |
[7] |
Dirac PAM (1930) Note on Exchange Phenomena in the Thomas Atom. Mathematical Proceedings of the Cambridge Philosophical Society, 26: 376–385. doi: 10.1017/S0305004100016108
![]() |
[8] |
Hohenberg P, Kohn W (1964) Inhomogeneous Electron Gas. Phys Rev 136: B864–B871. doi: 10.1103/PhysRev.136.B864
![]() |
[9] |
Kohn W, Sham LJ (1965) Self-Consistent Equations Including Exchange and Correlation Effects. Phys Rev 140: A1133–A1138. doi: 10.1103/PhysRev.140.A1133
![]() |
[10] | Becke AD (1996) Density-functional thermochemistry. IV. A new dynamical correlation functional and implications for exact-exchange mixing. J Chem Phys 104: 1040–1046. |
[11] |
Becke AD (1988) Density-functional exchange-energy approximation with correct asymptotic behavior. Phys Rev A 38: 3098–3100. doi: 10.1103/PhysRevA.38.3098
![]() |
[12] |
Lee C, YangW, Parr RG (1988) Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density. Phys Rev B 37: 785–789. doi: 10.1103/PhysRevB.37.785
![]() |
[13] |
Feynman RP, Metropolis N, Teller E (1949) Equations of State of Elements Based on the Generalized Fermi-Thomas Theory. Phys Rev 75: 1561–1573. doi: 10.1103/PhysRev.75.1561
![]() |
[14] |
Schwinger J (1981) Thomas-Fermi model: the second correction. Phys Rev A 24: 2353–2361. doi: 10.1103/PhysRevA.24.2353
![]() |
[15] |
Shakeshalf R, Spruch L (1981) Remarks on the existence and accuracy of the Z-1/3 expansion of the nonrelativistic ground-state energy of a neutral atom. Phys Rev A 23: 2118–2126. doi: 10.1103/PhysRevA.23.2118
![]() |
[16] |
Englert BG, Schwinger J (1985) Atomic-binding-energy oscollations. Phys Rev A 32: 47–63. doi: 10.1103/PhysRevA.32.47
![]() |
[17] |
Spruch L (1991) Pedagogic notes on Thomas-Fermi theory (and on some improvements): atoms, stars, and the stability of bulk matter. Rev Mod Phys 63: 151–209. doi: 10.1103/RevModPhys.63.151
![]() |
[18] |
Mermin ND (1965) Thermal Properties of the Inhomogeneous Electron Gas. Phys Rev 137: A1441–A1443. (Concerns the generalization to electrons at finite temperatures). doi: 10.1103/PhysRev.137.A1441
![]() |
[19] |
Levy M (1979) Universal variational functionals of electron densities, first-order density matrices, and natural spin-orbitals and solution of the v-representability problem. Proc Natl Acad Sci U S A 76: 6062–6065. doi: 10.1073/pnas.76.12.6062
![]() |
[20] | Becke AD (2014) Perspective: Fifty years of density-functional theory in chemical physics. J Chem Phys 140: 18A301. |
[21] | Lindhard J (1954) On the properties of a gas of charged particles. Kgl Danske Videnskab Selskab Mat Fys Medd 28: 8. |
[22] |
Jones RO, Young WH (1971) Density functional theory and the vonWeizsacker method. J Phys C 4: 1322–1330. doi: 10.1088/0022-3719/4/11/007
![]() |
[23] |
von Weizsäcker CF (1935) Zur Theorie der Kernmassen. Z Physik 96: 431–458 . doi: 10.1007/BF01337700
![]() |
[24] |
Jones RO, Gunnarsson O (1989) The density functional formalism, its applications and prospects. Rev Mod Phys 61: 689–746. doi: 10.1103/RevModPhys.61.689
![]() |
[25] |
Slater JC (1951) A Simplification of the Hartree-Fock Method. Phys Rev 81: 385–390. doi: 10.1103/PhysRev.81.385
![]() |
[26] |
Robinson JE, Bassani F, Knox BS, et al. (1962) Screening Correction to the Slater Exchange Potential. Phys Rev Lett 9: 215–217. doi: 10.1103/PhysRevLett.9.215
![]() |
[27] |
Wigner EP (1934) On the Interaction of Electrons in Metals. Phys Rev 46: 1002–1011. doi: 10.1103/PhysRev.46.1002
![]() |
[28] |
Gell-Mann M, Brueckner K (1957) Correlation Energy of an Electron Gas at High Density. Phys Rev 106: 364–368. doi: 10.1103/PhysRev.106.364
![]() |
[29] |
Ceperley DM (1978) Ground state of the fermion one-component plasma: A Monte Carlo study in two and three dimensions. Phys Rev B 18: 3126–3138. doi: 10.1103/PhysRevB.18.3126
![]() |
[30] |
Ceperley DM, Alder BJ (1980) Ground State of the Electron Gas by a Stochastic Method. Phys Rev Lett 45: 566–569. doi: 10.1103/PhysRevLett.45.566
![]() |
[31] |
Perdew JP, Wang Y (1992) Accurate and simple analytic representation of the electron-gas correlation energy. Phys Rev B 45: 13244–13249. doi: 10.1103/PhysRevB.45.13244
![]() |
[32] |
Levy M (1982) Electron densities in search of Hamiltonians. Phys Rev A 26: 1200–1208. doi: 10.1103/PhysRevA.26.1200
![]() |
[33] |
Lieb EH (1983) Density functionals for coulomb-systems. Int J Quantum Chem 24: 243–277. doi: 10.1002/qua.560240302
![]() |
[34] | Levy M, Perdew JP (1985) Hellmann-Feynman, virial, and scaling requisites for the exact universal density functionals. Shape of the correlation potential and diamagnetic susceptibility for atoms. Phys Rev A 32: 2010–2021. |
[35] |
Wigner E, Seitz F (1933) On the Constitution of Metallic Sodium. Phys Rev 43: 804–810. doi: 10.1103/PhysRev.43.804
![]() |
[36] |
Hellmann H (1933) Zur Rolle der kinetischen Elektronenenergie für die zwischenatomaren Kräfte. Z Physik 85: 180–190. doi: 10.1007/BF01342053
![]() |
[37] |
Feynman RP (1939) Forces in Molecules. Phys Rev 56: 340–343. doi: 10.1103/PhysRev.56.340
![]() |
[38] | Pupyshev VI (2000) The Nontriviality of the Hellmann-Feyman Theorem. Russ J Phys Chem 74: S267–S278. |
[39] | Perdew JP, Kurth S (2003) Density Functionals for Non-relativistic Coulomb Systems in the New Century, In: Fiolhais C, Nogueira F, Marques MAL, A Primer in Density Functional Theory, Berlin Heidelberg: Springer-Verlag, 1–55. |
[40] |
Gunnarsson O, Lundqvist BI (1976) Exchange and correlation in atoms, molecules, and solids by the spin-density-functional formalism. Phys Rev B 13: 4274–4298; Gunnarsson O, Lundqvist BI (1977) Erratum: Exchange and correlation in atoms, molecules, and solids by the spin-densityfunctional formalism. Phys Rev B 15: 6006. doi: 10.1103/PhysRevB.13.4274
![]() |
[41] |
Gunnarsson O, Jonson M, Lundqvist BI (1979) Descriptions of exchange and correlation effects in inhomogeneous electron systems. Phys Rev 20: 3136–3165. doi: 10.1103/PhysRevB.20.3136
![]() |
[42] |
Langreth DC, Perdew JP (1975) The Exchange-Correlation Energy of a Metallic Surface. Solid State Commun 17: 1425–1429. doi: 10.1016/0038-1098(75)90618-3
![]() |
[43] |
Langreth DC, Perdew JP (1977) Exchange-correlation energy of a metallic surface: Wave-vector analysis. Phys Rev B 15: 2884–2901. doi: 10.1103/PhysRevB.15.2884
![]() |
[44] |
Jones RO, Gunnarsson O (1989) The density functional formalism, its applications and prospects. Rev Mod Phys 61: 689–746. doi: 10.1103/RevModPhys.61.689
![]() |
[45] |
Charlesworth JPA (1996) Weighted-density approximation in metals and semiconductors. Phys Rev B 53: 12666–12673. doi: 10.1103/PhysRevB.53.12666
![]() |
[46] |
Gunnarsson O, Lundqvist BI (1976) Exchange and correlation in atoms, molecules, and solids by the spin-density-functional formalism. Phys Rev B 13: 4274–4298. doi: 10.1103/PhysRevB.13.4274
![]() |
[47] |
Alvarellos JE, Chacon E, Tarazona P (1986) Nonlocal density functional for the exchange and correlation energy of electrons. Phys Rev 33: 6579–6587. doi: 10.1103/PhysRevB.33.6579
![]() |
[48] | Chacon E, Tarazona P (1988) Self-consistent weighted-density approximation for the electron gas. I. Bulk properties. Phys Rev 37: 4020–4025. |
[49] | Gupta AK, Singwi KS (1977) Gradient corrections to the exchange-correlation energy of electrons at metal surfaces. Phys Rev B 15: 1801–1810. |
[50] |
Perdew JP, Burke K, Ernzerhof M (1996) Generalized Gradient Approximation Made Simple. Phys Rev Lett 77: 3865–3868. doi: 10.1103/PhysRevLett.77.3865
![]() |
[51] |
Perdew JP, Burke K, Wang Y (1996) Generalized gradient approximation for the exchangecorrelation hole of a many-electron system. Phys Rev B 54: 16533–16539. doi: 10.1103/PhysRevB.54.16533
![]() |
[52] |
Perdew JP, Ruzsinszky A, Csonka GI, et al. (2008) Restoring the Density-Gradient Expansion for Exchange in Solids and Surfaces. Phys Rev Lett 100: 136406. (Modified PBE subroutines are available from: http://dft.uci.edu./pubs/PBEsol.html.) doi: 10.1103/PhysRevLett.100.136406
![]() |
[53] |
Colle R, Salvetti O (1975) Approximate calculation of the correlation energy for the closed shells. Theoret Chim Acta 37: 329–334. doi: 10.1007/BF01028401
![]() |
[54] |
Leung TC, Chan CT, Harmon BN (1991) Ground-state properties of Fe, Co, Ni, and their monoxides: Results of the generalized gradient approximation. Phys Rev B 44: 2923–2927. doi: 10.1103/PhysRevB.44.2923
![]() |
[55] |
Singh DJ, Ashkenazi J (1992) Magnetism with generalized-gradient-approximation density functionals. Phys Rev B 46: 11570–11577. doi: 10.1103/PhysRevB.46.11570
![]() |
[56] |
Körling M, Häglund J (1992) Cohesive and electronic properties of transition metals: The generalized gradient approximation. Phys Rev B 45: 13293–13297. doi: 10.1103/PhysRevB.45.13293
![]() |
[57] |
Engel E, Vosko SH (1994) Fourth-order gradient corrections to the exchange-only energy functional: Importance of r2n contribution. Phys Rev B 50: 10498–10505. doi: 10.1103/PhysRevB.50.10498
![]() |
[58] |
Andersson Y, Langreth DC, Lundqvist BI (1996) van derWaals Interactions in Density-Functional Theory. Phys Rev Lett 76: 102–105. doi: 10.1103/PhysRevLett.76.102
![]() |
[59] |
Dobson JF, Dinte BP (1996) Constraint Satisfaction in Local and Gradient Susceptibility Approximations: Application to a van der Waals Density Functional. Phys Rev Lett 76: 1780–1783. doi: 10.1103/PhysRevLett.76.1780
![]() |
[60] |
Patton DC, PedersonMR (1997) Application of the generalized-gradient approximation to rare-gas dimers. Phys Rev A 56: R2495–R2498. doi: 10.1103/PhysRevA.56.R2495
![]() |
[61] |
Van Voorhis T, Scuseria GE (1998) A novel form for the exchange-correlation energy functional. J Chem Phys 109: 400–410. doi: 10.1063/1.476577
![]() |
[62] |
Tao J, Perdew JP, Staroverov VN, et al. (2003) Climbing the Density Functional Ladder: Nonempirical Meta–Generalized Gradient Approximation Designed for Molecules and Solids. Phys Rev Lett 91: 146401. doi: 10.1103/PhysRevLett.91.146401
![]() |
[63] | Van Doren VE, Van Alsenoy K, Greelings P (2001) Density Functional Theory and Its Applications to Materials, Melville, NY: American Institute of Physics. |
[64] | Gonis A, Kioussis N (1999) Electron Correlations and Materials Properties, New York: Plenum. |
[65] |
Perdew JP, Kurth S, Zupan A, et al. (1999) Accurate Density Functional with Correct Formal Properties: A Step Beyond the Generalized Gradient Approximation. Phys Rev Lett 82: 2544– 2547; Perdew JP, Kurth S, Zupan A, et al. (1999) Erratum: Accurate Density Functional with Correct Formal Properties: A Step Beyond the Generalized Gradient Approximation. Phys Rev Lett 82: 5179. doi: 10.1103/PhysRevLett.82.2544
![]() |
[66] |
Becke AD (2000) Simulation of delocalized exchange by local density functionals. J Chem Phys 112: 4020–4026. doi: 10.1063/1.480951
![]() |
[67] |
Becke AD (1993) A new mixing of Hartree–Fock and local density-functional theories. J Chem Phys 98: 1372–1377. doi: 10.1063/1.464304
![]() |
[68] | Becke AD (1993) Density-functional thermochemistry. III. The role of exact exchange. J Chem Phys 98: 5648–5652. |
[69] |
Seidl M, Perdew JP, Kurth S (2000) Simulation of All-Order Density-Functional Perturbation Theory, Using the Second Order and the Strong-Correlation Limit. Phys Rev Lett 84: 5070–5073. doi: 10.1103/PhysRevLett.84.5070
![]() |
[70] |
Fuchs M, Gonze X (2002) Accurate density functionals: Approaches using the adiabaticconnection fluctuation-dissipation theorem. Phys Rev B 65: 235109. doi: 10.1103/PhysRevB.65.235109
![]() |
[71] |
Seidl A, Görling A, Vogl P, et al. (1996) Generalized Kohn-Sham schemes and the band-gap problem. Phys Rev B 53: 3764–3774. doi: 10.1103/PhysRevB.53.3764
![]() |
[72] | Savin A (1996) On degeneracy, near degeneracy and density functional theory, In: Seminario JM, Recent Developments of Modern Density Functional Theory, Amsterdam: Elsevier, 327–357. |
[73] |
Sousa SF, Fernandes PA, Ramos MJ (2007) General Performance of Density Functionals. J Phys Chem A 111: 10439–10452. doi: 10.1021/jp0734474
![]() |
[74] |
Curtiss LA, Raghavachari K, Redfern PC, et al. (1997) Assessment of Gaussian-2 and density functional theories for the computation of enthalpies of formation. J Chem Phys 106: 1063–1079. (This paper contains a set of 148 molecules having well-established enthalpies of formation at 298 K.) doi: 10.1063/1.473182
![]() |
[75] |
Chan GKL, Handy NC (1999) Optimized Lieb-Oxford bound for the exchange-correlation energy. Phys Rev A 59: 3075–3077. (As an example, the exchange-correlation energy must satisfy the inegality |Exc|≤2.27|ELDAx| .) doi: 10.1103/PhysRevA.59.3075
![]() |
[76] |
Perdew JP, Ernzerhof M, Burke K (1996) Rationale for mixing exact exchange with density functional approximations. J Chem Phys 105: 9982–9985. doi: 10.1063/1.472933
![]() |
[77] |
Adamo C, Barone V (1998) Exchange functionals with improved long-range behavior and adiabatic connection methods without adjustable parameters: The mPW and mPW1PW models. J Chem Phys 108: 664–675. doi: 10.1063/1.475428
![]() |
[78] |
Wu Q, Yang W (2002) Empirical correction to density functional theory for van der Waals interactions. J Chem Phys 116: 515–524. doi: 10.1063/1.1424928
![]() |
[79] |
Becke AD, Johnson ER (2005) Exchange-hole dipole moment and the dispersion interaction. J Chem Phys 122: 154104. doi: 10.1063/1.1884601
![]() |
[80] |
Tkatchenko A, Scheffler M (2009) Accurate Molecular Van Der Waals Interactions from Ground-State Electron Density and Free-Atom Reference Data. Phys Rev Lett 102: 073005. doi: 10.1103/PhysRevLett.102.073005
![]() |
[81] |
Grimme S, Antony J, Ehrlich S, et al. (2010) A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu. J Chem Phys 132: 154104. doi: 10.1063/1.3382344
![]() |
[82] |
Dion M, Rydberg H, Schröder E, et al. (2004) Van der Waals Density Functional for General Geometries. Phys Rev Lett 92: 246401. doi: 10.1103/PhysRevLett.92.246401
![]() |
[83] |
Thonhauser T, Cooper VR, Li S, et al. (2007) Van der Waals density functional: Self-consistent potential and the nature of the van der Waals bond. Phys Rev B 76: 125112 . doi: 10.1103/PhysRevB.76.125112
![]() |
[84] |
Klimes J, Michaelides A (2012) Perspective: Advances and challenges in treating van der Waals dispersion forces in density functional theory. J Chem Phys 137: 120901. doi: 10.1063/1.4754130
![]() |
[85] |
Car R, Parrinello M (1985) Unified Approach for Molecular Dynamics and Density-Functional Theory. Phys Rev Lett 55: 2471–2474. doi: 10.1103/PhysRevLett.55.2471
![]() |
[86] |
Heyd J, Scuseria GE, Ernzerhof M (2003) Hybrid functionals based on a screened Coulomb potential. J Chem Phys 118: 8207–8215. doi: 10.1063/1.1564060
![]() |
[87] |
Heyd J, Scuseria GE (2004) Effcient hybrid density functional calculations in solids: Assessment of the Heyd–Scuseria–Ernzerhof screened Coulomb hybrid functional. J Chem Phys 121: 1187–1192. doi: 10.1063/1.1760074
![]() |
[88] |
Ernzerhof M, Scuseria GE (1999) Assessment of the Perdew–Burke–Ernzerhof exchangecorrelation functional. J Chem Phys 110: 5029–5036. doi: 10.1063/1.478401
![]() |
[89] |
Paier J, Marsman M, Kresse G (2007) Why does the B3LYP hybrid functional fail for metals? J Chem Phys 127: 024103. doi: 10.1063/1.2747249
![]() |
[90] |
Burke K (2012) Perspective on density functional theory. J Chem Phys 136: 150901. doi: 10.1063/1.4704546
![]() |
[91] |
Jones RO (2015) Density functional theory: Its origins, rise to prominence, and future. Rev Mod Phys 87: 897–923. doi: 10.1103/RevModPhys.87.897
![]() |
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