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WIREs Comput Mol Sci
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Spin‐restricted ensemble‐referenced Kohn–Sham method: basic principles and application to strongly correlated ground and excited states of molecules

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Ensemble density functional theory (DFT) is a novel theoretical approach that is capable of exact treatment of non‐dynamic electron correlation in the ground and excited states of many‐body fermionic systems. In contrast to ordinary DFT, ensemble DFT has not found so far a way to the repertoire of methods of modern computational chemistry, probably owing to the lack of practically affordable implementations of the theory. The spin‐restricted ensemble‐referenced Kohn–Sham (REKS) method represents perhaps the first computational scheme that makes ensemble DFT calculations feasible. The REKS method is based on the rigorous ensemble representation of the energy and the density of a strongly correlated system and provides for an accurate and consistent description of molecular systems the electronic structure of which is dominated by the non‐dynamic correlation. This includes the ground and excited states of molecules undergoing bond breaking/bond formation, the low‐spin states of biradicals and polyradicals, symmetry forbidden chemical reactions and avoided crossings of potential energy surfaces, real intersections between the energy surfaces of the ground and excited states (conical intersections), and many more. The REKS method can be employed in connection with any local, semi‐local and hybrid (global and range‐separated) functional and affords calculations of large and very large molecular systems at a moderate mean‐field cost. WIREs Comput Mol Sci 2015, 5:146–167. doi: 10.1002/wcms.1209

Conflict of interest: The author has declared no conflicts of interest for this article.

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Profile of the ground state PES of ethylene along the π‐bond torsion mode. The energies (solid curves) and the populations of the b3u orbital (dashed curves) as obtained by the BS‐UKS (yellow), RKS (green), and REKS (red) methods are shown. The relative energies are calculated with respect to the planar conformation. The cc‐pVTZ basis set is employed in connection with the CAM‐B3LYP density functional.
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Profile of the PES of H2 + H2 reaction and populations of the b2u orbital as obtained from the KS/CI (blue), BS‐UKS (yellow), RKS (green), and REKS (red) calculations. The relative energies are calculated with respect to two isolated H2 molecules. Solid curves show the energies and dashed curves show the occupation numbers as a function of R (see Scheme for definition). DFT calculations employ the CAM‐B3LYP functional.
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Geometric parameters and frontier orbitals of H2 + H2 system. (Reprinted with permission from Ref Copyright 2000 American Chemical Society)
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Dissociation curve of the LiH molecule in the 1Σ+ ground electronic state as obtained using the REKS(2,2) (red), RKS (green), BS‐UKS (yellow), and CCSD (blue) methods. The solid curves refer to the relative energy with respect to the dissociation limit of two neutral atoms, the dashed curves refer to the Mulliken charge on Li atom.
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Dissociation curve of the H2 molecule in the $1Σg+$ ground electronic state as obtained using the REKS(2,2) (red), RKS (green), BS‐UKS (yellow), and CCSD (blue) methods. The solid curves refer to the relative energy with respect to the dissociation limit of two neutral atoms, the dashed curves refer to the population of the 1σg bonding orbital (HOMO) obtained as the natural orbital population for the respective method.
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Optimal value of the parameter x = na/2 as a function of the ratio Δ on the right‐hand side of Eq. .
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RMSD (Å) of the geometries at the minimum energy CI points of a number of organic molecules optimized using the SI‐SA‐REKS method from the target geometries obtained using ab initio multi‐reference methods, CASPT2 and MRSDCI. (Reprinted with permission (ACS AuthorChoice) from Ref . Copyright 2013 American Chemical Society)
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Profile of the potential energy surfaces of the ground 1A1 and excited 1B1 states (under the D2 symmetry) along the double bond torsion mode of C2H4. Black lines—CASPT2/6‐31G* results from Ref solid colored lines—SI‐SA‐REKS results, and dashed colored lines—TD‐DFT results. DFT calculations employ the CAM‐B3LYP density functional and 6‐31G* basis set.
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Potential energy curves (upper panel) of the x1Σ+ and a1Σ+ states of LiH and the x1Σ+ ← a1Σ+ excitation energy (lower panel) as a function of the Li‐H distance. Solid curves—SI‐SA‐REKS results, dashed curves—TD‐DFT results. DFT calculations employ the CAM‐B3LYP functional and aug‐cc‐pVTZ basis set. The reference RASCI excitation energy curve (solid black) is taken from Ref .
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Potential energy curves (upper panel) of the $1Σg+$ and $1Σu+$ states of H2 and the $1Σu+←1Σg+$ excitation energy (lower panel) as a function of the H–H distance. Solid colored curves (blue for the ground state and red for the excited state) represent the results of the SA‐REKS calculations, dashed colored curves refer to TD‐DFT, and black curve is the exact excitation energy obtained from the results of Ref . DFT calculations employ the CAM‐B3LYP density functional and the cc‐pV5Z basis set modified as in Ref .
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Relaxed potential energy curves of TME diradical along the torsion mode ϑ. The REKS/ROKS calculations are carried out using the CAM‐B3LYP density functional in connection with the cc‐pVTZ basis set.
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