# Effective local potentials for density and density-matrix functional
approximations with non-negative screening density

Thomas C. Pitts  Department of Physics, Durham University, South Road, Durham,
DH1 3LE, United Kingdom  Sofia Bousiadi  Theoretical and Physical Chemistry
Institute, National Hellenic Research Foundation, Vass. Constantinou 48,
GR-11635 Athens, Greece  Faculty of Physics, National and Kapodistrian
University of Athens, Panepistimiopolis, Zografos, Athens 157 84, Greece
Nikitas I. Gidopoulos  Department of Physics, Durham University, South Road,
Durham, DH1 3LE, United Kingdom  Nektarios N. Lathiotakis<EMAIL_ADDRESS>Theoretical and Physical Chemistry Institute, National Hellenic Research
Foundation, Vass. Constantinou 48, GR-11635 Athens, Greece

###### Abstract

A way to improve the accuracy of the spectral properties in density functional
theory (DFT) is to impose constraints on the effective, Kohn-Sham (KS), local
potential [J. Chem. Phys. 136, 224109 (2012)]. As illustrated, a convenient
variational quantity in that approach is the “screening” or “electron
repulsion” density, $\rho_{\rm rep}$, corresponding to the local, KS Hartree,
exchange and correlation potential through Poisson’s equation. Two
constraints, applied to this minimization, largely remove self-interaction
errors from the effective potential: (i) $\rho_{\rm rep}$ integrates to $N-1$,
where $N$ is the number of electrons, and (ii) $\rho_{\rm rep}\geq 0$
everywhere. In this work, we introduce an effective “screening” amplitude,
$f$, as the variational quantity, with the screening density being $\rho_{\rm
rep}=f^{2}$. In this way, the positivity condition for $\rho_{\rm rep}$ is
automatically satisfied and the minimization problem becomes more efficient
and robust. We apply this technique to molecular calculations employing
several approximations in DFT and in reduced density matrix functional theory.
We find that the proposed development is an accurate, yet robust, variant of
the constrained effective potential method.

††preprint: AIP/123-QED

##  I INTRODUCTION

Replacing the Kohn-Sham (KS) potential Kohn and Sham (1965) for approximate
functionals in Density Functional Theory (DFT) with a variational, constrained
potential that fulfills exact properties is one of the proposed ways to
improve spectral properties Gidopoulos and Lathiotakis (2012) and more
specifically the interpretation of KS orbital energies as a meaningful
electronic spectrum of the system. One can minimize the total energy of any
Density Functional Approximation (DFA) under the constraint that the optimal
effective local potential exhibits the correct asymptotic behavior. It was
argued that, in this way, the effect of self-interactions (SI) on the local
potential is suppressed and the ionization energies obtained by the orbital
energies of the highest occupied molecular orbital (HOMO) are significantly
closer to the experiment than those obtained from the unconstrained,
approximate KS potential. Gidopoulos and Lathiotakis (2012); Pitts,
Gidopoulos, and Lathiotakis (2018); Gidopoulos and Lathiotakis (2015a); Callow
_et al._ (2020)

In single particle theories, like KS, SIs can appear when the exact exchange
energy functional is replaced with approximations that violate the complete
cancellation of the Hartree SI terms. Characteristic examples are local/semi-
local approximations like LDA Kohn and Sham (1965); Perdew and Zunger (1981);
Ceperley and Alder (1980); Vosko, Wilk, and Nusair (1980) or GGAs. Perdew  _et
al._ (1992); Becke (1988); Lee, Yang, and Parr (1988); Perdew, Burke, and
Ernzerhof (1996) The seminal work of Perdew and Zunger Perdew and Zunger
(1981) aimed at correcting this problem. Several other attempts were made to
address the same problem. Lundberg and Siegbahn (2005); Toher  _et al._
(2005); Goedecker and Umrigar (1997); Gidopoulos and Lathiotakis (2012);
Pitts, Gidopoulos, and Lathiotakis (2018); Van Leeuwen and Baerends (1994);
Legrand, Suraud, and Reinhard (2002); Tsuneda and Hirao (2014); Pederson,
Ruzsinszky, and Perdew (2014); Gidopoulos and Lathiotakis (2015b); Clark  _et
al._ (2017) All these attempts aim at correcting the total energy. Contrarily,
the main concept of Ref. Gidopoulos and Lathiotakis, 2012, is to correct the
effective potential instead through additional constraints. A typical
manifestation of SIs is the wrong asymptotic behavior of the KS potential. On
the other hand, the correct behavior of the potential at large distances plays
a crucial role in the determination of several molecular properties. The
Coulomb potential, $V_{\rm H}$, behaves like $N/r$ at large distances. In
order to obtain the correct overall behavior, the exchange and correlation
(xc) potential, $V_{\rm xc}$ should behave as $-1/r$, which does not hold for
many approximations. Indeed, in most semilocal DFAs, the xc potential decays
exponentially fast. In Ref. [Gidopoulos and Lathiotakis, 2012], the aim was to
replace the Hartree-exchange and correlation (HXC) part of the KS potential
with the energetically optimal effective potential $V_{\rm rep}$ (that
minimizes the total energy functional) but which also satisfies two subsidiary
conditions. These conditions or constraints are expressed in terms of the
screening (or electron repulsion) density, Görling (1999) $\rho_{\rm rep}$;
this density corresponds to $V_{\rm rep}$ through Poisson’s equation. The
first is that $\rho_{\rm rep}$ integrates to $N-1$ electrons and the second is
that it is everywhere non-negative ($\rho_{\rm rep}({\bf r})\geq 0$). With
this approach, the effect of SIs on the effective potential $V_{\rm rep}$ was
largely corrected and molecular properties were improved. Gidopoulos and
Lathiotakis (2012); Pitts, Gidopoulos, and Lathiotakis (2018); Gidopoulos and
Lathiotakis (2015a); Callow  _et al._ (2020)

A recent publication Callow  _et al._ (2020) explored the relaxation of the
positivity constraint, $\rho_{\rm rep}({\bf r})\geq 0$; this leads to a
mathematical optimization problem that is strictly not well-posed.
Nevertheless, in practice, the mathematical problems became evident only with
large auxiliary basis sets, or for systems with few electrons. In addition, a
density inversion method, yielding the corresponding KS potential that
satisfies the correct asymptotic behavior was introduced recently.Callow,
Lathiotakis, and Gidopoulos (2020); Bousiadi, Gidopoulos, and Lathiotakis
(2022) Finally, a hybrid scheme was proposed Pitts, Lathiotakis, and
Gidopoulos (2021), where the local part of the KS potential is optimized under
the aforementioned subsidiary conditions, yielding single-particle energies in
excellent agreement with experimental ionization energies, even for the core
electrons. The hybrid potential scheme in Ref. Callow  _et al._ , 2020 is
applicable to any semilocal DFA.

The idea of optimizing the energy functional with respect to a constrained
local potential has also been applied to approximations within the framework
of the reduced, density-matrix functional theory (RDMFT), Lathiotakis  _et
al._ (2014a, b) leading to an approach that was called local RDMFT (LRDMFT).
In RDMFT, functionals are usually explicitly expressed in terms of the natural
orbitals and their corresponding occupation numbers, i.e. the eigenfunctions
and eigenvalues of the one-body, reduced, density-matrix (1RDM). Although the
foundations of RDMFT were laid a long time ago, Gilbert (1975); Zumbach and
Maschke (1985); Levy (1979) it has received significant attention in the last
couple of decades. Goedecker and Umrigar (1997, 1998); Buijse and Baerends
(2002); Gritsenko, Pernal, and Baerends (2005); Lathiotakis  _et al._ (2009);
Marques and Lathiotakis (2008); Piris and Ugalde (2009); Pernal, Gritsenko,
and Baerends (2007); Baldsiefen, Cangi, and Gross (2015); Liebert  _et al._
(2022) One of its main advantages is that the kinetic energy can be written as
an explicit functional of the 1-RDM, so the exchange-correlation term is not
“contaminated" by kinetic energy contributions. Fractional occupation numbers
are introduced through Coleman’s $N$-representability conditions. Coleman
(1972) RDMFT provides the opportunity to express energy terms exactly as
functionals of 1-RDM except for the electron-electron interaction energy. For
this term, many approximate functionals have been proposed, Goedecker and
Umrigar (1998); Buijse and Baerends (2002); Csányi, Goedecker, and Arias
(2002); Lathiotakis  _et al._ (2009); Kollmar (2004); Piris (2006); Sharma
_et al._ (2008); Marques and Lathiotakis (2008); Müller (1984); Piris  _et
al._ (2011); Gritsenko, Pernal, and Baerends (2005); Piris (2021) offering a
good description of electronic correlations. In these approaches, energy
minimization is performed iteratively in two steps: (1) Minimization with
respect to natural occupation numbers, and (2) minimization with respect to
natural orbitals. The first of these steps is relatively inexpensive
computationally, but the second is very demanding. Although significant
progress has been achieved in improving its efficiency, Piris and Ugalde
(2009) this minimization still remains a considerable bottleneck.

In LRDMFT,Lathiotakis  _et al._ (2014a, b) functionals of the 1RDM are
employed, but the minimization with respect to the orbitals is replaced with a
constrained minimization that restricts the domain of orbital variation to
those that are obtained from a single particle Schrödinger equation with local
potential. Thus, it becomes a direct application of the constrained potential
optimization of Refs. Gidopoulos and Lathiotakis, 2012; Pitts, Gidopoulos, and
Lathiotakis, 2018; Gidopoulos and Lathiotakis, 2015a; Callow  _et al._ , 2020
to 1RDM functionals. One important attribute of this approach is, of course,
the relative efficiency in orbital optimization compared to full RDMFT
minimization. A second one is that single-particle properties are offered
through the spectrum of the single-particle Hamiltonian. It was found that
energy eigenvalues of the highest occupied orbitals (HOMO) obtained in that
way reproduce accurately the ionization potentials (IPs).Lathiotakis  _et al._
(2014a, b) LRDMFT should not be considered as an approximation of RDMFT
because the true natural orbitals cannot be obtained (even approximately) by a
local potential and the optimal orbitals from LRDMFT are KS-like orbitals
Theophilou  _et al._ (2015). It can be considered as a DFA with fractional
occupation numbers introduced through the minimization of the energy. LRDMFT
incorporates good aspects of both RDMFT and DFT, combining the low
computational cost of DFT and the description of the static correlation of
RDMFT through the introduction of fractional occupation numbers.

A hurdle in the application of the constrained minimization method is the
enforcement of the positivity condition for the effective density, $\rho_{\rm
rep}({\bf r})$. The fact that this condition has to be satisfied at each point
in space, e.g. by a penalty term, has been proven computationally inefficient.
In this work, we present a variant of the constrained minimization method with
the positivity condition enforced by expressing the screening density as the
square of an orbital-like “screening-density amplitude”, $\rho_{\rm rep}({\bf
r})=f({\bf r})^{2}$, and using this amplitude, $f({\bf r})$, as the
minimization variable. In that way, the positivity condition is automatically
satisfied, however, there is a price to pay, since the variation with respect
to $f({\bf r})$ does not lead to linear equations. Hence, in this work, we
determine $f$ through a minimization procedure instead of solving any
variational equations. In this first application, for simplicity, we assume
that $f$ is expanded on the same orbital basis as the molecular orbitals. We
apply the new method to the LDA functional in DFT as well as several
approximations in LRDMFT.

This paper is organized as follows: In Section II, we present the methodology
to determine the screening-density amplitude. Section III is devoted to
applications. In Section III.1, we include the results of the application of
the present method in optimizing total energies for a few local, semi-local,
and hybrid DFAs and focus on the agreement of the obtained IPs with the
experiment. Finally, in Section III.2, we show the results of applying the
present method to RDMFT functionals, i.e. as a variant of the LRDMFT
methodology.

##  II THEORY

In KS theoryKohn and Sham (1965), the local potential $V_{\rm KS}$ is the sum
$V_{\rm KS}=V_{\rm ext}+V_{\rm Hxc}$ of the external potential, $V_{\rm ext}$,
and the Hartree, exchange, and correlation term, $V_{\rm Hxc}$. It is obtained
from the functional derivative of the total electronic energy with respect to
the electron density. This potential is inserted in the single particle Kohn-
Sham equations,

$\Big{[}-\frac{\nabla^{2}}{2}+V_{\rm ext}({\bf r})+V_{\rm Hxc}({\bf
r})\Big{]}\phi_{i}({\bf r})=\epsilon_{i}\,\phi_{i}({\bf r})\,,$ (1)

yielding the KS orbitals, hence the density, in a self-consistent cycle.

Following Refs. Gidopoulos and Lathiotakis, 2012; Pitts, Gidopoulos, and
Lathiotakis, 2018, $V_{\rm Hxc}$, in Eq. (1), is replaced by a variational
effective repulsive potential $V_{\rm rep}$. The total energy, $E$, depending
explicitly on the orbitals, $\\{\phi_{i}\\}$, can be considered a functional
of $V_{\rm rep}$ if we assume that the orbitals are obtained from the solution
of an equation like Eq. (1), but with the variational potential, $V_{\rm
rep}$, replacing $V_{\rm Hxc}$. For any DFA, this energy functional can be
minimized with respect to $V_{\rm rep}$, under additional constraints
(subsidiary conditions), e.g. in order to enforce exact properties for the
potential. Without any such constraint, it is easy to see that the optimal
$V_{\rm rep}$ will be equal to $V_{\rm Hxc}$. Since the considered additional
constraints concern the "screening" or "electron repulsion" density,
$\rho_{\rm rep}$, associated with $V_{\rm rep}$ through Poisson’s equation, it
is convenient to express $V_{\rm rep}$ in terms of $\rho_{\rm rep}$, as
Görling (1999); Gidopoulos and Lathiotakis (2012); Pitts, Gidopoulos, and
Lathiotakis (2018); Lathiotakis  _et al._ (2014a)

$V_{\rm rep}({\bf r})=\int d\bf{r}^{\prime}\,\frac{\rho_{\rm
rep}(\bf{r}^{\prime})}{|r-r^{\prime}|}\,.$ (2)

An exact property of the KS potential is that the asymptotic behavior of the
electron-electron repulsion part, $V_{\rm Hxc}$, is

$\lim_{r\rightarrow\infty}V_{\rm Hxc}({\bf r})=\frac{N-\alpha}{r},\mbox{with\
}\alpha=1\,,$ (3)

i.e. an electron at infinity “feels" the repulsion of the remaining $N-1$
electrons. One of the manifestations of SIs, which are present in most DFAs,
is that this property is not satisfied by the approximate $V_{\rm Hxc}$,
typically with $0\leq\alpha<1$. In Ref. Gidopoulos and Lathiotakis, 2012, it
was proposed that properties of the exact KS potential, like Eq. (3), can be
enforced in the minimization with respect to $V_{\rm rep}$, thus the optimal
$V_{\rm rep}$ differs from $V_{\rm Hxc}$, i.e. it is no longer the functional
derivative of the total energy with respect to the density. For such an
optimization, it is convenient to express $V_{\rm rep}$ in the formGörling
(1999); Gidopoulos and Lathiotakis (2012) of Eq. (2).

Assuming the form of Eq. (2) one can minimize the total DFA energy with
respect to $\rho_{\rm rep}$, instead of the effective potential. $\rho_{\rm
rep}$ can be expanded in an auxiliary basis set, in principle different than
the basis used for the expansion of the KS orbitals. In terms of $\rho_{\rm
rep}$, the condition (3), concerns the total repulsive charge, $Q_{\rm rep}$,

$Q_{\rm rep}=\int d{\bf r}\>\rho_{\rm rep}({\bf r})=N-1.$ (4)

In Refs. Gidopoulos and Lathiotakis, 2012; Pitts, Gidopoulos, and Lathiotakis,
2018, it was argued that this condition alone does not lead to a
mathematically well-posed optimisation problem and is not sufficient to obtain
well-behaved solutions in general. The reason is that with this condition
alone, the constrained minimization of the total energy of a DFA that is
contaminated with SIs (i.e., whose unconstrained screening density corresponds
to $Q_{\rm rep}^{\rm DFA}=N$) would not lead to well-converged solution. It
would be energetically favorable for the constrained minimization to converge
to a screening density that locally, near the system, matches the
unconstrained screening density, integrating locally to screening charge $N$;
the constraint $Q_{\rm rep}=N-1$ would then be satisfied by distributing
negative screening charge equal to $-1$ away from the system, depending on the
size of the auxiliary basis. In Ref. Callow  _et al._ , 2020, an investigation
of the constrained minimization is shown, imposing the constraint just on the
norm of the screening charge (4). A proposed successful solution in Refs.
Gidopoulos and Lathiotakis, 2012; Pitts, Gidopoulos, and Lathiotakis, 2018;
Gidopoulos and Lathiotakis, 2015a is to supplement this condition with the
positivity of $\rho_{\rm rep}$ at all points in space:

$\rho_{\rm rep}({\bf r})\geq 0,\quad\forall{\bf r}.$ (5)

This condition is sufficient to yield well-behaved solutions, although it is
not necessarily satisfied by the exact KS potential. It concerns the whole Hxc
screening density, which is mostly positive because the Hartree repulsion
dominates. So, according to this condition, the xc part of the screening
density, can not exceed in absolute size the electronic density at any point
in space. Unfortunately, this supplementary condition results in an additional
computational cost since it has to be verified and enforced for all points in
space.

A choice that renders the positivity condition redundant, is to introduce a
new variational parameter, namely an effective screening-density amplitude,
$f$, such that

$\rho_{\rm rep}({\bf r})=f({\bf r})^{2}.$ (6)

Table 1:  Ionization potentials, in eV, using cc-pVTZ basis sets, for various molecules obtained with different DFAs (a) with the standard KS scheme and (b) using the optimization of the screening-density amplitude proposed here, compared with vertical experimental (Exp) ionization potentials. nis  Hartree-Fock (HF) Koopmans’ results are also shown. For each DFA, the average absolute percentage error, $\Delta=({100}/{N})\sum_{i}|{(\chi_{i}-\chi_{i}^{\rm(Exp)})}/{\chi_{i}^{\rm(Exp)}}|$, is also included. System | LDA | PBE | B3LYP | HF | Exp
---|---|---|---|---|---
| (a) | (b) | (a) | (b) | (a) | (b) |  |
He | 15.30 | 23.13 | 15.63 | 23.65 | 17.91 | 23.77 | 24.97 | 24.59
Ne | 13.01 | 18.96 | 12.79 | 19.15 | 15.18 | 19.66 | 23.01 | 21.60
Be | 5.62 | 8.54 | 5.65 | 8.74 | 6.36 | 8.62 | 8.42 | 9.32
H2 | 10.21 | 15.50 | 10.38 | 15.83 | 11.84 | 15.79 | 16.21 | 15.43
H2O | 6.85 | 11.28 | 6.72 | 10.93 | 8.39 | 11.26 | 13.73 | 12.78
NH3 | 5.9 | 9.82 | 5.84 | 9.52 | 7.23 | 9.82 | 11.64 | 10.80
CH4 | 9.36 | 12.90 | 9.38 | 12.60 | 10.72 | 12.83 | 14.82 | 13.60
O3 | 8.03 | 10.86 | 7.83 | 11.00 | 9.60 | 11.46 | 13.18 | 12.73
C2H2 | 7.26 | 10.53 | 7.10 | 10.24 | 8.09 | 10.51 | 11.07 | 11.49
C2H4 | 6.86 | 9.85 | 6.69 | 9.60 | 7.58 | 9.61 | 10.24 | 10.68
C2H5 | 5.75 | 8.61 | 5.69 | 8.31 | 6.92 | 8.67 | 10.73 | 9.85
SiH4 | 8.47 | 11.28 | 8.51 | 11.39 | 9.68 | 11.35 | 13.23 | 11.00
H2O2 | 6.10 | 9.78 | 5.97 | 9.31 | 7.68 | 9.82 | 13.08 | 11.70
O2 | 5.95 | 9.63 | 5.70 | 9.63 | 7.04 | 9.95 | 12.78 | 12.30
CO2 | 9.19 | 12.37 | 8.97 | 12.25 | 10.35 | 12.41 | 14.74 | 13.78
CO | 9.09 | 12.56 | 9.03 | 12.6 | 10.56 | 12.96 | 15.14 | 14.01
Li2 | 3.23 | 5.14 | 3.22 | 5.16 | 3.65 | 5.10 | 4.89 | 5.11
CH3OH | 6.07 | 9.42 | 5.99 | 9.13 | 7.50 | 9.31 | 12.22 | 10.96
C2H6 | 8.01 | 11.01 | 8.07 | 10.80 | 9.36 | 11.3 | 13.22 | 11.99
CH3NH2 | 5.24 | 8.43 | 5.20 | 7.99 | 6.52 | 8.27 | 10.66 | 9.65
C2H5OH | 6.03 | 8.85 | 5.95 | 8.39 | 7.43 | 8.84 | 11.99 | 10.00
$\Delta$ | 38.87 | 9.74 | 39.4 | 11.21 | 28.21 | 9.14 | 8.31 |
Table 2: Same as in Table 1, but for cc-pVQZ basis sets. System | LDA | PBE | B3LYP | HF | Exp
---|---|---|---|---|---
| (a) | (b) | (a) | (b) | (a) | (b) |  |
He | 15.32 | 23.13 | 15.66 | 23.59 | 17.93 | 23.73 | 24.98 | 24.59
Ne | 13.23 | 19.06 | 13.03 | 18.77 | 15.38 | 19.34 | 23.10 | 21.60
Be | 5.62 | 8.52 | 5.65 | 8.77 | 6.36 | 8.62 | 8.42 | 9.32
H2 | 10.22 | 15.48 | 10.39 | 15.81 | 11.84 | 15.80 | 16.21 | 15.43
H2O | 7.07 | 11.33 | 6.94 | 11.11 | 8.57 | 11.49 | 13.82 | 12.78
NH3 | 6.06 | 9.88 | 5.99 | 9.66 | 7.36 | 9.96 | 11.69 | 10.80
CH4 | 9.37 | 12.98 | 9.40 | 12.67 | 10.73 | 13.01 | 14.83 | 13.60
O3 | 8.12 | 11.41 | 7.93 | 11.36 | 9.68 | 11.73 | 13.22 | 12.73
C2H2 | 7.29 | 10.54 | 7.14 | 10.13 | 8.12 | 10.52 | 11.10 | 11.49
CO2 | 9.24 | 12.86 | 9.03 | 12.23 | 10.41 | 12.84 | 14.79 | 13.78
Li2 | 3.23 | 5.13 | 3.22 | 5.17 | 3.65 | 5.12 | 4.90 | 5.11
N2 | 10.34 | 14.10 | 10.22 | 13.77 | 11.94 | 14.16 | 16.33 | 15.58
$\Delta$ | 37.15 | 7.19 | 37.54 | 8.55 | 27.21 | 6.54 | 6.02 |

In order to get the derivative of the total energy, $E$, with respect to $f$,
we apply a chain rule and obtain

$\frac{\delta E[f]}{\delta f({\bf x})}=2\,f({\bf x})\left[\tilde{b}({\bf
x})-\int d{\bf y}\;\tilde{\chi}({\bf x},{\bf y})\,f^{2}({\bf y})\right]\,,$
(7)

where

$\tilde{b}({\bf x})=\int d{\bf r}\frac{1}{|{\bf r}-{\bf x}|}\int d{\bf
r^{\prime}}\;\chi({\bf r},{\bf r^{\prime}})\;V_{\rm Hxc}({\bf r^{\prime}})\,,$
(8)

and

$\tilde{\chi}({\bf x},{\bf y})=\iint d{\bf r}\,d{\bf
r^{\prime}}\frac{\chi({\bf r},{\bf r^{\prime}})}{|{\bf r}-{\bf x}||{\bf
r}^{\prime}-{\bf y}|}\,.$ (9)

The quantity $\chi({\bf r},{\bf r^{\prime}})$ is the KS density-density
response function given by

$\chi({\bf r},{\bf r}^{\prime})=2\sum_{i}\sum_{\alpha}\frac{\phi_{i}({\bf
r})\phi_{i}({\bf r})\phi_{\alpha}({\bf r}^{\prime})\phi_{\alpha}({\bf
r}^{\prime})}{\epsilon_{i}-\epsilon_{\alpha}}\,.$ (10)

Due to its similarity with an orbital, we can expand the amplitude $f$ in the
same local basis set employed to expand the molecular orbitals, $f({\bf
r})=\sum_{k}f_{k}\,\xi_{k}({\bf r})$ and we obtain:

$\frac{\partial E}{\partial
f_{n}}=2\sum_{k}B_{nk}\,f_{k}-2\sum_{kml}f_{k}\,A_{nkml}\,f_{m}\,f_{l}\,,$
(11)

where,

$A_{nkml}=\iint d{\bf x}\,d{\bf y}\,\xi_{n}({\bf x})\,\xi_{k}({\bf
x})\,\tilde{\chi}({\bf x},{\bf y})\,\xi_{m}({\bf y})\,\xi_{l}({\bf y})\,,$
(12) $B_{nk}=\int d{\bf x}\;\xi_{n}({\bf x})\,\tilde{b}({\bf x})\,\xi_{k}({\bf
x})\,.$ (13)

Eq. (11) no longer leads to a linear equation unlike the equation derived in
our previous work Gidopoulos and Lathiotakis (2012). Here, we solve directly
the minimization problem of $E$ using standard minimization techniquesgsl
which require the gradient with respect to $f_{n}$ of Eq. (11).

The optimization with respect to $f$ has to be carried out in the domain of
amplitudes normalized to $N-1$ according to the constraint of Eq. (4). This
requirement can be incorporated by Lagrange multipliers technique. However, in
the present work, we chose an alternative equivalent way by introducing an
auxiliary amplitude, $f^{\prime}({\bf r})$, as our variational quantity that
is not normalized, i.e.,

$f_{n}=s\;f_{n}^{\prime}\,,\quad
s=\frac{\sqrt{N-1}}{\left\|f^{\prime}\right\|}\,,$ (14)

where $\left\|f^{\prime}\right\|^{2}=\int d{\bf x}\,\big{(}f^{\prime}({\bf
x})\big{)}^{2}=\sum_{kl}f_{k}^{\prime}\,S_{kl}\,f_{l}^{\prime}$ and
$S_{kl}=\int d{\bf r}\;\xi_{k}({\bf r})\,\xi_{l}({\bf r})$. Then the gradient
with respect to $f^{\prime}$ is

$\frac{\partial E}{\partial f_{n}^{\prime}}=s\frac{\partial E}{\partial
f_{n}}-\frac{s}{\left\|f^{\prime}\right\|^{2}}\sum_{m}\frac{\partial
E}{\partial
f_{m}}f_{m}^{\prime}\sum_{\lambda}S_{n\lambda}f_{\lambda}^{\prime}\,.$ (15)

Through Eq. (14), $E$ become as functional of $f^{\prime}$,
$E=E[f]=E[s\,f^{\prime}]$, and can be minimized through an unconstrained
variation with respect to $f^{\prime}$, using the gradient with respect to
$f^{\prime}$ given in Eq. (15).

In principle, one can expand the auxiliary amplitude $f^{\prime}$ (or
equivalently $f$) in any set of basis functions, different from that of the
orbitals. In this first implementation, however, we chose to expand it in the
same basis set as the orbitals. This is a restriction compared to our previous
work, where $\rho_{\rm rep}$ was expanded in a different (auxiliary) basis
set. However, due to the similarity of the screening-density amplitude to an
orbital, as we will demonstrate, this choice does not affect the accuracy of
our results. We implemented the present method in the HIPPO computer code. hip

##  III Applications

###  III.1 Application to DFAs

As a first application, we use the effective screening density amplitude
method for the constrained optimization of DFAs for molecular systems. The
effect of applying this method to common DFAs is to improve the values of the
ionization potentials as calculated from the highest occupied molecular
orbital (HOMO) energies in the same manner as in Refs. Gidopoulos and
Lathiotakis, 2012; Pitts, Gidopoulos, and Lathiotakis, 2018.

Our results for several molecules and different DFAs are presented in Tables 1
and 2 using the cc-pVTZ and cc-pVQZ basis sets, respectively, compared with
results from plain DFAs and the experiment values of (vertical) IPs obtained
from the NIST WebBook nis . As we see, the corrected DFAs have an error of
roughly half that of the uncorrected functionals. These corrected IPs have a
similar error to those from Hartree-Fock without needing to perform a non-
local calculation. As mentioned the screening-density amplitude is expanded on
the same basis as the orbitals and that introduces an additional basis set
dependence of the IPs. Reasonably, the results of the larger, cc-pVQZ basis
sets, are superior.

The present results are in line with those of our previous work Gidopoulos and
Lathiotakis (2012); Pitts, Gidopoulos, and Lathiotakis (2018) where the
$\rho_{\rm rep}$ was the variational parameter and the positivity had to be
enforced additionally. This is shown in Fig. 1 where we compare the average
absolute percentage errors for the present method with those of Ref. Pitts,
Gidopoulos, and Lathiotakis, 2018. However, we should note that the results of
Ref. Pitts, Gidopoulos, and Lathiotakis, 2018 are obtained with cc-pVDZ
orbital basis and an auxiliary uncontracted cc-pVDZ basis for the screening
density. The present method has enhanced convergence compared to that of Ref
Gidopoulos and Lathiotakis, 2012 in the objective functional minimization and
provides a more robust framework.

Figure 1:  Average absolute percentage error, $\Delta$, defined in the caption
of Table 1, in the IPs for various DFAs calculated with (a) the standard KS
scheme, (b) the present method, and (c) results from Refs. Gidopoulos and
Lathiotakis, 2012; Pitts, Gidopoulos, and Lathiotakis, 2018.

###  III.2 Application to Local-RDMFT

As a second application of the present constrained minimization we considered
LRDMFT, i.e. we minimized functionals of the 1RDM under the subsidiary
condition that the orbitals are eigenfunctions of a single particle
Hamiltonian with local potential. The repulsion part of this potential is
assumed to be of the form of Eq. (2) where $\rho_{\rm rep}$ is given by the
quadratic form of Eq. (6). The occupation numbers are minimized directly as in
standard RDMFT calculations.

For the calculations, we employed five popular approximate RDMFT functionals
namely Mueller, Müller (1984) BBC3, Gritsenko, Pernal, and Baerends (2005)
Power, Sharma  _et al._ (2008) ML, Marques and Lathiotakis (2008) and PNOF5.
Piris  _et al._ (2011) As far as the basis sets are concerned the cc-pVTZ and
cc-pVQZ were employed, for expanding both the orbitals and the screening-
density amplitude, for selected systems. However, due to limitations of the
HIPPO code, g-functions were removed from the cc-pVQZ basis set, a
modification that is not expected to affect our results.

We compare energy eigenvalues of the HOMOs with the values of (vertical)
experimental IPs (from NIST WebBook nis ), in Tables 3 and 4 for the two
different basis sets, respectively. For comparison, we also calculated IPs
from standard Hartree-Fock through Koopman’s theorem. As we see, there is a
general trend of improvement of the results with the size of the basis set,
i.e. the results of the cc-PVQZ basis set are better than those of cc-pVTZ.
Nevertheless, our results are of similar quality to those of Refs. Lathiotakis
_et al._ , 2014a, b, as seen in Fig. 2 and Table 5. As we see, BBC3, Power and
PNOF5 functionals are yielding very good results compared with the experiment.

Table 3:  Calculated ionization potentials, in eV, for cc-pVTZ basis sets, for various molecules, using the present variant of LRDMFT of optimizing the screening-density amplitude and different functionals of the 1RDM, compared with vertical experimental ionization potentials nis . Hartree-Fock Koopmans’ results are also shown. The average absolute percentage errors, defined in the caption of Table 1, are also included. System | Müller | BBC3 | POW | ML | PNOF5 | HF | Exp.
---|---|---|---|---|---|---|---
He | 24.67 | 24.62 | 24.78 | 25.05 | 24.35 | 24.97 | 24.59
Ne | 21.76 | 22.34 | 22.22 | 22.76 | 22.58 | 23.01 | 21.6
Be | 9.41 | 8.93 | 8.80 | 8.74 | 8.72 | 8.42 | 9.32
H2 | 16.27 | 16.18 | 16.16 | 16.23 | 16.14 | 16.21 | 15.43
H2O | 13.19 | 12.9 | 12.89 | 13.39 | 13.39 | 13.73 | 12.78
NH3 | 11.62 | 11.12 | 11.13 | 11.36 | 11.31 | 11.64 | 10.80
CH4 | 14.11 | 14.10 | 13.91 | 14.23 | 14.12 | 14.82 | 13.60
O3 | 14.02 | 12.87 | 14.10 | 13.19 | 12.42 | 13.18 | 12.73
C2H2 | 12.20 | 11.26 | 11.47 | 11.62 | 11.50 | 11.07 | 11.49
C2H4 | 10.98 | 10.59 | 10.73 | 10.92 | 10.74 | 10.24 | 10.68
C2H5N | 11.33 | 10.40 | 10.53 | 10.37 | 10.37 | 10.73 | 9.85
SiH4 | 11.33 | 11.72 | 10.39 | 10.38 | 11.07 | 13.24 | 11.00
H2O2 | 11.31 | 11.70 | 11.18 | 12.25 | 11.53 | 13.08 | 11.70
O2 | 13.04 | 11.9 | 12.98 | 11.90 | 12.24 | 12.78 | 12.30
CO2 | 14.29 | 14.82 | 14.61 | 15.23 | 14.86 | 14.74 | 13.78
CO | 14.44 | 14.57 | 14.79 | 13.73 | 14.37 | 15.14 | 14.01
Li2 | 5.96 | 4.99 | 5.06 | 4.91 | 4.87 | 4.89 | 5.11
CH3OH | 11.00 | 10.93 | 11.00 | 11.20 | 11.07 | 12.22 | 10.96
C2H6 | 12.01 | 12.29 | 12.11 | 12.53 | 12.24 | 13.22 | 11.99
CH3NH2 | 10.24 | 9.69 | 9.88 | 10.00 | 9.81 | 10.66 | 9.65
C2H5OH | 10.35 | 9.78 | 10.80 | 10.61 | 10.00 | 11.97 | 10.00
$\Delta$ | 5.04 | 2.98 | 4.17 | 4.70 | 3.30 | 8.5 |
Table 4:  Same as in Table 3, for cc-pVQZ basis sets.  System | Müller | BBC3 | POW | PNOF5 | HF | Exp
---|---|---|---|---|---|---
He | 24.67 | 24.61 | 24.78 | 24.41 | 24.98 | 24.59
Ne | 21.59 | 22.3 | 22.24 | 22.72 | 23.10 | 21.60
Be | 9.24 | 8.80 | 8.59 | 8.49 | 8.42 | 9.32
H2 | 16.13 | 16.11 | 16.11 | 16.13 | 16.21 | 15.43
H2O | 12.90 | 13.15 | 12.60 | 13.09 | 13.82 | 12.78
NH3 | 11.60 | 11.15 | 10.97 | 11.60 | 11.69 | 10.80
CH4 | 13.68 | 13.92 | 13.83 | 14.14 | 14.83 | 13.60
O3 | 13.54 | 12.42 | 13.58 | 12.53 | 13.22 | 12.73
C2H2 | 11.94 | 10.72 | 10.73 | 10.43 | 11.10 | 10.68
CO2 | 14.15 | 14.47 | 14.13 | 13.90 | 14.79 | 13.78
Li2 | 5.98 | 5.00 | 5.04 | 4.88 | 4.90 | 5.11
N2 | 15.91 | 15.99 | 16.36 | 15.60 | 16.33 | 15.58
$\Delta$ | 4.56 | 2.87 | 3.06 | 3.55 | 6.10 |
Figure 2:  Average absolute percentage error, $\Delta$ defined in the caption of Table 1, in the IPs calculated with LRDMFT for various functionals with (a) the method of Ref. Lathiotakis  _et al._ , 2014a and (b) the present method. Table 5:  Ionization potentials, in eV, calculated with LRDMFT, for various 1-RDM functionals, and cc-pVTZ basis sets, using (a) the method of Ref. Lathiotakis  _et al._ , 2014a and (b) the present method. | Müller (a) | Müller (b) | BBC3 (a) | BBC3 (b) | Power (a) | Power (b) | ML (a) | ML (b) | Exp. |
---|---|---|---|---|---|---|---|---|---|---
He | 24.69 | 24.67 | 24.57 | 24.62 | 24.84 | 24.78 | 25.15 | 25.05 | 24.59 |
Be | 9.51 | 9.41 | 8.73 | 8.93 | 8.58 | 8.80 | 8.55 | 8.74 | 9.32 |
Ne | 22.9 | 21.76 | 20.92 | 22.34 | 21.65 | 22.22 | 21.32 | 22.76 | 21.6 |
H2 | 16.24 | 16.27 | 16.15 | 16.18 | 16.15 | 16.16 | 16.28 | 16.23 | 15.43 |
H2O | 12.59 | 13.19 | 12.03 | 12.9 | 12.1 | 12.89 | 12.64 | 13.39 | 12.78 |
NH3 | 11.03 | 11.62 | 10.65 | 11.12 | 10.74 | 11.13 | 10.95 | 11.36 | 10.8 |
CH4 | 13.55 | 14.11 | 13.72 | 14.1 | 13.43 | 13.91 | 13.84 | 14.23 | 13.6 |
C2H2 | 11.67 | 12.2 | 11.12 | 11.26 | 11.46 | 11.47 | 11.59 | 11.62 | 11.49 |
C2H4 | 10.68 | 10.98 | 10.45 | 10.59 | 10.47 | 10.73 | 10.9 | 10.92 | 10.68 |
CO2 | 13.81 | 14.29 | 13.67 | 14.82 | 13.3 | 14.61 | 14.42 | 15.23 | 13.78 |

##  IV conclusion

In conclusion, we present a variant of the constrained local potential
optimization method, Gidopoulos and Lathiotakis (2012) where an effective
screening-density amplitude is introduced, replacing the screening density as
the variational parameter. With this choice, the positivity condition of the
screening density is automatically satisfied, and thus, it does not need to be
imposed. This condition was proven necessary in many cases to obtain
reasonable effective local potentials without certain pathologies in the
asymptotic region. However, its explicit imposition is a formidable task and
in this work, we circumvent this problem by assuming a quadratic form for the
screening density. The price to pay is that the optimization with respect to
the effective amplitude is no longer equivalent to the solution of a set of
linear equations. Although linearization schemes can be applied to solve these
equations, we chose, in this first application, to solve the problem by a
direct minimization algorithm. In addition, for simplicity, we expanded the
effective amplitude in the same basis set as the orbitals.

We demonstrate that, with the present method, accurate local potentials are
obtained, as evidenced by the agreement of the obtained IPs for several DFAs
with those of our previous work and the experiment. Thus the choice of
expanding the effective amplitude in the same basis set as the orbitals is not
affecting the method’s accuracy. This accuracy, as expected, improves with the
size of the basis set. In addition, similarly, we applied the effective
amplitude method to local RDMFT, i.e. we minimized functionals of the 1RDM in
terms of the effective amplitude. We found that the obtained IPs are of
similar quality to those of Refs Lathiotakis  _et al._ , 2014a, b, i.e. the
present minimization technique offers a more efficient and robust method in
LRDMFT.

The pursuit of high-quality spectral properties coming from single particle
Kohn-Sham theoretical models is of central importance and there exist numerous
efforts addressing this issue. Hamel  _et al._ (2002); Chong, Gritsenko, and
Baerends (2002); van Meer, Gritsenko, and Baerends (2014); Verma and Bartlett
(2012); Bartlett (2019); Ranasinghe  _et al._ (2019) The methodology of
applying additional constraints to the potential and in that way enforcing
exact properties is an elegant and promising solution that is applicable to
any DFA by modifying the potential but not the energy functional. Gidopoulos
and Lathiotakis (2012); Pitts, Gidopoulos, and Lathiotakis (2018); Gidopoulos
and Lathiotakis (2015a); Callow  _et al._ (2020) As such, it is fairly easy to
implement in existing computer codes. If tuned we obtain very accurate
ionization potentials compared to other theoretical models and experiment.
Pitts, Lathiotakis, and Gidopoulos (2021) With the present proposition, this
methodology becomes simpler, more robust, and more efficient than previous
implementations yet equally accurate.

###### Acknowledgements.

We acknowledge funding for this work by the Hellenic Foundation for Research
and Innovation (HFRI) under the HFRI Ph.D. Fellowship grant (Fellowship
Number: 1310) and by The Leverhulme Trust, through a Research Project Grant
with number RPG-2016-005.

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