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SUMMARY:Reduced Density-Matrix Functional Theory: Improving its foundation
DTSTART;VALUE=DATE-TIME:20221003T070000Z
DTEND;VALUE=DATE-TIME:20221014T170000Z
DTSTAMP;VALUE=DATE-TIME:20221206T163400Z
UID:indico-event-153@indico.ectstar.eu
CONTACT:driessen@ectstar.eu
DESCRIPTION:Speakers: Carlos L. Benavides-Riveros (Max-Planck Institute fo
r Complex Systems)\, Christian Schilling (LMU Munich)\, Eberhard K. U. Gro
ss (The Hebrew University of Jerusalem)\n\nPlease note that in presence at
tendance is by invitation only\n\nThe aim of this international workshop i
s to discuss and explore new aspects and challenges in\nReduced Density Ma
trix Functional Theory (RDMFT). For this\, we invite up to 25 experts in R
DMFT to\nTrento for an intensive and informal meeting.\n\nThis in-person w
orkshop during 3-14 October will be complemented by five mini-symposia (hy
brid\nformat)\, open to a broad international audience:\n\nSymposium 1 - O
ctober 3\, 14:30-18:00 CEST\nExact results in RDMFT: properties of univers
al functionals\, role of N-representability\, etc.\n\nSymposium 2 - Octobe
r 5\, 9:30-13:00 CEST\nRDMFT for quantum chemistry: Computational and theo
retical state-of-the-art and open challenges\n \nSymposium 3 - October 7\
, 9:30-13:00 CEST\nExtending the scope of RDMFT: bosons\, ultracold gases\
, superconductors\, relativistic QM\, polarons\,\netc.\n\nSymposium 4 - Oc
tober 10\, 9:30-13:00 CEST\nRDMFT for excited states and time-evolution\n\
nSymposium 5 - October 12\, 9:30-13:00 CEST\nRDMFT for translational invar
iant systems\n\nDescription\n\nIn 1959\, at the Colorado conference on Mol
ecular Quantum Mechanics\, Charles Coulson pointed out that the descriptio
n of atomic and molecular ground states involves the two-electron reduced
density matrix (2RDM)\, only [1\,2]. Indeed\, since electrons interact onl
y by pairwise interactions\, the energies and other electronic properties
of atoms and molecules can be computed directly from the 2RDM. This crucia
l insight has defined the starting point for the development of new theore
tical approaches to the ground state problem\, avoiding the use of the exp
onentially complex N-electron wave function. Since each subfield of the Qu
antum Sciences typically restricts to systems all characterized by a fixed
pair interaction V (for instance Coulomb interaction in quantum chemistry
\, contact interaction in quantum optics and Hubbard interaction in solid
state physics) the ground state problem should de facto involve only the o
ne-particle reduced density matrix (1RDM). Indeed\, for Hamiltonians of th
e form H(h) = h + V\, where h represents the one-particle terms and V the
fixed pair interaction\, the conjugate variable to H(h) and h\, respective
ly\, is the 1RDM. The corresponding exact one-particle theory is known as
Reduced Density Matrix Functional Theory (RDMFT) which is based on the exi
stence of an exact energy functional of the 1RDM [3]. This interaction fun
ctional is universal in the sense that it depends only on the fixed intera
ction V but not on the one-particle terms h [3\,4].\n\nMost notably\, comp
ared to the widely used Density Functional Theory (DFT)\, RDMFT has some s
ignificant conceptual advantages and is therefore expected to overcome at
some point the fundamental limitations of DFT. On the one hand\, the kinet
ic energy is described in an exact way due to the access of the full 1RDM
and all the effort can be spent to derive good approximations to the inter
action energy. On the other hand\, RDMFT allows explicitly for fractional
occupation numbers and has therefore great prospects of describing systems
with strong correlations\, particularly static correlations [5]. For inst
ance\, benchmark calculations revealed that the common functionals in RDMF
T yield correlation energies for closed shell atoms and molecules which ar
e by one order of magnitude more accurate than B3LYP (one of the most popu
lar density functionals in quantum chemistry) and a precision comparable t
o Møller-Plesset second-order perturbation theory [5-8]. RDMFT has also s
ucceeded in predicting more accurate gaps of conventional semiconductors t
han semi-local DFT and demonstrated the insulating behavior of Mott-type i
nsulators [9-11]. At the same time\, involving the full 1RDM lies\, howeve
r\, also at the heart of possible disadvantages of RDMFT relative to DFT.
The 1RDM involves quadratically more degrees of freedom than the spatial d
ensity and enforcing the orthonormality of the natural orbitals is computa
tionally highly demanding [5]. Furthermore\, for open-shell systems\, the
domain of the universal functional seems to be constrained by the more inv
olved generalized Pauli constraints. All these aspects are still hampering
RDMFT from reaching its full potential and effectiveness.\n\nSignificant
recent progress on reduced density matrices and the theory of fermion corr
elation provides new ways for overcoming the problems of the recent realiz
ation of RDMFT. For instance\, it has been observed that for many solid ma
terials\, one can exploit either the spatial locality or the translational
symmetry in an efficient way\, leading to more accurate functionals and r
emarkable additional insights [12-14]. Moreover\, it has been shown that t
he one- and two-body N-representability constraints strongly shape the exa
ct functional [15\,16]. Due to the effort of several research groups in th
e world the last two years\, we have witnessed a remarkable progress along
several lines:\n\n\n In the form of the DoNOF\, the first RDMFT software
code has been made public just about one year ago [17].\n In a series of v
ery recent works [18\,19]\, RDMFT has been extended to excited states by g
eneralizing the work by Gross\, Oliviera and Kohn on ensemble DFT. This ha
s revealed in particular a generalization of Pauli’s exclusion principle
to mixed states [20].\n Novel ideas were proposed to improve the numerica
l minimization of functionals which may help to overcome the most severe o
bstacle of the recent version of RDMFT [21].\n RDMFT has been extended to
bosonic quantum systems with remarkable new insights into its fermionic co
unterpart [22\,23].\n The cusp condition due to the Coulomb interaction ha
s finally been translated into a property of the 1RDM providing new insigh
ts into the role and accurate treatment of dynamic correlation in RDMFT [2
4].\n Quite recently\, modern machine learning techniques were put forward
for improving functional approximations such as in [25\,26].\n\n\nMost of
these novel results and ideas are rather challenging\, fundamental\, and
of potentially high impact. It will be one of the crucial challenging for
the next few years to combine all those new ideas to improve the foundatio
n of RDMFT\, overcome its recent limitations and extent its scope by inclu
ding also non-singlet\, finite-temperature\, and time-evolving systems [27
\,28].\n\n\n\n\nBibliography\n\n[1] C. A. Coulson\, “Present state of mo
lecular structure calculations”\, Rev. Mod. Phys. 32\, 170 (1960).\n[2]
D. A. Mazziotti (ed.)\, Reduced Density-Matrix mechanics: with application
s to many-electron atoms and molecules\, John Wiley & Sons (2007).\n[3] T.
L. Gilbert\, “Hohenberg-Kohn theorem for nonlocal external potentials
”\, Phys. Rev. B 12\, 2111 (1975).\n[4] M. Levy\, “Universal variation
al functionals of electron densities\, first-order density matrices\, and
natural spin-orbitals and solution of the v-representability problem”\,
PNAS 76\, 6062 (1979).\n[5] K. Pernal and K. J. Giesbertz\, “Reduced Den
sity Matrix Functional Theory (RDMFT) and Linear Response Time-Dependent R
DMFT (TD-RDMFT)”\, Top. Curr. Chem. 368\, 125 (2016).\n[6] N. Lathiotaki
s and M. A. L. Marques\, “Benchmark calculations for reduced density-mat
rix functional theory”\, J. Chem. Phys. 128\, 184103 (2008).\n[7] M. Pir
is\, “A natural orbital functional based on an explicit approach of the
two-electron cumulant”\, Int. J. Quantum Chem. 113\, 620 (2012).\n[8] M.
Piris\, “Global Method for Electron Correlation”\, Phys. Rev. Lett. 1
19\, 063002 (2017).\n[9] S. Sharma\, J. K. Dewhurst\, S. Shallcross\, and
E. K. U. Gross\, “Spectral Density and Metal-Insulator Phase Transition
in Mott Insulators Within Reduced Density Matrix Functional Theory”\, Ph
ys. Rev. Lett. 110\, 116403 (2013).\n[10] Y. Shinohara\, S. Sharma\, J. K.
Dewhurst\, S. Shallcross\, N. N. Lathiotakis\, and E. K. U. Gross\, “Do
ping induced metal-insulator phase transition in NiO. A reduced density ma
trix functional theory perspective”\, New J. Phys. 17\, 093038 (2015).\n
[11] K. Pernal\, “Turning reduced density matrix theory into a practical
tool for studying the Mott transition”\, New J. Phys. 17\, 111001 (2015
).\n[12] M. Saubanère\, M. B. Lepetit\, and G. M. Pastor\, “Interaction
-energy functional of the Hubbard model: Local formulation and application
to low-dimensional lattices”\, Phys. Rev. B 94\, 045102 (2016).\n[13] C
. Schilling and R. Schilling\, “Diverging exchange force and form of the
exact density matrix functional”\, Phys. Rev. Lett. 122\, 013001 (2019)
.\n[14] J. Schmidt\, C. L. Benavides-Riveros\, and M. A. L. Marques\, “R
educed density matrix functional theory for superconductors”\, Phys. Rev
. B 99\, 224502 (2019).\n[15] M. Piris\, “The Role of the N-Representabi
lity in One-Particle Functional Theories”\, chapter in “Many-body Appr
oaches at Different Scales” (Springer\, New York\, 2016) Chapter 22.\n[1
6] C. Schilling\, “Communication: Relating the pure and ensemble density
matrix functional”\, J. Chem. Phys. 149\, 231102 (2018).\n[17] M. Piris
\, I. Mitxelena\, Comput. Phys. Commun. 259\, 107651 (2021).\n[18] C. Schi
lling\, S. Pittalis\, Phys. Rev. Lett. 127\, 023001 (2021)\n[19] J. Lieber
t\, F. Castillo\, J.-P. Labbé\, C. Schilling\, arXiv:2106.03918\, to appe
ar in J. Chem. Theory Comput.\n[20] F. Castillo\, J.-P. Labbé\, J. Lieber
t\, A.Padrol\, E.Philippe\, C. Schilling\, arXiv:2105.06459\n[21] D. Gibne
y\, J.N. Boyn\, D.A. Mazziotti\, J. Phys. Chem. Lett. 12\, 385 (2021)\n[22
] C.L. Benavides-Riveros\, J. Wolff\, M.A.L. Marques\, C. Schilling\, Phys
. Rev. Lett. 124\, 180603 (2020)\n[23] J. Liebert\, C. Schilling\, Phys. R
ev. Research 3\, 013282 (2021)\n[24] J. Cioslowski\, J. Chem. Phys. 153\,
154108 (2020)\n[25] J. Schmidt\, C.L. Benavides-Riveros\, and M.A.L. Marqu
es\, J. Phys. Chem. Lett. 10\, 6425 (2019)\n[26] J. Schmidt\, M. Fadel\, C
.L. Benavides-Riveros\, Phys. Rev. Research 3\, L032063 (2021).\n[27] K. G
iesbertz\, O. Gritsenko\, and E. Baerends\, “Response calculations with
an independent particle system with exact one-particle density matrix”\,
Phys. Rev. Lett. 105\, 013002 (2010).\n[28] K. Giesbertz and M. Ruggentha
ler\, “One-body reduced density-matrix functional theory in finite basis
sets at elevated temperatures”\, Phys. Rep. 806\, 1 (2019).\n\nhttps://
indico.ectstar.eu/event/153/
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LOCATION:Sala Leonardi (ECT*)
URL:https://indico.ectstar.eu/event/153/
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