Density-density functionals and effective potentials in many-body electronic structure calculations
aa r X i v : . [ c ond - m a t . o t h e r] M a r October 25, 2018
Density-density functionals and effective potentials in many-body electronic structure calculations
F. A. Reboredo and P. R. C. Kent Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA
We demonstrate the existence of different density-density functionals designed to retain selected propertiesof the many-body ground state in a non-interacting solution starting from the standard density functional theoryground state. We focus on diffusion quantum Monte Carlo applications that require trial wave functions withoptimal Fermion nodes. The theory is extensible and can be used to understand current practices in severalelectronic structure methods within a generalized density functional framework. The theory justifies and stimu-lates the search of optimal empirical density functionals and effective potentials for accurate calculations of theproperties of real materials, but also cautions on the limits of their applicability. The concepts are tested andvalidated with a near-analytic model.
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I. INTRODUCTION
Density Functional Theory (DFT) is based on theHohenberg-Kohn proof of a functional correspondence be-tween the ground state energy and the ground state density E [ ρ ( r )] . In the formulation of Kohn and Sham (K-S), the in-teracting electron gas is replaced by non-interacting electronsmoving in an effective potential. In this construction, the non-interacting density ¯ ρ ( r ) is equal to the interacting one ρ ( r ) butno other property of the interacting ground state is in principleretained in the non-interacting wave function. Initially DFTwas formulated to describe the total ground state energy of aninteracting system and ρ ( r ) . Although progress towardsmore accurate density functionals is ongoing, current approxi-mations such as the local density approximation (LDA) andmore recent gradient-based extensions are already success-ful in predicting many electronic properties of real materials.This success has led to the use of DFT beyond its formal scopeand unfortunately tempted some to believe that if we had theexact ground state density functional, we would only need tosolve non-interacting problems even for properties not relatedto the ground state energy and density. While the virtues andlimitations of the Kohn-Sham eigenvalues are discussed intextbooks, the possible reasons for the success or failure ofKohn-Sham wave-functions in many-body problems are littleunderstood and widely dispersed throughout the literature.It is often assumed, without a formal proof, that the Kohn-Sham non-interacting ground state wave-function forms agood description of the interacting ground state wave-functionto be used as the foundation of theories that go beyond DFTsuch as GW-Bethe Salpeter Equation (GW-BSE), QuantumMonte Carlo (QMC), or even configuration interaction (CI).This leads to an apparent contradiction in the literature sincedensity functionals that provide wave-functions that are agood starting points in one field (as judged by comparisonwith experiment) are found inadequate in others. Broadlysummarizing: for structural properties gradient corrected den-sity functionals are nowadays preferred over LDA. Struc-tural properties depend essentially of atomic forces which inturn are related to the density. However, for GW-BSE cal- culations of optical properties, an LDA-based ground stateis preferred . In this approach a good initial approximationfor the Green function is required. In QMC calculations a(non-interacting) Hartree-Fock (HF) ground state might bepreferred over LDA, but the subject is still under debate. InCI calculations, instead, it is empirically claimed that the con-vergence with natural orbitals is more rapid than HF orbitals.In Diffusion Quantum Monte Carlo (DMC) a trial wavefunction enforces the antisymmetry of the electronic many-body wave function and the nodal structure of the solution.The accuracy of the trial wave-function is critical and deter-mines the success or failure of the method to accurately pre-dict properties of real materials. The trial wave-function isusually a product of a Slater determinant Φ T ( R ) and a Jastrowfactor e J ( R ) . Φ T ( R ) is often constructed with single parti-cle Kohn-Sham orbitals or from other mean field approachessuch as HF. The Jastrow, in turn, is a symmetric factor whichdoes not change the nodes, but accelerates convergence andimproves the algorithm’s numerical stability. The DMC algo-rithm finds the lowest energy of the set of all wave-functionsthat share the nodes of Ψ T ( R ) . The exact ground state en-ergy is obtained only if the exact nodes are provided. Sinceany change to an antisymmetric wave-function must resultin a higher energy than the antisymmetric ground state, theenergy obtained with arbitrary nodes is an upper bound tothe exact ground state energy. Only in small systems is itpossible to improve the nodes or even avoid thetrial wave-function approach altogether . Consequently,a general formalism that could alleviate the nodal error inlarge systems is highly desired. Quite recently it has beenshown that within the single Slater determinant approach thecomputational cost of the DMC algorithm can have an al-most linear scaling with the number of electrons . It isclaimed, if not formally proved, that the nodes of the many-body wave-function are not too far from those of a wave func-tion obtained via a mean field approach. However, this mightnot continue to hold as electron-electron interactions becomemore important. To improve the accuracy of these approachesand increase the range of materials to which they can be ap-plied it is important to examine the advantages of differentmean-field wave-functions.In this paper, we demonstrate that density-density function-als can be obtained by finding the minimum of different costfunctions relating the set of non-interacting v -representableground state with an interacting many-body state. The mini-mum of these cost functions establishes a correspondence be-tween the non-interacting and the interacting wave-functionsand their associated densities and potentials. The cost func-tion can be designed to retain selected properties of the many-body wave function in the non-interacting one. Crucially, forDMC applications the nodes can be optimized. Under cer-tain conditions density-density functionals exist that can leadto standard scalar-density functionals. As in the case of stan-dard DFT, this proof does not mean that we know the expres-sion of each functional or associated potential but it will cer-tainly stimulate the search of methods to find or approximatethem. For DMC applications it is enough to prove that an op-timal mean field potential for nodes exists. In order to testthe theory, we find the ground state wave function of a modelinteracting system. Then we obtain (i) the exact DFT effec-tive exchange correlation potential associated with the groundstate density, (ii) the potential that maximizes the projectionof Φ T ( R ) with the ground state. Finally, (iii) we optimizea potential to match the nodes and find that surprisingly, forthis model, the non-interacting solution in the same potentialas the interacting problem is a very good approximation forthe nodes while the exact non-interacting Kohn-Sham groundstate is particularly poor.This paper is organized as follows: In Section II we demon-strate the existence of an different density-density correspon-dences associated with cost functions. We prove the exis-tence of this functional correspondence for the case of optimalnodes required in DMC. In Section III we solve an interact-ing problem up to numerical precision and find its many-bodyground state wave-function. Subsequently we optimize differ-ent cost functions to retain specific properties of the groundstate. Finally in Section IV we discuss the relevance of ourresults for many-body electronic structure and give our con-clusions. II. GENERALIZED DENSITY-DENSITY FUNCTIONALS
Given an interaction in a many-body system, theHohenberg-Kohn theorem establishes a functional cor-respondence between densities ρ ( r ) , external potentials V ( r )[ ρ ( r )] and ground state wave-functions Ψ( R )[ ρ ( r )] ;where [ ρ ( r )] denotes a functional dependence on the groundstate density, and R denotes a point in the many-body N space. Since the density changes according to the strengthand functional form of the interaction, this correspondenceis different for different interactions. For a fixed interac-tion, the subset of densities ρ ( r ) corresponding to a groundstate of an interacting system under an external potential V ( r ) are denoted as pure state v -representable. A non-interacting pure state v -representable density is given instead by ¯ ρ ( r ) = P ν | φ ν ( r ) | where φ ν ( r ) are the Kohn-Sham-like single particle orbitals, or eigenvectors, of the Hamiltonian: (cid:20) − ∇ + ¯ V ( r ) (cid:21) φ ν ( r ) = ε ν φ ν ( r ) , (1)where ¯ V ( r ) is an effective single particle potential. For sim-plicity we denote here a density to be v -representable if it isboth pure state non-interacting and pure state v-representable.In the following we also imply pure state when we write only v -representable.Each point in the sets of v -representable densities is asso-ciated with two different points in the wave-functions Hilbertspace. In figure 1 we schematize the subset of v -representabledensities and the functional correspondence with the sub-sets of the interacting and non-interacting ground state wave-functions. Note that, in principle, the two subsets of groundstate wave-functions do not necessarily overlap. In the non in-teracting case the wave-function is given by a Slater determi-nant of Kohn-Sham-like orbitals but for interacting problemsthis simplification is not longer possible.The Kohn-Sham scheme for density functional theory es-tablishes a correspondence between interacting and non-interacting wave-functions represented as line (1) in Fig 1.This Kohn-Sham correspondence between wave-functions isimplicit in the Khon-Sham construction for the external effec-tive potential. Figure 1 emphasizes that the wave-functionsjoined by line (1), while different, give the same electronicdensity. In more technical terms they both belong to thesame Percus-Levy partition of the Hilbert space but theyare the minimum energy wave-function for different interac-tions. The exchange-correlation potential is by constructionthe difference that one has to add to the external potential in anon-interacting problem so that its ground state density is thesame as the interacting one. If the energy-density functional E [ ρ ( r )] is known, the effective non-interacting potential canbe obtained following the standard Kohn-Sham approach. Ifthe ground state density ρ ( r ) is known, the same correspon-dence between interacting and non-interacting densities canbe achieved by minimizing the following function K ρ = 12 Z dr [¯ ρ ( r ) − ρ ( r )] . (2)within the subset of non-interacting v -representable densities.Formally, this could be done by exploring all values ¯ V ( r ) inEq (1).In practice, if the density of the interacting ground state isknown, the potential ¯ V K ρ ( r ) that minimizes Eq. (2) can beobtained numerically with a procedure similar in spirit to theoptimized effective potential (OEP) method. The change inthe density required to minimize Eq. (2) is ∆ ρ = − [¯ ρ ( r ) − ρ ( r )] . (3)Within linear response, the change in the potential required toproduce ∆ ρ is ∆ ¯ V K ρ ( r ) = Z dr ′ [ ρ ( r ′ ) − ¯ ρ ( r ′ )] δV ( r ′ ) δρ ( r ) (4)Adding recursively ∆ ¯ V K ρ ( r ) we can find the potential ¯ V K ρ ( r ) associated with K ρ = 0 (see an example below). FIG. 1: (Color online) a) Representation of the sets of pure state v -representable interacting densities. b) Sets of interacting and non-interacting ground state wave-functions. The Kohn-Sham formula-tion of DFT relates a v − representable density with a pair interact-ing and a non-interacting wave-function. The same functional corre-spondence can be obtained minimizing Eq. (2) (line 1 in the figure).Different cost functions relate an interacting v -representable densitywith a different non-interacting- v representable density (see lines 2and 3). A. Other density-density correspondences
It is often desirable o preserve properties in addition tothe density of the many-body ground state Ψ( R ) in a non-interacting wave-function Φ T ( R ) to be used as a starting pointfor theories that go beyond DFT. This task involves exploringall the non-interacting v -representable set in order to find awave-function that best describes a given property. This isa typical optimization problem. One of the most commonstrategies in optimization is the design of a cost function. Oneexample is Eq (2), a measure of the difference in two densities.Another example of a cost function is K Det = − |h Ψ | Φ T i| (5)which involves a projection of the interacting ground state Ψ in the set of non-interacting v -representable wave-functions { Φ T } . The minimum of Eq. (5) is the non-interacting groundstate Slater determinant with maximum projection in the in-teracting ground state. We have claimed above that the inter-acting wave-functions might be in general very different froma single non-interacting Slater determinant. Accordingly, weexpect K Det > − .We expect to find a different minimum in the non-interacting ground state set, if we change the functional formof the cost function from Eq. (2) to Eq. (5) for the followingreasons:1) We can visualize the cost function as a scalar potentialdefined in the full Hilbert-space. Although different cost func-tions can share the same minimum in the complete Hilbertspace, in the restricted subset of non-interacting ground statewave-functions, different cost functions can have a differentminimum: the optimal point found depends on the functionalform of the cost function. Accordingly, while all the costfunctions we propose here would be minimized if we could reach the interacting many body state Ψ ( where, of course,every property is retained exactly), because our search is con-strained to non-interacting v -representable subset the mini-mum we would find will depend on the properties we wishto retain.2) The Hohenberg Kohn theorem, when applied to the non-interacting v -representable case implies that, in the absenceof degeneracy, there is at most one wave-function that has thesame density as the interacting case. Therefore, once the min-imum of an arbitrary cost function is found, its associatednon-interacting density can no longer be equal to the interact-ing density unless the property enforced by the cost functioncan be related back to density. Enforcing the non-interactingdensity to remain equal to the interacting ground state den-sity prevents all other properties of the non-interacting wave-function from being further improved. If we intend to opti-mize other properties, we have to relax the density constraintfinding a different wave-function associated with a differentdensity.The minimization of different cost-functions, relatingthe interacting ground state Ψ( R ) with the non-interacting v -representable set, provide in-principle different corre-spondences between interacting and non-interacting wave-functions represented as different lines in figure 1. Eachcost function K defines a correspondence different thanthe identity between pure state v -representable densities andnon-interacting pure-state v − representable densities. As aconsequence, the idealized optimization processes outlinedhere defines an operator U K that turns each ρ ( r ) into anon-interacting density corresponding to the wave-functions Φ T ( R ) which is the minimum of a cost function K . ¯ ρ K ( r ) = U K [ ρ ( r )] . (6)Note that if the minimum of a given cost function K is asingle Φ T ( R ) for every v -representable density, then U K de-fines a density-density functional. When more than one non-interacting v -representable wave-function give the same opti-mal value for K , the degeneracy can be broken by additionalrequirements in the cost function [e.g. also minimizing Eq.(2), the difference between the current and pure state densities]. Since we only need one optimal wave-function, any from adegenerate minimum can be chosen to construct the density-density functional U K . When minimization of a cost functiondefines a one to one correspondence with an inverse, a moreusual energy-density functional of the form E { U − K [¯ ρ K ( r )] } can be constructed. Only a restricted class of cost functionslead to density transformations with an inverse. Minimizationof the cost functions among all pure-state-non-interacting v -representable densities defines the optimal effective potentialwhich is a function of this density.Given a cost function, K , finding an approximation for thedensity transformation operator U K could certainly be as de-manding as finding an approximation for the energy-densityfunctional E [ ρ ( r )] required by DFT based methods. This taskis beyond the goal of this paper. However, we will show thatwe can expect the operator associated to the best nodes forDMC ( U DMC ) to be non-local and very different from theidentity. Accordingly we can expect non-interacting wave-functions with good nodes to be a poor source of densities.Moreover, for the example considered below, we find, that thedirection we might have to explore to optimize the potentialmight be surprisingly different than the attempts consideredso far . B. The Diffusion Monte Carlo case
We next show that optimization of the nodes for DMCamong the set of v − representable wave-functions leads to acorrespondence between pure state v -representable densitiesand pure state non-interacting v -representable densities of theclass described above. These in turn demonstrate the existenceof an optimal effective non-interacting nodal potential.Since, the ground state density ρ ( r ) determines the groundstate wave-function Ψ( R )[ ρ ( r )] , ρ ( r ) defines also the points R of the nodal surface S ( R )[ ρ ( r )] where Ψ( R )[ ρ ( r )] =0 . We can also classify the nodal surfaces in pure state v-representable and pure-state-non-interacting v -representable.The DMC algorithm in the fixed node approximation findsthe lowest energy of the set of all wave-functions that sharethe nodes or the trial wave-function. For Slater determinantJastrow wave-functions, the nodes of the trial wave-functionare by construction those of Φ T ( R ) ; that is they are pure-statenon-interacting v -representable. The DMC energy, E DMC isalso a function of the external potential which in turn is afunction of the interacting ground state density V ( r )[ ρ ( r )] .Thus minimization of E DMC [Φ T ( R ) , ρ ( r )] in the set ofnon-interacting v -representable wave-functions Φ T ( R ) deter-mines one Φ T ( R ) with the best nodes. Every optimal Φ T ( R ) defines an optimal auxiliary density ¯ ρ DMC ( r ) . As a conse-quence optimizing the nodes of the trial wave-function by per-turbing the nodes of pure state non-interacting wave-functionsimplies finding another correspondence between interactingand non-interacting densities (another line in figure 1). Thebest cost function for optimal nodes is ultimately the DMCenergy itself.Since we restrict the search to pure-state non-interacting v -representable nodes, the minimum energy E DMC [ ρ ( r ] will belarger than the true ground state energy E [ ρ ( r ] , because ofthe upper bound theorem, unless S ( R ) is non-interacting v -representable.Note that for an arbitrary interaction S ( R ) is not expectedto be, in general, pure-state-not-interacting v-representable.However, if S ( R ) were non-interacting v-representable, thebest Slater Determinant Φ T ( R ) for DMC could be formallyfound by finding the minimum of the cost function K S = Z S dS | Φ T ( R ) | . (7)where R S denotes a surface integral over the interacting nodalsurface. III. COST FUNCTION MINIMIZATION
To demonstrate the theoretical concepts above we solve asimple non-trivial interacting model as a function of the in-teracting potential strength and shape. We then optimize thewave-functions to minimize the cost functions in Eqs. (2),(5) and (7) so as to find the exact DFT wave-function, thewave-functions that maximize the projection on the interact-ing ground state and minimize the projection on the nodes.Subsequently, we estimate the volume of the Hilbert spaceenclosed between the nodes of the interacting wave-functionsand the optimized non-interacting ones.
A. A model interacting ground state
For illustrative purposes we choose the interacting prob-lem to be as simple as possible and yet not trivial. Wesolve the ground state of two spin-less electrons mov-ing in a two dimensional square of side length 1 witha repulsive interaction potential of the form V ( r , r ′ ) =8 γ cos [ απ ( x − x ′ )] cos [ απ ( y − y ′ )] . While this potential isdifferent than the Coulomb interaction, it shares some of itsproperties. For positive γ and | α | < the interaction is repul-sive with a repulsion that increases monotonically when forshorter distances. The amplitude of the repulsion as comparedto the kinetic energy can be changed by adjusting γ . Since theCoulomb interaction is self similar, changing γ mimics whathappens in a real system when we change the size of the sys-tem. The shape of the potential within the confined region canbe altered by changing α . In the limit of α → the interactionpotential is separable which allows several limits to be tested(such as the nodes). The functional form facilitates an analyt-ical treatment of the problem by removing the singularity ofthe Coulomb interaction at short distances.We expanded the many-body wave-function in a full CIon non-interacting Slater determinants with the same sym-metry as the ground state. The ground state is degeneratebecause there are only two electrons. We chose one of theground state wave-functions according to the D subgroupof the D symmetry of the Hamiltonian. With this choice, ρ ( r ) has D symmetry ( x is not equivalent to y ). The basisof Slater determinants was constructed with functions of theform nπx )sin( mπy ) with n and m < . Since parity ispreserved by the interaction and Slater determinants of iden-tical functions are zero, the size of the basis set is reduced toonly 300 in our calculations.Most of the calculations reported here were done analyt-ically with the help of the Mathematica package, includingall of the electron-electron interaction integrals. The onlysource of errors are numerical truncation and the size of thebasis, which was tested for convergence.In Figure 2 we show the quadrants of densities correspond-ing to wave-functions that are even for reflections in the y direction and odd in the x direction for γ = 2 and α = 1 .Figure 2(a) shows the upper left quadrant of the density of theinteracting ground state of two spin-less electrons obtainedwith full CI. Figure 2(b) shows the upper right quadrant of the FIG. 2: Ground state densities, in particles per unit area, for two in-teracting spin-less electrons in a square box. The complete densitycan be obtained by reflections (see also Fig 3). (a) Full CI groundstate interacting density. (b) Exact DFT solution obtained minimiz-ing Eq. (2). (c) Slater determinant with maximum projection with CIthe ground state [see Eq. (5)]. (d) Slater determinant with minimumamplitude on the nodes of the CI ground state [see Eq. (7)]. non-interacting density corresponding to the effective poten-tial obtained by adding recursively ∆ V K ρ ( r ) [Eq. (4)], whichis exactly the reflection of ρ ( r ′ ) Fig. 2(a) up to numerical pre-cision [ K ρ = 0 in Eq. (2)]. Because of the Hohenberg-Kohntheorem, V K ρ and the Kohn-Sham potential V KS ( ρ ) can onlydiffer by a constant and thus the wave-functions coming fromthis potential are the exact DFT wave-functions for our inter-action. The properties of the wave-functions will be discussedlater in the text. The densities in Figs. 2(d) and 2(d) corre-spond to the minimum of the cost functions given in Eqs (5)and (7) obtained as described below. B. Effective potential optimization
We now consider more difficult cost functions than den-sity differences, Eq. (2). For non-interacting v -representabledensities there are also functional correspondences betweenground state wave-functions, potentials and densities. Thisconcept has been exploited in the optimized effective poten-tial (OEP) for exact exchange. The exchange poten-tial can be calculated in OEP as: V x ( r ) = δE x δρ ( r ) (8) = occ X ν ZZ dr ′ dr ′′ (cid:20) δE x δφ ν ( r ′′ ) δφ ν ( r ′′ ) δV KS ( r ′ ) + c.c. (cid:21) δV KS ( r ′ ) δρ ( r ) . In Eq. (8), the functional derivative δE x /δφ ν ( r ) is evaluateddirectly from the explicit expression for the exchange energy E x in terms of φ ν ( r ) . Next δφ ν ( r ) /δV KS ( r ′ ) is evaluatedusing first-order perturbation theory from Eq. (1). Finally δV KS ( r ′ ) /δρ ( r ) is the inverse of the linear susceptibility op-erator. If there are fixed boundary conditions such as the num-ber of particles, the susceptibility operator is singular . Ex-cluding these null spaces it can be inverted numerically. Weuse earlier this susceptibility operator in Eq. (4). Equation(8) is by construction the gradient of the exchange energy inthe set of pure-state-non-interacting v -representable densities.While we are not going to attempt an exact exchange approachin this paper, the ability to calculate gradients allow us to min-imize cost functions as long as the cost function K can be ex-pressed in terms of non-interacting ground state wave wave-functions or eigenvalues. The potential that minimizes K canbe obtained by recursively applying the formula δV K ( r ) = ǫ occ X ν Z dr ′ δKδφ ν ( r ′ ) δφ ν ( r ′ ) δV KS ( r ) + c.c. (9)Equation (9) gives the direction we need to change the po-tential to minimize the cost function. The magnitude ofthe change is controlled by ǫ , which can be adjusted as onereaches the minimum.Replacing K by K Det in Eq. (9) and using Eq (5) and firstorder perturbation theory we find δV K Det ( r ) = ǫ h Ψ | Φ T i o X ν u X n h Ψ | c † n c ν | Φ T i φ n ( r ) φ nu ( r ) ε ν − ε n + c.c. (10)In equation (10) P on ( P un ) means sum over occupied (un-occupied) states, while c † n and c ν are creation and destruc-tion operators on the non-interacting ground state | Φ T i . Onecan understand also the state c † n c ν | Φ T i as the many bodywave-function Φ n,νT ( R ) resulting from replacing the occupiedstate φ ν by the φ n . This is equivalent to creating an electronhole pair excitation in a non-interacting ground state. In Eq.(10) a term in the potential is added every time an electronhole pair excitation has no zero projection to the interactingground state. Since the basis of products of wave-functions φ n ( r ) φ ν ( r ) is over-complete, there are linear combinationswith non-zero coefficients that add up to zero. A minimumis found when the gradient of the cost function with respectto variations of the effective potential is zero. If the absoluteminimum is found, the wave function can only be improvedfurther by a multi-determinant expansion, that is, outside theset of pure-state non-interacting densities. Since we choose abasis expansion for the single particle orbitals to be sine func-tions, the products φ n ( r ) φ nu ( r ) are linear combinations ofsine products. These sine products can be transformed analyt-ically to cosines. The change in the potential is thus written ina cosine basis which is complete. All coefficients must van-ish in the cosine basis when a minimum is found. This allowsus to verify that the gradient in the potential can be minimizedup to numerical precision. The integrated effective potential isthus naturally expressed as a linear combinations of cosines,which allows the analytical calculation of the coefficients ofthe effective potential matrix in a basis of sines, where thekinetic energy is diagonal. The Slater determinant | Φ T > iswritten in the same basis as the interacting ground state | Φ > .The projections involved in Eq. (10) are then reduced to ascalar product of the vectors of coefficients.In figure 2(c) we show the ground state density associatedto minimization of Eq. (5) for the same parameters as the in-teracting ground state density in Fig 2(a). We see that whileoptimizing the cost function (2) allows matching the interact-ing density exactly, optimizing the wave-function projectionrequires a significant change in the resulting density.Similarly, replacing K by K S in Eq. (9) and using Eq (7)we get δV K S ( r ) = ǫ o X ν u X n Z S dS Φ n,νT ( R )Φ T ( R ) φ n ( r ) φ nu ( r ) ε ν − ε n + c.c. (11)Unlike Eqs.4 and 10, a complication appears when evaluatingthe integral over the nodal surface R S dS . This integral in-volves finding the points where the many-body wave-functionis zero. The problem is simplified because the derivativesof the many body wave-functions can be obtained analyti-cally. Consequently, starting from an arbitrary point R wecan find a zero recursively with the Newton-Raphson method, R n +1 = R n + Ψ( R ) ∇ Ψ( R ) / |∇ Ψ( R ) | . Next we makea random displacement ∆R in the hyper-plane perpendicu-lar to ∇ Ψ( R ) within a circle of radius 0.05 and find a nodeagain. We repeat this process times and select an elementof { R } S . With this parameters, the random position R n cantravel across the full size of the system so that the distributionis homogeneous. By repeating this process N = 500 times,excluding points at the boundaries which are zero by construc-tion, we generate an homogeneous distribution of points at thenodal surface { R } S . We approximate the integral in Eq. (7)as a sum on the values on the set { R } S . Note that while thetotal area of the surface would be in general involved as a fac-tor, the value of this area is not relevant since we are interestedin finding a minimum of the cost function and the position ofthe minimum of any function is not altered by a positive mul-tiplicative constant. The sum over random points introduces arelative error of order / √ . Replacing the integral with asummation creates also many local minima in the landscape ofEq. (11). Accordingly, we tested different initial conditions;the best results are obtained starting from V K S = 0 .The density resulting from minimization of Eq. (7) is plot-ted in Fig 2(d). We see here again a significant change ascompared with the fully interacting CI ground state [see Fig2(a)] and the the exact DFT non-interacting solution Fig 2(b).Figure 2 is a clear example that corroborates our claimin Section II A that enforcing different properties on thenon-interacting wave-function implies a density-density cor-respondence different than the identity between the interact-ing and non interacting systems. Similar results are observedas function of the strength and shape of the interaction [con-trolled by γ and α ]A comparison between Eqs. (10) and (11) clearlyshows that the relative values of the coefficient multiplying φ n ( r ) φ nu ( r ) depends fundamentally on the cost function.Therefore, even starting from the same effective potentialand Φ T ( R ) the coefficient affecting each individual product φ n ( r ) φ nu ( r ) depends on the functional form of the cost func-tion. This change in the potential remains present when thepotential is written in the complete cosine basis. Thus, the ef-fective potential must change in accord with the property ofthe interacting ground state that one aims to enforce in thenon-interacting ground state with a cost function.Figure 3 shows the effective potentials used for the calcula-tions shown in Fig 2. We show in Fig 3(a) a constant, since inthe interacting problem solved with full CI no effective exter-nal potential was added. Figures 3(b) [minimum of Eq. (2)],3(c) [minimum of Eq. (5)] and 3(d) [minimum of Eq. (7)]show a clear change in the effective potential depending onthe cost functions. As argued earlier Fig. 3(b) shows the exactKohn-Sham DFT potential for this interaction which impliesthat a different density functional must be used to obtain non-interacting wave-functions preserving properties other thanthe density.Note that Eq. (7) could be zero only for a pure-state-non-interacting v -representable nodal surface. However, if thenodes are not, replacing in K S could result in a potential thatsimply prevents the non-interacting wave-function to reach re-gions of space where the nodes are more troublesome. Thepotential shown in Fig. 3(d) presents a maximum in regionswhere instead Figs. 3(b) and 3(c) develop a minimum. Theseare the regions where the electrons in the many-body wave-functions tend to localize because of correlation effects. Themaximum in Fig 3(d) suggest the possibility of non- v repre-sentability by a non-interacting wave-function in this model. C. Wave-function internal structure
In order to quantitatively test the quality of the nodes of thewave-functions found by minimization of Eqs. (2), (5) and (7)
FIG. 3: Optimized effective potentials corresponding to the densitiesin Fig. 2. The complete potentials can be obtained by reflectionon the black lines. Gray level values are given on the right. Theoptimal effective potentials are strongly dependent on the propertywe target to retain in the wave-function. and to test the convergence of the nodes of the full CI calcula-tions, we take advantage of the homogeneous distribution ofpoints { R } S at the nodal surface S ( R ) described earlier.For each point R in { R } S we can find the distance ℓ i tothe node of another wave-function Φ T ( R ) in the direction of ∇ Ψ( R ) . Thus ∆ V = 1 N X i ℓ i (12)is an approximated measure of the fraction of the Hilbertspace volume between the nodal surface of Φ T ( R ) and Ψ( R ) and δρ = 13 N X i ℓ i | Φ T ( R i ) | (13)measures the probability density inside ∆ V .We can use Eqs. (12) and (13) to test the convergence ofthe CI ground state nodes as a function of the size of the basisset. While the ground state energy requires basis func-tions, the nodes are more difficult to converge requiring fourtimes as many. The size of the basis required to converge thenodes was determined for γ = 2 plotting δρ between the CIground state with 300 wave-functions and the CI ground stateobtained using a reduced basis.The quantities in Eqs. (12) and (13) can be used also tocharacterize the nodes of different wave-functions as com-pared with the exact node. In figure 4 we show the volume en-closed between the nodes ∆ V of different optimized Φ T ( R ) and the interacting ground state Ψ( R ) as a function of thestrength of the interaction potential γ . Note that the Kohn-Sham DFT solution gives a significantly larger volume thanother optimized wave-functions. The difference increases asthe interaction strength increases. The wave-function that re-sults from minimizing Eq. (5) which targets wave-functionprojection fares very well over the range explored. In turn,minimization of K S [see Eq. (7)] results in nodes thatare only sometimes marginally better. Surprisingly, the non-interacting solution, that is the non-interacting ground statein the absence of any effective potential, is remarkably good.Similar results are found by altering the shape of the potentialwith α .In figure 5 we plot the values δρ for different optimizedwave-functions as a function of γ . We see again that the exactKohn-Sham DFT solution is not the best. The quantity δρ isa measure on how much the error in the nodes would affectthe probability density and thus it can be understood as theas a measure of the nodal error in the ground state energy.Again in this case the non-interacting ground state withoutany effective potential is the best approximation. IV. DISCUSSION
Although the numerical investigation of the differentdensity-density functionals described above required numer-ical representation of the many body ground state wave-function, the conclusions that we draw have general value.
FIG. 4: (Color online) Fraction of the Hilbert space ∆ V between thefull CI node and the nodes of different optimized wave-functions.Triangles correspond to the exact DFT wave-function [Eq. (2)],squares to maximum projection [Eq. (5)], rhombi to the minimumamplitude at the nodes [Eq. (7)] and circles to the non-interactingground state. The inset shows the method used to estimate ∆ V FIG. 5: (Color online) Probability density inside the volume betweenthe nodes of the full CI wave-function and optimized wave-functions.Same conventions and symbols as in Fig. 2
The model we explore is simplified but has the advantagesthat the results can be converged and are free of significantapproximations. The simplified interaction used in the modelretains essential features of the Coulomb interaction.We have shown (Fig. 2 and 3) that the effective densi-ties and potentials are explicit functions of a cost function.Potentials and densities very different to the exact DFT so-lutions are obtained if we enforce properties beyond ρ ( r ) inthe cost function. The exact DFT wave-function matches ρ ( r ) with complete disregard to other elements of the many bodywave-function structure. Since the Hohenberg-Kohn theorem is valid, optimizing other properties of the non-interactingground state in general requires changing the potential witha resulting impact on the density.We find that mean field methods, while giving an accuratedescription of the density can mislead us in other aspects ofthe wave-function structure such as the nodal surface. In thispaper we argue that, among the pure-state-non-interacting v -representable densities, there is at least one that more accu-rately describes the interacting ground state nodes. We canoptimize the wave-function associated with this density witha cost function. The cost function form depends on the prop-erty we target to retain and also the optimal density we find.For the nodes, the optimal cost function is clearly the DMCenergy. A fixed cost function establishes a density-densitycorrespondence which can be described as an operator U thattransforms interacting densities into non-interacting ones.While finding the functional form of U DMC is a task be-yond the scope of this paper, we argue that we can expect thisoperator to be highly non-local and very different from theidentity, in particular for strong electron-electron correlations.We find that we can improve the nodes with some simple costfunctions, but the best nodes we found were obtained solv-ing the non-interacting problem without the addition of anyeffective potential. We cannot exclude the possibility that thisresult might well be an accident of the model. Our result, how-ever, shows that the popular expectation that the DFT solutionis a good starting point for nodes is not valid in general.Optimal wave-functions can be found by altering anexternal potential. This idea is not new. In prac-tice wave-functions are optimized with the trial wave-function only minimizing the ground state energy K V MC = h Φ T | e − J He − J | Φ T i / ( h Φ T | e − J | Φ T i ) , or the variance ofthe ground state energy. Replacing K V MC into Eq. (9)leads to a procedure similar to the optimization of Filippi andFahy providing additional support to that method. In thecase of Refs. the nodes are selected by adjusting the mixof density functionals that gives the lowest DMC energy fora small system. The same mix is then used in a larger sys-tem. This procedure is in fact equivalent to optimizing theshape of the effective potential with the restriction of remain-ing a linear combination of two of more exchange-correlationpotentials. We find that the change in the effective potentialrequired to optimize the nodes could be of the order of the Hartree potential, since the wave-function with the best nodesis the non-interacting ground state, without any effective po-tential, for all the range of interaction strengths and shapesexplored. Our results suggest that counter intuitive directionsfor potential optimizations should be explored to improve thenodes.Potential optimization has also been applied for the predic- tion of electronic excitations. Since ρ ( r ) determines V ( r ) (butfrom a constant), the excitation spectra { E ν,n } is a functionof ρ ( r ) . This allows defining cost functions K ex to matchthe spectra of a non-interacting system. In order to minimize K ex one should do the derivatives δε ν /δV KS ( r ) as in Ref .When { E ν } is taken from experiment, the search of a potentialgiving a non-interacting density that minimizes K ex is equiv-alent to the empirical potential method. . Unfortunately, inthis case the electronic density can no longer be used to ob-tain the forces on the atoms. The existence of a single densityfunctional that can be used to obtain the excitation spectra ofany system is then a subject of debate.In summary, although the popular languages of electronicstructure theory all share the same quantum mechanical un-derpinnings, when applied by experts to physical systems weoften reach different conclusions. Many experts in QMC pre-fer HF wave-functions, while in contrast calculations donewithin the GW-BSE approach often rely on LDA derivedwave-functions and energies , while some hybrid densityfunctionals obtain single particle excitations in direct agree-ment with excitation spectra . We argue that as different the-ories need to retain different properties of the same groundstate wave-function to minimize errors, different functionalsshould also be used. 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