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Quantum fluctuations and kinetic correlation in the strongly Interacting Limit of
Density Functional Theory
Grossi, J.
2020
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Grossi, J. (2020). Quantum fluctuations and kinetic correlation in the strongly Interacting Limit of Density Functional Theory.
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5
F U N C T I O N A L D E R I VAT I V E O F T H E Z E R O P O I N T E N E R G Y F U N C T I O N A L
We used to think that if we knew one, we knew two, because one and one are two. We are finding that we must learn a great deal more about "and".
— Arthur Eddington[69]
5.1 introduction
Besides the xc functional itself, another quantity that plays an impor-tant role in KS Density Functional Theory is its functional derivative with respect to the density, which is the xc potential entering in the KS equations. The exact (or very accurate) xc potential has been studied for small systems in several works, using various
reverse-engineering procedures[70–72]: these works have shown that for
strongly-correlated systems the xc potential must display very peculiar
features, such as “peaks” and “steps”[73–75]. While the functional
derivative of the SCE leading term has been evaluated and used as an approximation for the xc potential in the self-consistent KS equations in various works[16,76–78], the potential associated to the next lead-ing term has never been investigated in an exact manner. Only very recently, a semi-local approximation for the ZPE has been used to look at KS potentials coming from functionals that interpolate between
the weak- and strong-coupling limits of the xc functional[79]. It is
the purpose of this paper to fill this gap, by starting an investigation of the exact ZPE functional derivative. The SIL functionals have a density dependence that is rather complicated and unusual, making it actually difficult to evaluate functional derivatives. The reason why the functional derivative of the leading SIL term (the SCE term) could be easily computed is that it can be obtained from an exact shortcut[76,
77], which seems to be missing at the next leading order. For this
reason, our investigation starts from a simple, yet non trivial, case: two electrons confined in one dimension (1D). Similar 1D models have been widely used to investigate features in exact KS Density Functional Theory. Such models have provided a good qualitative
description of the relevant features of their 3D counterparts[61, 74,
80–83]. Besides its interest as an xc potential at strong coupling, the ZPE functional derivative that we compute here is also a crucial in-gredient to analyse the third term in the large-l expansion of the exact Levy–Lieb functional. This next term, in fact, requires solving a hierarchy of Schrödinger equations for which knowledge of the
asymp-totic expansion at strong coupling of vl (the 1-body potential that
52 functional derivative of the zero point energy functional
keeps the density fixed at each l) is needed; the potential vl at orders
l1/2 should be given by minus the ZPE functional derivative[14]. The
Chapter is organized as follows: we first briefly cover the key concepts
of SCE and ZPE formalism in section 5.1.1. The core of the Chapter,
section5.2, hosts an analytical expression of the functional derivative
of the ZPE functional (5.12). Its features are discussed and numerical
calculation is provided to verify the consistency of our results. Last, in
section5.4we draw our conclusions and outline future steps.
5.1.1 Cliff Notes on SCE + ZPE for N =2 electrons in 1D
This brief paragraph is meant for the hasty reader in search of a quick summary of the basic quantities used in this Chapter. As such, it will be particularly concise, and can be skipped.
sce Defining f1(s)⌘s, f2(s)⌘ f(s), we have[16]
f[$](s) = 8 < : N 1 e Ne(s) +1 s< Ne 1(1) N 1 e Ne(s) 1 s> Ne 1(1), (5.1) where Ne(s)⌘ Z s •dx $(x). (5.2)
The comotion function is such that the integral of the density between
x and f(x)always integrates to 1 independently of x. Therefore, when
x<0, for a symmetric density f(x)must necessarily be positive, and vice versa. As the reference electron approaches 0 from the left, the
second electron is pushed towards+•. When the reference electron
crosses the origin, the second electron must "jump" to •. Moreover, equations (3.8) and (3.7) read
f f(s) =s =) f0 f(s) = 1
f0(s) (5.3a)
f0(s) = $(s)
$ f(s) (5.3b)
zpe The only non-zero frequency (eigenvalue of the 2⇥2 matrix
H(s), eq. (3.28)) is given by[16]
w(s)⌘w2[$](s) = s v00ee s f(s) ✓ f0(s) + 1 f0(s) ◆ . (5.4)
Notice that vee(x)is convex, v00ee(x) >0, and that f0(x) >0, see (5.3b).
FZPE[$], equation (3.61), reads explicitly
FZPE[$] = 1
4 Z +•
• ds $(s)w(s). (5.5)
5.2 functional derivative of FZPE[$] for N=2, D=1
5.2.1 Explicit expression
Inserting (3.50) and (3.60) in (2.40) and comparing the terms propor-tional topl, we have
dFZPE[$] d$(x) = v
ZPE(x). (5.6)
The derivation of an explicit form for dFZPE/d$(x)starts from noticing
that dFZPE[$] d$(x) = 1 4 d d$(x) Z +• • dy $(y)w(y) = w(x) 4 + 1 4 Z +• • dy $(y) dw(y) d$(x). (5.7)
The frequency function w(x)is an implicit functional of the density,
via the co-motion function and its derivative. Even for 2 electrons in
D=1, computing the functional derivatives of f(x)can be delicate,
as it changes sign when Ne(s) =1: perturbing the density in this point
implies taking into account a step function, for which the chain rule does not apply (see Appendix in[84] for further details). Step functions
are also expected whenever there is a step in $(x)or a difference in
the values of the density at the boundaries in a compact support. This
is not our case however, since we assume $(x) to be a continuous
integrable function defined on the whole real axis. As a consequence, limx!N 1
e (1)"w(x) = limx!Ne1(1)#w(x), there is no step to be taken
into account. Hence, we can simply apply the chain rule and write
dw⇥f[$], f0[$]⇤(y) d$(x) = ∂w ∂f df[$](y) d$(x) + ∂w ∂f0 df0[$](y) d$(x) , (5.8) which reads dw⇥f[$], f0[$]⇤(y) d$(x) = w(x)(f0(x)2 1) 2(f0(x) + f0(x)3) df0[$](y) d$(x) + ⇣ f0(x) + 1 f0(x) ⌘ v000 ee x f(x) 2w(x) df[$](y) d$(x) . (5.9) For the chain rule, only the regular part of the functional derivative of
f(x), which can be found in[84], is relevant, and reads in 1D df[$](y)
d$(x) =
Q y x Q f(y) x
$ f(y) , (5.10)
54 functional derivative of the zero point energy functional [ 1](x) [ 2](x) [ 3](x) -4 -2 0 2 4 -2 -1 0 1 2 x
Figure 5.1: L(y)for the densities in (5.16) below. of f0[$](x), we make use of (5.3b), df0(y) d$(x) = d d$(x) $(y) $ f(y) ! = d(y x) f0(y)d(f(y) x) $(f(y)) f0(y)$0(f(y)) $(f(y)) df(y) d$(x). (5.11)
In the appendix, we show that, using (5.10) and (5.11) in (5.8) and
inserting the result in (5.7), dFZPE/d$(x) can be expressed as (see
AppendixCfor details)
dFZPE[$] d$(x) = w(x) 4 + 1 4 Z f(x) x dy L(y) | {z } =I(x) , (5.12)
where L(y) is an odd, well behaved function (see also fig.5.1) and
reads explicitly L(y) = v000ee f(y) y w(y) + v00 ee f(y) y w(y) $0 f(y) $ f(y) 3 f0(y)2+1 f0(y)2+1 . (5.13)
Equation (3.58) implies a sum rule on dFZPE/d$(x). Inserting (5.6)
in (3.58), and remembering that w(s) =w f(s) , we see that we must have dFZPE[$] d$(s) + dFZPE[$] d$ f(s) = w(s) 2 . (5.14)
5.2.2 Numerical results for selected densities
In this section, we are going to verify (5.12) numerically, using the
effective convex Coulomb interaction renormalized at the origin vee(x) = 1+1
|x|. (5.15)
(See Chapter.4for a brief discussion on the importance of convexity
of the interaction in SCE-Density Functional Theory.) We pick 3 test
densities, peaked at x =0 $1(x) = p2 pe x2 x 2R, (5.16a) $2(x) = 2 p 1 cosh(x) x 2R, (5.16b) $3(x) = 2 p 1 1+x2 x 2R. (5.16c)
All the respective co-motion functions can be evaluated analytically since the inverse function of (5.2) can be written explicitly. In fig. 5.2, we provide the profile of dFZPE/d$(x)for the test densities (5.16). The
plots show that the shape of the curve can vary drastically depending on the density chosen. In particular, in all the densities we chose
(excluding $3) the functional derivative shows divergences both in the
origin and in the large x limit. The nature of this divergences shall be investigated deeper in Sec.5.2.3.
Since the derivation of (5.12) was cumbersome, we decided to verify
it numerically to exclude any possible error. We thus simply use the definition of functional derivative
FZPE[$+ef] FZPE[$]⇠ eZ dx dFZPE[$]
d$(x) f(x) e⌧1. (5.17)
If our expression for the functional derivative is correct, we should
have that the slope of the l.h.s. of (5.17) at e = 0 coincides with the
straight line on the r.h.s. of (5.17). For the numerical verification, we consider the following perturbations
f1(x) =e 3x2 ✓ x2 5 36 ◆ cos(x), (5.18a) f2(x) =e 3x4 x2 0.171617 cos(x). (5.18b)
The shape of these functions has been chosen arbitrarily, though they are symmetric, integrate to 0 (thus not changing the number of particles) and, thanks to their fast decay at large x, are such that
$i(x) +ef(x) >0 8x2 R, for at least e2 [ 0.5, 0.5]for the chosen
densities. In fig.5.3we show the l.h.s. of (5.17) as a function of e and the corresponding r.h.s., linear in e. In all cases the tangent of the l.h.s. of (5.17) shows an excellent agreement with (5.12).
FZPE [1] 1(x) (x) 4 I(x) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 -2 -1 0 1 2 x (a) FZPE[ 2] 2(x) (x) 4 I(x) 0 2 4 6 8 10 12 -0.4 -0.2 0.0 0.2 0.4 0.6 x (b) FZPE[ 3] 3(x) (x) 4 I(x) 0 2 4 6 8 10 -0.2 -0.1 0.0 0.1 0.2 0.3 x (c)
Figure 5.2: Functional derivative as from (5.12) for the first three densi-ties (5.16).
-0.4 -0.2 0.0 0.2 0.4 -0.010 -0.005 0.000 0.005 0.010 1 1+12 1 1-12 1 (a) -0.4 -0.2 0.0 0.2 0.4 -0.02 -0.01 0.00 0.01 0.02 1 1+12 2 1-12 2 (b) -0.4 -0.2 0.0 0.2 0.4 -0.006 -0.004 -0.002 0.000 0.002 0.004 0.006 2 2+12 1 2-1 2 1 (c) -0.4 -0.2 0.0 0.2 0.4 -0.010 -0.005 0.000 0.005 0.010 2 2+1 2 2 2-1 2 2 (d) -0.4 -0.2 0.0 0.2 0.4 -0.004 -0.002 0.000 0.002 0.004 3 3+12 1 3-12 1 (e) -0.4 -0.2 0.0 0.2 0.4 -0.006 -0.004 -0.002 0.000 0.002 0.004 0.006 3 3+12 2 3-12 2 (f)
Figure 5.3: For variation f1 (left column) and f2 (right column), and the
58 functional derivative of the zero point energy functional 5.2.3 Divergences of dFZPE/d$(x)in 1D
In what follows, we study the behaviour of the functional derivative at large x. The same behaviour can be deduced for small x, due to the fact that w(x) = w f(x) and that limx!0± f(x) = ⌥• (see text
after (5.2)). Keeping in mind that limx!•I(x) = const, it is clear
from (5.12) that for x 1
dFZPE[$] d$(x) ⇠ w(x) 4 ) vZPE(x)⇠ w(x) 4 . (5.19)
The behaviour of dFZPE/d$(x)at large x is dominated by w(x), which
in turn is determined by the interplay between the electron-electron
interaction and the density decay at large x, cf. (5.4). With
interac-tion (5.15) v00
ee(x)⇠ x 3 at large x, hence the frequency will diverge
whenever $(x) =o(x 3)for x 1. This is the case for densities $1,2
in (5.16) which both decay exponentially (or faster). Such a divergence
of w(x)makes the interpretation of the expansion of vl less
straight-forward: for what just stated in (5.19), at large distances, its asymptotic expansion reads
vl[$](x)
⇠lvSCE(x) plw(x)
4 x 1. (5.20)
At first glance, it seems that the expansion at large l for vl is not
consistent with the requirement vl 2 L3/2+L•: if w(x)diverges to
+• then, for every fixed l, there is a point x after which the second
term in (5.20) becomes dominant and the minimum of vl(x) is at
x = ±• (since vSCE(x) ⇠ (N 1)/|x| for large x for the chosen
interaction). To make sense of (5.20), one has to be careful in taking
the correct order of limits: what we mean here is that for each fixed x the expansion of vl as a function of l follows (5.20).
On the other hand, 1D models often assume an effective electron-electron interaction depending on the physics they aim to describe, often leading to short range interactions. From the preceding discus-sion, it is clear that a short-range interaction should lead to a better
behaviour of w(x)and hence, the behaviour of vZPE. We have tried
two different short range interactions, namely a modified Yukawa potential
vYukee (x) = e
a|x|
1+|x| (5.21a)
and a purely exponential one, popular in DMRG calculations[85],
vexpee (x) =Ae k|x|, (5.21b)
with k 1 =2.385345 and A=1.071295. As an example, in fig.5.4we
plot how the profile of w[$2](x)varies as we pick different interactions.
If we pick a sufficiently high a, w(x)is damped (and consequently the
Coul[ 2](x) Yuk[ 2](x) exp[ 2](x)
0
1
2
3
4
5
6
0.0
0.2
0.4
0.6
0.8
1.0
x
Figure 5.4: Different frequency-profiles for $2with the regularized Coulomb
interaction (5.15) (wCoul), the Yukawa interaction (5.21a) with a=2 (wYuk) and the exponential interaction (5.21b) (wexp).
to a convergent frequency (w[$2](x) ⇠ p pa 2 x 1 2e(1 a)x). Notice that
neither (5.21a) nor (5.21b) would provide a finite w(x)when using
density $1, as the Gaussian decay would prevail on both interactions
with any choice of parameters. A faster decaying interaction would be
needed, e.g.⇠e x2
.
5.3 exchange-correlation potential for a 1d dimer It is known[73,83,86–89] that the exact exchange-correlation (xc) po-tential of a homo-nuclear dimer builds a peak in the mid-bond region that, in the dissociating limit, must be proportional in height to the ionization potential of each fragment. Although some GGA
function-als build peak-like features in the bond mid-point[88], they miss its
peculiar scaling properties[89] which in general are not recovered by
local, semilocal or hybrid functionals[89]. Using only vSCE as an
approximation to the true xc correlation potential does not allow to recover exactly this feature, which is of purely kinetic nature[16,89,
90]. It is the purpose of this section to investigate whether the expres-sion obtained so far can help in reproducing, at least qualitatively, this characteristic.
Consider the density $D
$D(R; x) = 12 ⇣ e |x R 2|+e |x+R2| ⌘ . (5.22)
Having two equal maxima located at±R
2, $D can be considered as a
1D model for a homo-nuclear dimer whose density profile is paramet-rically dependent on the internuclear distance R. This model has been used several times[74,80,81,83,85] since it has been proved to mimic
60 functional derivative of the zero point energy functional
Figure 5.5: The SCE xc potential (solid) and the effect of the ZPO correc-tion (dashed) for R =15. Inset: functional derivative of FZPE[$]
as from (5.12) for $D(R; x) calculated numerically at different internuclear distances R.
many exact features of the exact KS potential for real molecules; in particular, it gives us the opportunity to model the bond stretching and analyse the kinetic contributions to the xc potential.
To write an expression for the xc potential, we start from the
adia-batic connection formalism[28]. The xc energy can be written exactly
in terms of an integral over the coupling l Exc[$] =
Z 1
0 dl Wl[$], (5.23)
where
Wl[$] =hYl[$]|ˆVee|Yl[$]i UH[$], (5.24)
UH[$]being the Hartree functional. Using the large l expansion of
the adiabatic connection integrand[91]
Wl[$]⇠VeeSCE[$] UH[$] +F ZPE[$]
2pl l 1, (5.25)
we obtain
Exc[$]⇠ EZPExc [$] =VeeSCE[$] UH[$] +FZPE[$], (5.26)
and
vxc[$](x)⇠ vSCE(x) vH(x) + dF ZPE[$]
d$(x) . (5.27)
In fig. 5.5 we show the potential in (5.27) for R = 15. Via (5.19),
dFZPE/d$(x)indeed introduces a correction in the mid-bond region.
In fact, since we have[84] f[$D](x!0+) ⇠log(x) R+log ✓ 2 1+e R ◆ , (5.28)
from our treatment in section 5.2.3, the divergence in the mid-bond region can be readily evaluated inserting (5.28) in (5.4)
dFZPE[$D] d$D(R; x) ⇠
(8x) 1/2
(1+|R+log(1+e R) log(2x)|)3/2. (5.29)
For any fixed x 6=0, we find dFZPE/d$D(R; x) !0 as R !• while
similarly, due to the fact that w(x) = w f(x) , a divergence of the
xc potential appears also at large x. Thus, the kinetic correlation energy introduced by the ZPE creates a divergence instead of a finite peak in the bond mid-point, and this divergence occurs on a region
that shrinks when R ! •. This divergence is due to the extreme
correlation between the two electrons: when, say, electron 1 oscillates around the origin, electron 2 jumps from plus to minus infinity. In the exact wavefunction, when one electron crosses the bond mid-point, the conditional position of the other electron also “jumps” from one atom to the other (which is the origin of the peak[83,86,89]), but it is distributed according to the one-electron density on each atom.
The ZPE correction to the SCE approximation of vxc in (5.27)
in-cludes a positive contribution from the region l ⇠ 0 that, although
integrable, is too large to provide a reasonable estimate of the xc
energy Exc[$]. This is due to the fact that we are using only pieces
of information from the high coupling limit to approximate Wl[$]. A
way to improve this approximation is to include also exact ingredients
from the l ! 0 limit[8, 16, 42, 92], by writing an expression that
reproduces the correct behaviour of Wl at small and strong couplings;
among these, the interaction strength interpolation (ISI)[8] has been
subject of study in recent years[79, 92, 93]. In this final paragraph,
we investigate the effect of one (simplified ISI) as proposed in[16],
which is size consistent for the dissociation of a system into two equal
fragments. Hence we approximate Wl[$]to
WlisiZPE[$] =VeeSCE[$] UH[$] + F ZPE[$] 2pl+a[$], (5.30) with a[$] = ✓ FZPE[$] 2(Ex[$] (VeeSCE[$] UH[$])) ◆2 . (5.31)
The xc energy reads then
EISIxc[$]⇠VeeSCE[$] UH[$] +FZPE[$]
✓q 1+a[$] qa[$] ◆ | {z } FZPE ISI [$] (5.32)
While the xc potential is changed considerably at small R, for large internuclear distances the effect of the ISI becomes negligible: already
at R=5 we see that the effect is small and at R=15 the two curves
becomes indistinguishable (see fig. 5.6). The term dFZPE
d$ completely
5.4 conclusions
In this work we worked out an explicit expression for the functional derivative of the subleading term of the generalized universal func-tional FZPE[$]in the strong coupling limit of Density Functional Theory
for 2 electrons in 1D. Our expression respects the sum rules deduced
first in[14] on physical grounds, and has been verified numerically.
We found that the asymptotic behaviour of dFZPE/d$(x)for x !
• is dictated by the asymptotic behaviour of the ZPE frequency
w(x). The asymptotic behaviour of w(x) is dominated by the ratio
v00
ee(x)/$(x)for large x, so typically depends on the relative decay of
the density compared to the interaction. For relatively fast decaying
densities, w(x) and hence vZPE diverges for x ! • and x ! Ne(1).
We expect similar features to be present in more general cases (higher dimensions and more particles). Though we do not have an explicit
expression of FZPE to directly evaluate its functional derivative, the
sum rule (3.58) is generally valid and indicates that vZPE should have
at least the same divergences as the ZPE frequencies wµ(x). So in the
general 1D case, we expect divergences of the ZPE potential at the points where the density integrates to an integer particle number.
By studying the dissociation of a symmetric dimer, we have demon-strated that the ZPE correctly generates a peak in the mid-point re-gion, properly purely built by the kinetic energy. Unfortunately, the
diverging features of w(x)also make the peak diverging for Coulomb
systems, instead of reaching a finite value as in the exact case.
In the future, we aim to investigate the next leading term of the generalized universal functional. This should include exact pieces of information on the ionization energy, hence “curing” the divergences
appearing at the ZPE order[90]. Another promising research line is the
calculation of the kernel of FZPE[$], i.e. its second functional derivative,
which can be used as an adiabatic but spatially non-local xc-kernel in the response formulation of Time Dependent Density Functional Theory.
