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

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

[1]  H. Englisch,et al.  Exact Density Functionals for Ground-State Energies , 1984, June 1.

[2]  R. C. Morrison.  Electron correlation and noninteracting v-representability in density functional theory: The Be isoelectronic series , 2002 .

[3]  W. Goddard,et al.  Generalized valence bond description of bonding in low-lying states of molecules , 1973 .

[4]  F. Aryasetiawan,et al.  Excitation energies from time-dependent density-functional formalism for small systems , 2002 .

[5]  Satyender Goel,et al.  Pairwise Spin-Contamination Correction Method and DFT Study of MnH and H2 Dissociation Curves , 2009, ICCS.

[6]  Jochen Schirmer,et al.  A consistent third-order propagator method for electronic excitation , 1999 .

[7]  P. Hohenberg,et al.  Inhomogeneous Electron Gas , 1964 .

[8]  E. J. Baerends,et al.  Benchmark calculations of chemical reactions in density functional theory: comparison of the accurate Kohn-Sham solution with generalized gradient approximations for the H2+H and H2+H2 reactions. , 1999 .

[9]  Louis Noodleman,et al.  Valence bond description of antiferromagnetic coupling in transition metal dimers , 1981 .

[10]  S. Alvarez,et al.  About the calculation of exchange coupling constants using density-functional theory: the role of the self-interaction error. , 2005, The Journal of chemical physics.

[11]  Stefan Grimme,et al.  Semiempirical GGA‐type density functional constructed with a long‐range dispersion correction , 2006, J. Comput. Chem..

[12]  O. Franck,et al.  Generalised adiabatic connection in ensemble density-functional theory for excited states: example of the H2 molecule , 2013, 1308.4596.

[13]  S. Shaik,et al.  Spin-restricted density functional approach to the open-shell problem , 1998 .

[14]  Michael Filatov,et al.  Assessment of Density Functional Methods for Obtaining Geometries at Conical Intersections in Organic Molecules , 2013, Journal of chemical theory and computation.

[15]  C. A. Coulson,et al.  XXXIV. Notes on the molecular orbital treatment of the hydrogen molecule , 1949 .

[16]  W. T. Zemke,et al.  Spectroscopy and Structure of the Lithium Hydride Diatomic Molecules and Ions , 1993 .

[17]  W. Kohn,et al.  Kohn-Sham theory for ground-state ensembles. , 2001, Physical review letters.

[18]  Michael Filatov,et al.  Understanding the dynamics behind the photoisomerization of a light-driven fluorene molecular rotary motor. , 2010, The journal of physical chemistry. A.

[19]  E. Gross,et al.  Time-dependent density functional theory. , 2004, Annual review of physical chemistry.

[20]  F. Illas,et al.  Extent and limitations of density-functional theory in describing magnetic systems , 2004 .

[21]  R. Baer,et al.  Reliable prediction of charge transfer excitations in molecular complexes using time-dependent density functional theory. , 2009, Journal of the American Chemical Society.

[22]  M. Filatov,et al.  Obtaining stable solutions of the optimized-effective-potential method in the basis set representation , 2010 .

[23]  Elliott H. Lieb.  Density functionals for coulomb systems , 1983 .

[24]  Francesc Illas,et al.  Restricted Ensemble-Referenced Kohn-Sham versus Broken Symmetry Approaches in Density Functional Theory:  Magnetic Coupling in Cu Binuclear Complexes. , 2007, Journal of chemical theory and computation.

[25]  Walter Thiel,et al.  Surface Hopping Excited-State Dynamics Study of the Photoisomerization of a Light-Driven Fluorene Molecular Rotary Motor. , 2011, Journal of chemical theory and computation.

[26]  J. Autschbach,et al.  Applications of Time Dependent and Time Independent Density Functional Theory to the First π to π* Transition in Cyanine Dyes. , 2014, Journal of chemical theory and computation.

[27]  N. Handy,et al.  Dynamic correlation , 2001 .

[28]  J. Paldus,et al.  Electronic structure of organic diradicals: Evaluation of the performance of coupled-cluster methods. , 2008, The Journal of chemical physics.

[29]  H. Englisch,et al.  Exact Density Functionals for Ground-State Energies II. Details and Remarks , 1984, July 1.

[30]  S. Shaik,et al.  A spin-restricted ensemble-referenced Kohn Sham method and its application to diradicaloid situations , 1999 .

[31]  Klaus Ruedenberg,et al.  Potential energy surfaces near intersections , 1991 .

[32]  E. Davidson,et al.  Ligand spin polarization and antiferromagnetic coupling in transition metal dimers , 1986 .

[33]  Oliveira,et al.  Density-functional theory for ensembles of fractionally occupied states. I. Basic formalism. , 1988, Physical review. A, General physics.

[34]  P. Löwdin.  Quantum Theory of Many-Particle Systems. I. Physical Interpretations by Means of Density Matrices, Natural Spin-Orbitals, and Convergence Problems in the Method of Configurational Interaction , 1955 .

[35]  M. Filatov.  Antiferromagnetic interactions in the quarter-filled organic conductor (EDO-TTF)2PF6. , 2011, Physical chemistry chemical physics : PCCP.

[36]  Ichiro Hanazaki,et al.  Vapor-phase electron donor-acceptor complexes of tetracyanoethylene and of sulfur dioxide , 1972 .

[37]  Elfi Kraka,et al.  Ortho-, meta-, and para-benzyne. A comparative CCSD (T) investigation , 1993 .

[38]  M. Filatov.  Theoretical study of the photochemistry of a reversible three-state bis-thiaxanthylidene molecular switch. , 2011, Chemphyschem : a European journal of chemical physics and physical chemistry.

[39]  J Schirmer,et al.  Intermediate state representation approach to physical properties of electronically excited molecules. , 2004, The Journal of chemical physics.

[40]  M. E. Casida,et al.  Progress in time-dependent density-functional theory. , 2011, Annual review of physical chemistry.

[41]  Massimo Olivucci,et al.  Designing conical intersections for light-driven single molecule rotary motors: from precessional to axial motion. , 2014, The Journal of organic chemistry.

[42]  N. Handy,et al.  A new hybrid exchange–correlation functional using the Coulomb-attenuating method (CAM-B3LYP) , 2004 .

[43]  Björn O. Roos,et al.  Second-order perturbation theory with a complete active space self-consistent field reference function , 1992 .

[44]  W. Kohn,et al.  Self-Consistent Equations Including Exchange and Correlation Effects , 1965 .

[45]  E. J. Baerends,et al.  One - determinantal pure state versus ensemble Kohn-Sham solutions in the case of strong electron correlation: CH2 and C2 , 1998 .

[46]  Á. Rubio,et al.  Time-dependent density-functional theory. , 2009, Physical chemistry chemical physics : PCCP.

[47]  W. C. Lineberger,et al.  Ultraviolet Photoelectron Spectroscopy of the o-, m-, and p- Benzyne Negative Ions. Electron Affinities and Singlet−Triplet Splittings for o -, m -, and p -Benzyne , 1998 .

[48]  M. Filatov.  On multiferroicity of TTF-CA molecular crystal. , 2011, Physical chemistry chemical physics : PCCP.

[49]  S. Shaik,et al.  Diradicaloids: Description by the Spin-Restricted, Ensemble-Referenced Kohn-Sham Density Functional Method , 2000 .

[50]  J. C. Slater,et al.  Nonintegral Occupation Numbers in Transition Atoms in Crystals , 1969 .

[51]  Jochen Schirmer,et al.  An efficient polarization propagator approach to valence electron excitation spectra , 1995 .

[52]  S. Wang,et al.  SIMULATION OF NONDYNAMICAL CORRELATION IN DENSITY FUNCTIONAL CALCULATIONS BY THE OPTIMIZED FRACTIONAL ORBITAL OCCUPATION APPROACH : APPLICATION TO THE POTENTIAL ENERGY SURFACES OF O3 AND SO2 , 1996 .

[53]  Eileen P. Clifford,et al.  Properties of tetramethyleneethane (TME) as revealed by ion chemistry and ion photoelectron spectroscopy , 1998 .

[54]  K. Hirao,et al.  General SCF operator satisfying correct variational condition , 1973 .

[55]  Parr,et al.  From electron densities to Kohn-Sham kinetic energies, orbital energies, exchange-correlation potentials, and exchange-correlation energies. , 1994, Physical review. A, Atomic, molecular, and optical physics.

[56]  J. Gräfenstein,et al.  Development of a CAS-DFT method covering non-dynamical and dynamical electron correlation in a balanced way , 2005 .

[57]  M. Seth,et al.  Introducing constricted variational density functional theory in its relaxed self-consistent formulation (RSCF-CV-DFT) as an alternative to adiabatic time dependent density functional theory for studies of charge transfer transitions. , 2014, The Journal of chemical physics.

[58]  I. Ciofini,et al.  Performance of the tau-dependent functionals in predicting the magnetic coupling of ionic antiferromagnetic insulators. , 2004, The Journal of chemical physics.

[59]  R. Hoffmann.  Selection Rules for Concerted Cycloaddition Reactions (福井謙一とフロンティア軌導理論) -- (参考論文) , 1965 .

[60]  A. Becke.  Correlation energy of an inhomogeneous electron gas: A coordinate‐space model , 1988 .

[61]  Arvi Rauk,et al.  On the calculation of multiplet energies by the hartree-fock-slater method , 1977 .

[62]  K. Thygesen,et al.  Static correlation beyond the random phase approximation: dissociating H2 with the Bethe-Salpeter equation and time-dependent GW. , 2013, The Journal of chemical physics.

[63]  A. Savin,et al.  Combining multideterminantal wave functions with density functionals to handle near-degeneracy in atoms and molecules , 2002 .

[64]  Locked alkenes with a short triplet state lifetime , 2000 .

[65]  Francesc Illas,et al.  A unified view of the theoretical description of magnetic coupling in molecular chemistry and solid state physics. , 2006, Physical chemistry chemical physics : PCCP.

[66]  E. Baerends,et al.  Aufbau derived from a unified treatment of occupation numbers in Hartree-Fock, Kohn-Sham, and natural orbital theories with the Karush-Kuhn-Tucker conditions for the inequality constraints n(i)or=0. , 2010, The Journal of chemical physics.

[67]  H. Englisch,et al.  Exact Density Functionals for Ground-State Energies. I. General Results , 1984 .

[68]  W. C. Lineberger,et al.  The synergy between qualitative theory, quantitative calculations, and direct experiments in understanding, calculating, and measuring the energy differences between the lowest singlet and triplet states of organic diradicals. , 2011, Physical chemistry chemical physics : PCCP.

[69]  C. C. J. Roothaan,et al.  Self-Consistent Field Theory for Open Shells of Electronic Systems , 1960 .

[70]  Francesco A Evangelista,et al.  High-order excitations in state-universal and state-specific multireference coupled cluster theories: model systems. , 2006, The Journal of chemical physics.

[71]  G. Herzberg.  Molecular Spectra and Molecular Structure IV. Constants of Diatomic Molecules , 1939 .

[72]  Roland Lindh,et al.  Shape of Multireference, Equation-of-Motion Coupled-Cluster, and Density Functional Theory Potential Energy Surfaces at a Conical Intersection. , 2014, Journal of chemical theory and computation.

[73]  N. Ferré,et al.  Assessment of Density Functional Theory for Describing the Correlation Effects on the Ground and Excited State Potential Energy Surfaces of a Retinal Chromophore Model. , 2013, Journal of chemical theory and computation.

[74]  Lionel Salem,et al.  The Electronic Properties of Diradicals , 1972 .

[75]  Michael Filatov,et al.  Density functional study of the ground and excited state potential energy surfaces of a light-driven rotary molecular motor (3R,3'R)-(P,P)-trans-1,1',2,2',3,3',4,4'-octahydro-3,3'-dimethyl-4,4'-biphenanthrylidene. , 2009, The journal of physical chemistry. A.

[76]  Troy Van Voorhis,et al.  Nonlocal van der Waals density functional: the simpler the better. , 2010, The Journal of chemical physics.

[77]  J. McDouall.  Combining two-body density functionals with multiconfigurational wavefunctions: diatomic molecules , 2003 .

[78]  Roald Hoffmann,et al.  Stereochemistry of Electrocyclic Reactions (福井謙一とフロンティア軌導理論) -- (参考論文) , 1965 .

[79]  M. Filatov,et al.  Assessment of density functional theory based ΔSCF (self-consistent field) and linear response methods for longest wavelength excited states of extended π-conjugated molecular systems. , 2014, The Journal of chemical physics.

[80]  Roald Hoffmann,et al.  Conservation of orbital symmetry , 1968 .

[81]  S. Shaik,et al.  Tetramethyleneethane (TME) Diradical: Experiment and Density Functional Theory Reach an Agreement , 1999 .

[82]  P. Löwdin.  Quantum Theory of Many-Particle Systems. III. Extension of the Hartree-Fock Scheme to Include Degenerate Systems and Correlation Effects , 1955 .

[83]  K. Burke,et al.  Why semilocal functionals work: Accuracy of the on-top pair density and importance of system averaging , 1998 .

[84]  Francesco A Evangelista,et al.  Coupling term derivation and general implementation of state-specific multireference coupled cluster theories. , 2007, The Journal of chemical physics.

[85]  D. Cremer.  Density functional theory: coverage of dynamic and non-dynamic electron correlation effects , 2001 .

[86]  J. Malrieu,et al.  Remarks on the Proper Use of the Broken Symmetry Approach to Magnetic Coupling , 1997 .

[87]  O. Sǐnanoğlu,et al.  Many‐Electron Theory of Atoms, Molecules and Their Interactions , 2007 .

[88]  Walter Thiel,et al.  Benchmarks for electronically excited states: time-dependent density functional theory and density functional theory based multireference configuration interaction. , 2008, The Journal of chemical physics.

[89]  L. Wolniewicz,et al.  Potential-Energy Curves for the X1Sg+, b3Su+, and C1Pu States of the Hydrogen Molecule , 1965 .

[90]  M. Filatov,et al.  Excitation energies from spin-restricted ensemble-referenced Kohn-Sham method: a state-average approach. , 2008, The journal of physical chemistry. A.

[91]  Sason Shaik,et al.  Implicit and Explicit Coverage of Multi-reference Effects by Density Functional Theory , 2002 .

[92]  Walter Thiel,et al.  Assessment of approximate computational methods for conical intersections and branching plane vectors in organic molecules. , 2014, The Journal of chemical physics.

[93]  Filipp Furche,et al.  Molecular tests of the random phase approximation to the exchange-correlation energy functional , 2001 .

[94]  K. Yamaguchi,et al.  Resonating broken-symmetry approach to biradicals and polyradicals , 2006 .

[95]  C. Adamo,et al.  Comment on "about the calculation of exchange coupling constants using density-functional theory: the role of the self-interaction error" [J. Chem. Phys. 123, 164110 (2005)]. , 2006, The Journal of chemical physics.

[96]  Jochen Schirmer,et al.  Beyond the random-phase approximation: A new approximation scheme for the polarization propagator , 1982 .

[97]  Á. Rubio,et al.  Assessment of dressed time-dependent density-functional theory for the low-lying valence states of 28 organic chromophores , 2010, 1101.0291.

[98]  L. Wolniewicz,et al.  PotentialEnergy Curves for the X 1 Σg + , b 3 Σu + , and C 1 Πu States of the Hydrogen Molecule , 2013 .

[99]  F. Illas,et al.  Spin Symmetry Requirements in Density Functional Theory: The Proper Way to Predict Magnetic Coupling Constants in Molecules and Solids , 2006 .

[100]  B. Dunlap.  Symmetry and Degeneracy in Xα and Density Functional Theory , 2007 .

[101]  M. Filatov.  Description of electron transfer in the ground and excited states of organic donor-acceptor systems by single-reference and multi-reference density functional methods. , 2014, The Journal of chemical physics.

[102]  E. Baerends,et al.  Excitation energies with linear response density matrix functional theory along the dissociation coordinate of an electron-pair bond in N-electron systems. , 2014, The Journal of chemical physics.

[103]  D. Pines,et al.  The theory of quantum liquids , 1968 .