Density-Matrix Functional Theory (DMFT)[Zumbach, Maschke, J.Chem. Phys. 82 (1985) 5604 and further developments]

Matrix g:In the literature, it is denoted also as G or r1) called also one-matrix, one-particle spinless reduced density matrix,or just « density matrix ».

DMFT, T. Wesolowski, University of Geneva, 2016

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Ho = T + Vee

Vext

Notation:

Admissible density matrices g:

set of matrices which are well-behaving in variational principle.Note the analogy with N-representability of electron density r1.

DMFT, T. Wesolowski, University of Geneva, 2016

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Density-Matrix Functional Theory (DMFT)[Zumbach, Maschke, J.Chem. Phys. 82 (1985) 5604 and further developments]

Levy-Lieb universal functional

(note the similarity to the constrained search definitio of the functional FHK[r])

DMFT, T. Wesolowski, University of Geneva, 2016

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Density-Matrix Functional Theory (DMFT)[Zumbach, Maschke, J.Chem. Phys. 82 (1985) 5604 and further developments]

Universal « Hohenberg-Kohn-like »functional in DMFT:

F[g] = T[g] + Vee[g] = T[g] + J[g] + Exc[g]

DMFT, T. Wesolowski, University of Geneva, 2016

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Or to avoid confusion with Exc[r]=T[r] - Ts[r] + Vee[r]- J[r]:

F[g] = T[g] + Vee[g] = T[g] + J[g] + Oxc[g]

Density-Matrix Functional Theory (DMFT)[Zumbach, Maschke, J.Chem. Phys. 82 (1985) 5604 and further developments]

Theorem:

where

DMFT, T. Wesolowski, University of Geneva, 2016

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g g

Density-Matrix Functional Theory (DMFT)[Zumbach, Maschke, J.Chem. Phys. 82 (1985) 5604 and further developments]

Variational principle

where,

The Lagrange multiplyier represents the constraint:

DMFT, T. Wesolowski, University of Geneva, 2016

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Or as derived by Zumbach and Maschkein the appendix [J. Chem. Phys. 82, (1985) 5694]

Density-Matrix Functional Theory (DMFT)[Zumbach, Maschke, J.Chem. Phys. 82 (1985) 5604 and further developments]

Euler-Lagrange Equation for g

Approximating the functional Exc[g]

Admissible density matrices g can be represented as:

g(r’,r) = S ni fi*(r’)fi(r)

E[g] can be represented as a functional depending on {ni} and {fi}.

E[g] = E[{ni},{fi}]

A typical approximation to the Exc[g] component of E[g] has the form:

DMFT, T. Wesolowski, University of Geneva, 2016

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Approximating the functional Exc[g] (1)

Müller, 1984:

Goeddeker & Umrigar, 1998:

Csányi&Arias, 2000:

Csányi, Goeddeker&Arias, 2002:

DMFT, T. Wesolowski, University of Geneva, 2016

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DMFT, T. Wesolowski, University of Geneva, 2016

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Piris, 2006:

Piris, 2006:

Approximating the functional Exc[g] (2)

DMFT, T. Wesolowski, University of Geneva, 2016

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Baerends and collaborators, 1991-2005:

Approximating the functional Exc[g] (3)

DMFT results (1)atomization energies with approximated Exc[g]

N.N. Lathiotakis, M.A.L. Marques, J. Chem. Phys. vol. 128 (2008) 184103.

DMFT, T. Wesolowski, University of Geneva, 2016

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DMFT results (2) correlation energies with approximated Exc[g]

N.N. Lathiotakis, M.A.L. Marques, J. Chem. Phys. vol. 128 (2008) 184103.

DMFT, T. Wesolowski, University of Geneva, 2016

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DMFT results (3) waekly occupied levels

N.N. Lathiotakis, M.A.L. Marques, J. Chem. Phys. vol. 128 (2008) 184103.

DMFT, T. Wesolowski, University of Geneva, 2016

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DMFT results (4) correlation energies with approximated Exc[g]

N.N. Lathiotakis, M.A.L. Marques, J. Chem. Phys. vol. 128 (2008) 184103.

DMFT, T. Wesolowski, University of Geneva, 2016

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DMFT results (5) Benchmarking the Piris’ functional

Piris et al., . J. Chem. Phys. vol. 132 (2010) 031103

DMFT, T. Wesolowski, University of Geneva, 2016

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