One - determinantal pure state versus ensemble Kohn-Sham solutions in the case of strong electron correlation: CH 2 and C 2

Abstract

The possibility that the Kohn-Sham (KS) solution for a noninteracting auxiliary electron system is not the conventional one-determinantal pure state but a few-determinantal ensemble has been investigated. The KS solutions (the exchange-correlation potential v xc and the orbitals) have not been approximated by localdensity or density-gradient approximations but have been constructed from an accurate ab initio electron density. The lowest singlet states of the CH 2 and C 2 molecules have been selected for this investigation since for these cases the ground-state wave function W is nondegenerate but has an essentially multideterminantal character (electron correlation is strong). For C 2 the dependence of the type of KS solution on the bond distance R(C±C) has been studied at the QZ level. For the shortest distance considered, R(C±C) 1X8 a.u., a pure-state KS solution has been obtained. For the equilibrium distance e C±C 2X348 a.u. and at larger distances ensemble solutions have been obtained with widely varying weights of the individual determinants, depending on the bond distance. For CH 2 the dependence of the type of KS solution on the basis has been studied: calculation in the triple zeta (TZ) basis for the KS orbitals yields an ensemble solution, while the purestate KS solution has been obtained in the quadruple zeta (QZ) basis. The form of the KS orbitals has been compared with that of the natural orbitals (NOs). It has been shown for the model example of the stretched H 2 molecule as well as for CH 2 and C 2 Y that the KS orbitals of the pure state may be rather dierent from the corresponding NOsY while the occupied KS orbitals of the ensemble solution can be considered as plausible approximations to the corresponding NOs.

Assessment of approximate computational methods for conical intersections and branching plane vectors in organic molecules

“…The REKS method employs an ensemble representation for the KS reference state 58,59 to describe non-dynamic electron correlation in the context of KS DFT. [60][61][62][63] The ensemble representation for the density leads to fractional occupation numbers (FONs) of a few frontier KS orbitals. 60,61,63,64 The REKS ground-state energy is minimized with respect to the KS orbitals and the FONs of the frontier active orbitals.…”

“…[60][61][62][63] The ensemble representation for the density leads to fractional occupation numbers (FONs) of a few frontier KS orbitals. 60,61,63,64 The REKS ground-state energy is minimized with respect to the KS orbitals and the FONs of the frontier active orbitals. The currently employed version of the REKS method treats two fractionally occupied frontier orbitals, as, for example, in a diradical state resulting from (near) degeneracy of the bonding π and the anti-bonding π * frontier orbitals of an alkene near a ca.…”

Beyond Kohn–Sham Approximation: Hybrid Multistate Wave Function and Density Functional Theory

“…5 Equation 7 is consistent with the ensemble v s -representability in EDFT because the determinants 26 in MSDFT can easily be orthogonalized. 27 Importantly, the density and functional in eqs 7 and 8 are not the “ensemble” density and energy but rather they represent the electron densities and energies of the corresponding adiabatic ground and excited states of the system. Note also that a thermally-assisted-occupation DFT method 28,29 that employs fractional occupations as in the grand canonical approach has been developed to represent the multideterminant character for systems with strong correlation with good results for the ground state.…”

C2 in a Box: Determining Its Intrinsic Bond Strength for the X1Σg+ Ground State

“…Because of its multireference character,C 2 hasb ecome ab enchmark in connection with method development and high accuracyc alculations, which establishes its second way of importance in theoretical chemistry.S tudies on C 2 include density functional theory, [16] coupled cluster, [17][18][19][20][21] FCI (full configuration interaction), [18,[22][23][24][25] multireference configurationi nteraction, [15,18,26,27] multireferencep erturbation theory, [28,29] quantum Monte Carlo, [30,31] variational reduced-density-matrix, [32] valence bond (VB), [24,25] and density matrix renormalization group calculations. [33] There is no doubt that the character of the bond in C 2 should be describedi nt erms of natural orbitals with fractional occupation numbers, rather than the MOs of the HF approach.…”