Quantum fluctuations and kinetic correlation in the strongly Interacting Limit of
Density Functional Theory
Grossi, J.
2020
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citation for published version (APA)
Grossi, J. (2020). Quantum fluctuations and kinetic correlation in the strongly Interacting Limit of Density
Functional Theory.
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3
D E N S I T Y F U N C T I O N A L T H E O R Y F O R S T R O N G LY C O R R E L AT E D S Y S T E M S
It will be quickly seen that infg7!$Tr{g ˆVee}must be extremely complicated,
and to say it is "non-local" is an understatement.1
— Elliot Lieb[43] 3.1 strongly interacting limit in density functional
theory
In this chapter we will focus on the Strongly Interacting Limit of eq. (2.37), i.e. lim l!•Fl[$] =llim!• ⇣ Tl[$] +lVl ee[$] ⌘ . (3.1)
As shown later in sec.3.3, arguments based on semi-classical analy-sis[14,44] seem to suggest that Tl[$]⇠ O⇣pl⌘for l 1, implying
that it can be neglected in the limit l !•. We therefore define the Strongly Interacting Limit functional:
VSIL
ee [$]:=Yinf7!$hY|ˆVee|Yi =llim!• Fll[$]. (3.2)
Notice that we wrote eq. (3.2) as an infimum rather than a minimum, since the minimizer will turn out to be a distribution rather than a proper function: we recall that for every fixed l the minimizer of Fl[$]
must be an eigenstate of the Hamiltonian ˆHl[$], eq. (2.39).
In ˆHl[$], the one-body potential ˆVl[$]is supposed to fix the
den-sity at all l, counterbalancing the effects of the operators ˆT and ˆVee.
However, since we believe that the expectation value of the kinetic energy grows only aspl, in the limit l!• we must have
ˆVl[$] =l
Â
N i=1 vSIL[$](ri) | {z } ˆVSIL[$] +o(l), (3.3)in order to counter the infinite repulsion of the electron-electron oper-ator. We therefore end up minimizing a classical operator:
min Y7!$hYl|ˆT+l ˆVee+ ˆV l |Yli )l inf Y7!$ Z ⇣ ˆVee+ ˆVSIL ⌘ |Y|2. (3.4)
1 ˜I($) =inf{(G)|G7!$}in the original text.
We see that the infimum in eq. (3.4) is achieved for a distribution
|YSIL(r1, . . . , rN)|2 which is 0 everywhere except for the subspace W0
in which
ESILpot(r1, . . . , rN):= ˆVee+ ˆVSIL (3.5)
is minimum. Typically, the minimum of a manifold is a discrete set of points {˜s1, . . . , ˜sN}. This, however, would be inconsistent with a
smooth density, which cannot be obtained if|Y|2 ⇡Â d(ri ˜si).
Impos-ing the constraint Y7! $requires W0to be degenerate on a subspace
of RdN of dimension no less than d. If d is indeed the dimension of
W0 we can write
W0(s1) ={s1, s2= ˜f2(s1), s3= ˜f3(s1), . . . , sN = ˜fN(s1)}, s1 2 P ✓Rd.
(3.6) This is consistent with a deterministic solution to eq. (3.4) (see sec.3.2
for further details), and it is what became known in the Chemistry community as Strictly Correlated Electrons, or SCE2.
In fact, eq. (3.6) is coherent with a physical picture in which, being the kinetic energy suppressed, the electron-electron interaction fixes the equilibrium position of the particles. The deterministic maps , or co-motion functions ˜fi satisfy therefore
$ ˜fi(r) d˜fi(r) =$(r)dr, i2 [1, . . . , N]⇢N, (3.7)
and can be imposed to satisfy the group properties[16,50]: ˜f1(r)⌘r, ˜f2(r)⌘ ˜f(r), ˜f3(r)⌘ ˜f(˜f(r)), . . . ˜fN(r) = ˜f(˜f(. . . ˜f(r). . .)) | {z } N 1 times ˜f(˜f(. . . ˜f(r). . .)) | {z } N times =r (3.8)
From eq. (3.7) we can read the physical meaning of the comotion functions: since the number of particles in a volume dr must be the same in each of the volumes d˜fi(r), the role of the comotion functions
is to provide the position of N 1 electrons in terms of a reference electron.
Finally, the group properties (3.8) represented in fig.3.1are conve-nient, because they guarantee that the particles are indistinguishable.
2 It is worth noticing that that the problem (3.4) does not always have a deterministic solution[45,46]. Indeed, it is known that the minimizer of ESIL
pot is deterministic only in d=1 or N=2[47–49].
3.1 strongly interacting limit in density functional theory 17 s ˜f2(s) ˜f3(s) · · · ˜fN(s)
Figure 3.1: Schematic representation of the group properties. Each arrow rep-resents the application of the first non-trivial comotion function, ˜f2(s).
The minimizing SCE distribution then reads as a sum of configu-rations degenerate in energy which provides the correct density by means of eq. (2.5): |YSCE|2= 1 N!
Â
} Z N’
i=1 $(s) N d(ri ˜f}(i)(s))ds. (3.9)Using eq. (3.9) we can write an explicit form for the SCE functional: VSCE ee [$] = Z RdN ˆVee|YSCE[$]| 2 = 1 N N 1
Â
i=1 NÂ
j=i+1 Z Rd$(r)vee(|˜fi(r) ˜fj(r)|)dr = 1 2 N 1Â
i=1 Z Rd$(r)vee |r ˜fi(r)| dr. (3.10)The associate multiplicative potential vSCE[$](r)can be obtained in
principle by applying eq. (2.40), i.e.: dVeeSCE[˜$]
d ˜$(r) ˜$=$ = v
SCE[$](r). (3.11)
Being the comotion functions functionals of the density as well, via eq. (3.7), this computation is not the simplest. Luckily, the physical meaning of fi provides us with a powerful shortcut to compute the
effective external potential vSCE.
In fact, in the SCE regime the net force experienced by an electron in position r is given by N 1 electrons located at position fi(r).
This implies though that the force experienced by each particle is a function of the position of the particle itself. Imposing this force to be 0 at equilibrium implies that8i
riEpotSCE(r1, . . . , rN)W0 =ri " 1 2 N
Â
i6=j vee(|ri rj|) + NÂ
i=1 vSCE[$](ri) # W0 =0, (3.12) or rvSCE[$](r) = NÂ
i=2 [rvee(|r s|)] s=˜fi(r) . (3.13)3.1.0.1 The Seidl conjecture
In 1999, Seidl proposed[7] an explicit construction scheme for the comotion functions ˜fi based on integrals of the density for the d=1
cases. Later in 2007, Seidl Gori Giorgi and Savin (SGS)[50] generalized this construction to the spherically symmetric case in d=3.
The Seidl conjecture has been proved, in the case of d = 1 for bounded convex interactions or N =2, to provide the optimal
como-tion funccomo-tions[47–49]. On the other hand, there are cases in higher dimensions where, even for convex interactions, the optimal maps do not fit in the SGS scheme[46].
It is therefore both instructive and preparatory for the rest of this thesis to discuss the case for d=1. Eq. (3.7) in 1D implies that
d dx
Z fj(x)
fi(x)
$(x0)dx0 =0, (3.14)
meaning that the chunk of density comprised between any pair of positions given by fi(x) and fj(x)stays constant 8x. This, together
with the group properties (3.8), and to avoid any accidental clustering of the SCE state that would destroy the indistiguishability of the particles, necessarily imposes that
Z fi+1(x)
fi(x)
$(x0)dx0 =1. (3.15)
We illustrate this concept in fig.3.2.
The Seidl ansatz can be built starting from the cumulant function Ne(x):=
Z x
•$(x
0)dx0, (3.16)
we can combine eqs. (3.14), (3.15), get an explicit form for the comotion functions[16]: fi(x) = 8 < : N 1 e (Ne(x) +i 1) x ai N 1 e (Ne(x) +i 1 N) x> ai , (3.17)
3.1 strongly interacting limit in density functional theory 19
Figure 3.2: The condition (3.15) sketched for a general one dimensional den-sity with N =3. -8 -6 -4 -2 0 2 4 6 8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
(a) The density $g
-8 -6 -4 -2 0 2 4 6 8 -8 -6 -4 -2 0 2 4 6 8
(b) Comotion functions for $g
Figure 3.3: The relation between density and comotion functions. with ai = Ne 1(N+1 i). From now on, fi will always represent a
comotion function of the Seidl form.
In figure3.3 we report the comotion functions for a density $g(x)⇡ N
3
Â
i=1
gie (x x0,i)2 (3.18) normalized to N = 3 in the interval x 2 [ 8, 8], with g1 =0.5, g2 =
0.7, g3 = 0.3 and x0,1 = 1.5, x0,2 = 0.5, x0,3 = 3. For each fixed x,
one can read the position of the three particles from the plot on the right. When one electron reaches +•, it "reappears" at •. This is
3.2 strongly interacting limit as an optimal transport problem
Optimal Transport is a field in Mathematics which studies the prob-lem of transferring a mass µ into a mass n, while minimizing a cost function.
Despite being posed more than 250 years ago by Monge[51], it was not after 1942 when Kantorovich introduced a generalization to Monge’s problem, now known as Monge-Kantorovich formulation[52], that the field underwent a gargantuan development: in the past twenty years, interest in Optimal Transport has spawned new lines of research not only in Mathematics but also in Chemistry, Economics and Physics.
In the original formulation, assume that we are given two mass distributions, µ(x)and n(y), both normalised to one in a volume V.
Furthermore, assume that a function c(x, y)describes the cost to be paid in order to transfer a mass element from point x to point y (e.g., in its original form the cost function would read|x y|, accounting for the fact that the further the destination, the higher the cost to pay to transfer a mass element, e.g. of soil). We aim at devising a strategy, or optimal map ˜f, to transport the mass µ(x)into n(y), while minimising
the cost: inf
˜f
⇢Z
Vc(x, ˜f(x))dµ n(˜f(x)) =µ(x) 8x2V , (3.19)
where dµ=µ(x)dx for all our purposes. Proving the existence of such maps can be a daunting task, even for the simplest cost functions (e.g., c(x, y) =|x y|p, p >0).
Kantorovich therefore relaxed the problem in terms of probabilities g(x, y), by computing min g ⇢Z c(x, y)dg Z g(x, y)dx=n(y), Z g(x, y)dy=µ(x) , (3.20) where again in most cases of (our) interest dg=g(x, y)dxdy (as we
will outline in a few lines, it is often the case that we can identify g = |Y|2). The minimizer of eq. (3.20) is called optimal plan, and reduces to Monge’s problem if the support of g is a graph x, ˜f(x) .
Eq. (3.20) can be generalized to N mass distributions µi, in which
case it is called multimarginal optimal transport problem3
min
g
⇢Z
c(x1, . . . , xN)dg p#jg=µj . (3.21)
3 We use the customary shorthand notation p#jg =
R
3.3 zero point oscillations 21 The strength of Kantorovich’s relaxation method lies in the fact that it admits a dual formulation, namely:
min g ⇢Z c(x1, . . . , xN)dg p#jg=µj =max uj ⇢
Â
j Z ujdµjÂ
j uj(xj)c(x1, . . . , xN) . (3.22)Being the constraint on the left hand side linear, also the dual for-mulation allows the problem to be solved by linear programming techniques.
The first connection between Optimal Transport and Density Func-tional Theory was established independently in 2012[49] and 2013[48], by recognizing that the computation of VSIL
ee [$]is exactly a problem of
the form (3.22). Just set:
• c(x1, . . . , xN)⌘ 12Âi6=jvee(|xi xj|)
• dg(x1, . . . , xN)⌘ |Y(x1, . . . , xN)|2dx1. . . dxN
• µj(x)⌘ $(Nx)dx 8j
This implies that VSIL
ee [$] can be obtained by the maximum of the
Kantorovich dual problem with N equal marginals: VSIL ee [$] =maxu ⇢Z u(x)$(x)dx N
Â
i=1 u(xi) 12 NÂ
i6=j vee(|xi xj|) , (3.23) a formulation equivalent to the Legendre transform (2.14) which, in the SIL limit, has the formVeeSIL[$] =max v ⇢ inf Y hY|ˆVee+ N
Â
i=1 v(xi)|Yi Z $(x)v(x)dx . (3.24)The minimizers of eqns. (3.23) and (3.24) are connected by a simple shift according to u[$](x) = v[$](x) +C[$], since the minimizer in
the dual formulation is required to yield the minimum in 0, whereas the Lagrangian formulation sets the potential v(x) to 0 at infinity. When the support of g is a graph, the minimizer of VSIL
ee [$]is
Monge-type, or SCE. In the rest of this chapter, we shall always assume that the minimizers of the SIL functional are of SCE type.
3.3 zero point oscillations
We now shift our attention to the second leading term of the expansion of Fl[$]at high couplings. Before entering the mathematical details
of the computation, it is instructive to depict the physics that we are going to try to describe.
The physical assumptions behind the SCE formalism is that elec-trons form a system in which the kinetic energy is suppressed by an infinite interaction O(l). As explained in the previous sections, this lead the electrons to have deterministic positions provided by a set of N comotion functions, fi.
At finite couplings, Heisenberg’s principle suggests us that the elec-trons, being forced no more into fixed position by an infinite repulsive force, must perform zero point oscillations around the equilibrium positions provided by the comotion functions.
As argued in ref. [7], being the restoring forceO(l)we predict the
energy associated with the oscillations to be O(pl). Therefore, we
expect:
E0[$](l)=? lmin ESCEpot[$](r1, . . . , rN)
| {z }
:=ESCE
+plEZPE[$] +o(pl), l 1.
(3.25) The rest of this section will be devoted carry out this semi-classical analysis, following ref. [14].
To start with, rewrite ˆHl[$]as
ˆ
Hl[$] = ˆT+lESCE
pot[$](r1, . . . , rN) + ˆVl[$] l ˆVSCE[$]. (3.26)
Given that we expect small oscillations around the equilibrium posi-tion provided by W0, eq. (3.6), the electrons will explore a small subset
of the configurations space close to W0, which we denote We:
We ={r2W|min
s2Rd|r f(s)| <e}, (3.27)
where we introduced the notation r := {r1, . . . , rN}(see fig.3.4for
an illustration of We for the simplest case of 2 electrons in d =1 for
a symmetric density). Notice also that with the new notation we can write W0(s) = f(s).
3.3.0.1 The curvilinear coordinate system
It seems natural to consider only the region Wefor our computation,
for which we define the Hessian of ESCE
pot, ˜H(r), and carry on the
expansion: ESCEpot[$](r)⇡ l ✓ ESCE+1 2(r f(s))TH˜ (r)r=f(s) | {z } :=H(s) (r f(s)) ◆ = l ✓ ESCE+1 2 dN
Â
i,j=1 (ri fi(s))Hij(s)(rj fj(s)) ◆ . (3.28) The matrix H(s)retains the information about the curvature of ESCEpot(r)3.3 zero point oscillations 23
x f(x)
(a) The set Wespreads around the SCE graph
W0 (in green) as a subset of the whole configuration space W.
x f(x)
(b) The matrix H(s)determines the degen-erate direction (red, with null eigenvalue) and the orthogonal direction to W0 (in blue)
Figure 3.4: The arise of the curvilinear coordinate system of the ZPE formal-ism
paraboloid which locally approximates the behavior of ESCE
pot (r), and
suggest a natural system of coordinates.
The directions along the manifold W0will exhibit d null eigenvalues,
as the minimum is degenerate – hence flat – along the graph(s, f(s)).
The directions orthogonal to W0 will have a total of d(N 1)positive
eigenvalues (since we are considering the minimum of ESCE pot , H(s)
must be semi-positive definite). Therefore we collect the eigenvalues according to w2(s)⌘ {0, . . . , 0 | {z } d times , w2 d+1(s), . . . , w2dN(s) | {z } d(N 1)times }, (3.29)
and the corresponding eigenvectors can be organized accordingly: e(s)⌘ {e(s), . . . , ed(s), ed+1(s), . . . , edN(s)}. (3.30)
The eigenvectors are orthogonal to each other8 s since H(s)is sym-metric. The first d eigenvectors can be identified with the basis needed to write the minimizing s in eq. (3.27). The corresponding coefficients will be our first d coordinates in the new framework. The remaining d(N 1)coordinates can be obtained by imposing the orthogonality
to the manifold W0(s):
eµ(s)·∂f(s)
∂sa
=0, µ2 [d+1, . . . , dN]⇢N, a2 [1, . . . , d]⇢N.
Figure 3.5: The coordinates (3.33), illustrated.
The first d eigenvectors and the remaining d(N 1)decompose the
space We into two orthogonal subspaces which define the new set of
coordinates: r={r11, . . . , rd1 | {z } r1 , r1 2, . . . , rd2 | {z } r2 , . . . , r1 N, . . . , rdN | {z } rN } ) {s1, . . . , sd,qd+1, . . . , qdN | {z } :=q } (3.32) The curvilinear coordinates satisfy (see also fig.3.5)
8 < : (r f(s))·∂∂fs(ns) =0, n2 [1, . . . , d] qµ =eµ(s)· (r f(s)), µ2 [d+1, . . . , dN] , (3.33)
from which an explicit relation between Cartesian and curvilinear coordinate can be written as
r= f(s) + dN
Â
µ=d+1
qµeµ(s). (3.34) The hessian is diagonal in the coordinates(s, q) and reads H(s) =
diag[w2(s)]. The function EpotSCE is then readily written as a sum of
normal modes: ESCE
pot(r))EpotSCE(s, q)⇡ ESCE+12 dN
Â
µ=d+1
w2µ(s)q2µ. (3.35)
The main advantage accomplished by this coordinate transformation lies in the fact that we have been able to switch to a coordinate system in which particles move under the effect of uncoupled harmonic interactions, with a frequency dependent on their position s.
3.3 zero point oscillations 25 Notice that in eq. (3.26) the term (3.35) is multiplied by l. It follows from the basic properties of quantum harmonic oscillators that the effective scale of the fluctuations around the equilibrium position (or, equivalently, the width of the envelop We) must be O l 1/4 .
Therefore, we adjust the coordinates to take into account this scaling with l: (s, q)) (s, u):= (s, l1/4q), (3.36) which implies r = f(s) +l 1/4 dN
Â
µ=d+1 uµeµ(s). (3.37)3.3.0.2 The Hamiltonian in the curvilinear coordinates
The kinetic energy and the external potential in eq. (3.26) still need to be expressed in the new coordinates. To start with, we rewrite the Laplacian operator in its most general form, the Laplace-Beltrami operator[53]. Throughout this section, we will address the curvilinear coordinate set (s, q)with the letter x and the scaled coordinates(s, u)
with the letter ˜x, unless needed otherwise:
(s, q))x,
(s, u)) ˜x. We first define the metric tensor
Gµn(x):= ∂r
∂xµ ·
∂r
∂xn, µ, n 2 [1, . . . , dN]. (3.38)
The Laplace-Beltrami operator reads
dN
Â
i=1 ∂2 ∂r2i = dNÂ
µ,n=1 1 p det G(x) ∂ ∂xµ ✓q det G(x) ⇣G 1(x)⌘ µn ∂ ∂xn ◆ . (3.39) The operator (3.39) has a quite intimidating form. It is helpful in order to fix the ideas and get a feeling of how the general computation runs to consider the case of two electrons in d=1. The general case follows straightforward. In fact, we can write eq. (3.34) asr= r1 r2 ! = s f(s) ! + p q 1+ f0(s)2 f0(s) 1 ! . (3.40) The metric tensor (3.38) is therefore a 2⇥2 matrix whose components read g11(x) = ∂r ∂s · ∂r ∂s g12(x) = ∂r ∂s · ∂r ∂q g21(x) = ∂r ∂q· ∂r ∂s g22(x) = ∂r ∂q· ∂r ∂q (3.41)
or g11(x) =1+ f02 2q f 00 p 1+ f02 +q 2✓ f002 (1+ f02)2 ◆ g12(x) =g21(x) =0 g22(x) =1 . (3.42)
In terms of the scaled coordinates(s, u)the metric tensor ˜G is obtained through the matrix
S= 1 0
0 l 1/4
!
, (3.43)
from which ˜G(˜x) =S·G s, l 1/4u ·S. The scaled metric tensor can
be split now into contributions proportional to different orders in u: ˜G(˜x) = 0 @1+ f02 0 0 p1 l 1 A | {z } :=˜G0 + u l1/4 0 @ 2 f002 p 1+f02 0 0 0 1 A | {z } :=D +pu2 l 0 @ f002 (1+f02)2 0 0 0 1 A | {z } :=Z | {z } ”A” . (3.44) Use the fact that det(˜G0+A) =det(˜G0)det(1+A ˜G01)to write
p det ˜G ⇡ p g11(s, 0) l1/4 (3.45) 1 p det ˜G ⇡ l1/4 p g11(s, 0) (3.46) ˜G 1 ⇡ 0 @g11(1s,0) 0 0 pl 1 A . (3.47)
Using these results in eq. (3.39) (properly modified for the coordinates ˜x) provides ∂2 ∂r21 + ∂2 ∂r22 = p l ∂ 2 ∂u2 +o( p l). (3.48)
This result can be generalized (see AppendixA), allowing us to write ˆT⇠ p l 2 dN
Â
µ=d+1 ∂2 ∂u2µ +o(pl). (3.49)Thus both the second leading term of ESCE
pot and the kinetic energy
operator ˆT can be showed to be O(pl)in the curvilinear coordinates
˜x. The third term, namely the difference ˆVl ˆVSCEcannot be showed
to beO(pl). We are therefore compelled to make such an assumption, for which we write
ˆVl(r) l ˆVSCE(r)=? pl ˆVZPE(r) +
Â
• i=13.3 zero point oscillations 27 This, using eq. (3.37), implies that to leading order we should have on We
ˆVl(r) l ˆVSCE(r)⇠pl ˆVZPE(f(s)), l 1. (3.51)
To conclude, for l 1 we have ˆ Hl[$]⇠ lESCE+pl 1 2 dN
Â
µ=d+1 ∂2 ∂u2µ +1 2 dNÂ
µ=d+1 wµ(s)2uµ2 + ˆVZPE(f(s)) ! | {z } :=HˆZPE[$] . (3.52) The Hamiltonian ˆHZPE[$]describes a set of d(N 1)harmonicoscilla-tors which depend on the variable s whose dynamics, however, enters to a smaller order in l, as showed in eq. (3.49)
3.3.0.3 Ground state of the ZPE Hamiltonian
The ground state of ˆHZPE[$] will read formally as the product of
ground states of single particle harmonic oscillator with frequencies wµ(s) YZPE(˜x) =N (s) dN
’
µ=d+1 ✓ wµ(s) p ◆1 4 e wµ2(s)u2µ (3.53)with ground state
EZPE(s) = 1 2 dN
Â
µ=d+1 wµ(s) + ˆVZPE(f(s)). (3.54)The normalization can be obtained by imposing that Z
dsZ du ld1 N4 J(s, l 1/4u)|YZPE(˜x)|2=1, (3.55) J being the Jacobian for the change of coordinates r7!x. The fact that
lim l!• $l(s) N , $l(s) = Z du ld1 N4 J(s, l 1/4u)|YZPE(˜x)|2 (3.56) implies that, to leading order,
N (s) =
s $(s)
NJ(s, 0). (3.57)
At first, it might seem that something is wrong with eq. (3.54). In fact, on the right hand side we have a function of s which in general will admit a non-degenerate minimum, implying that at the orderplthe representability condition Y7!$cannot be fulfilled. Nevertheless, we still have the freedom to impose to ˆVZPE to be such that EZPE(s) =
const.,8s. In other words:
ˆVZPE s, f 2(s), . . . , fN(s) = N
Â
i=1 vZPE fi(s) = dNÂ
µ=d+1 wµ(s) 2 +const. (3.58)This also clarifies the role of the term ˆVZPE(f(s)), namely keeping
the degeneracy of W0 at the orderpl. The weight of each degenerate
configuration EZPE(˜s)will hence be proportional to the density $(s),
allowing to write EZPE = Z $(s) N 1 2 dN
Â
µ=d+1 wµ(s) + ˆVZPE(f(s)) ! ds. (3.59)Finally notice that since ˆHl[$]and ˆHZPE[$]differ only by a constant,
they have the same ground state wavefunction, allowing to write E0[$](l)⇠lESCE[$] +plEZPE[$]. This for the generalized functional
reads Fl[$] =E0[$](l) Z vl[$](r)$(r)dr ⇠lVeeSCE[$] +pl Z $(s) N dN
Â
µ=d+1 wµ(s) 2 ds | {z } :=FZPE[$] +o(pl). (3.60)By means of the Hellman-Feynman theorem we can infer also the asymptotics for Wl[$] at high couplings. Just differentiate on both
sides eq. (3.60) to obtain4
Vl ee[$] =Wl[$] +UH[$] ⇠VeeSCE[$] +p1 l 1 2 Z $(s) N dN
Â
µ=d+1 wµ(s) 2 ds | {z } :=W•0[$] +o(pl). (3.61)3.3.0.4 On the meaning of YZPE
In the last paragraph, we showed that the universal functional in the strongly interacting limit has an expansion in terms of powers ofpl:
hYl[$]|ˆT+l ˆVee|Yl[$]i ⇠lVSCE ee [$] + p l⇣˜T(1/2)[$] + ˜Vee(1/2)[$] ⌘ | {z } =2 ˜T(1/2)[$] . (3.62) The coefficient 2 ˜T(1/2)[$]was obtained by modifying the l-dependent
Hamiltonian ˆHl[$]into ˆHZPE[$]and correspondingly the ground state
from Yl[$]to YZPE[$]to obtain the same coefficients, to leading order:
hYZPE[$]|HˆZPE[$] ˆVZPE(f(s))|YZPE[$]i ⇠pl⇣˜TZPE[$] +W•0 [$]⌘,
(3.63)
4 A remark is in order. The notation W•0 [$]was originally chosen in analogy with the expansion at small couplings, eq. (2.47), for which the GL2 correction is indeed the derivative of the adiabatic connection integrand at l=0. The counterpart at l!• being clearly 0, as Wl approaches smoothly a constant, makes this an extremely
misleading choice. Nevertheless, it has become customary in literature and as such will be kept throughout this thesis.
3.3 zero point oscillations 29 with
˜T(1/2)[$] = ˜TZPE[$] =W0
•[$] = ˜Vee(1/2)[$]. (3.64)
It should be noted, however, that one needs to be careful when drawing conclusions while taking the expectation value of ˆT+l ˆVeeon the ZPE
wavefunction:
hYZPE[$]|ˆT+l ˆVee|YZPE[$]i ⇠pl⇣˜TZPE(1/2)[$] + ˜Vee,ZPE(1/2)[$]⌘. (3.65) While it is true in fact that
˜T(1/2)
ZPE [$] = ˜T(1/2)[$], (3.66)
it is not true that the same holds for the true electron-electron interac-tion operator. In other words, we have:
˜V(1/2)
ee,ZPE[$]6= ˜Vee(1/2)[$]. (3.67)
We believe it is better to clarify this via an example, which will be again subject of study in Chapter6. It consists of two electrons confined in an harmonic trap with repulsive harmonic interaction. We call this system "philharmonium". The Hamiltonian reads
ˆ Hl = 1 2 ✓ ∂2 ∂x21 + ∂2 ∂x22 ◆ +w 2 l 2 (x21+x22) l 2(x1 x2)2. (3.68) Again, introducing a rotation of coordinates
8 > < > : R = x1p+x2 2 r = x1px2 2 , (3.69)
the Hamiltonian reads ˆ Hl = 1 2 ∂2 ∂R2 + w2l 2 R2 1 2 ∂2 ∂r2 + w2l 2l 2 r2, (3.70) with a ground state given by
Yl(x1, x2) = 0 @wl q w2l 2l p2 1 A 1 4 e wl4 (x21+x22)e p w2l 2l 4 (x21+x22). (3.71)
The density coming from such a ground state is a Gaussian: $l(x) =2 r ˜wl p e ˜wlx2, ˜w l = 2wl q w2l 2l wl+ q w2l 2l . (3.72)
It is easy now to fix the density at all l, since it amounts to impose that ˜wl = ˜w= const) wl = g˜w(l). If we now expand the various
expectation values on the ground state wavefunction (3.71) with wl = g˜w(l), we obtain hYl|ˆT|Yli ⇠ l˜w+ p l 2p2 + ˜w 8 (3.73) hYl|ˆVee|Yli ⇠ l ˜w+ p l 2p2 (3.74) hYl|ˆVl|Yli ⇠ l˜w+ p l 2p2 + ˜w 8 (3.75)
(notice the virial theorem, satisfied to all orders).
On the other hand, the function g can be expanded for large l to yield g˜w(l)⇠p2l+ ˜w
2
8p2pl, (3.76)
a result that can be inserted directly in (3.71) to yield an approximate wavefunction, which we label YZPE:
YZPE = p l ˜w p 2p2 !1 4 e pl 2(x1+x2)2 41 ˜w(x1 x2)2. (3.77)
Using YZPE to compute the expectation values, we get hYl|ˆT|Yli ⇠ l ˜w+ p l 2p2 + ˜w 8 (3.78) hYl|ˆVee|Yli ⇠ l˜w (3.79) hYl|ˆVl|Yli ⇠ l˜w+ p l 2p2 (3.80)
So we see, comparing with eqs. (3.73), that indeed the expectation values of the external potential and the interaction energy are accurate only to first order, whereas the kinetic energy is correct through order
