Multiconfiguration Pair-Density Functional Theory

Abstract

We present a new theoretical framework, called Multiconfiguration Pair-Density Functional Theory (MC-PDFT), which combines multiconfigurational wave functions with a generalization of density functional theory (DFT). A multiconfigurational self-consistent-field (MCSCF) wave function with correct spin and space symmetry is used to compute the total electronic density, its gradient, the on-top pair density, and the kinetic and Coulomb contributions to the total electronic energy. We then use a functional of the total density, its gradient, and the on-top pair density to calculate the remaining part of the energy, which we call the on-top-density-functional energy in contrast to the exchange-correlation energy of Kohn-Sham DFT. Because the on-top pair density is an element of the two-particle density matrix, this goes beyond the Hohenberg-Kohn theorem that refers only to the one-particle density. To illustrate the theory, we obtain first approximations to the required new type of density functionals by translating conventional density functionals of the spin densities using a simple prescription, and we perform post-SCF density functional calculations using the total density, density gradient, and on-top pair density from the MCSCF calculations. Double counting of dynamic correlation or exchange does not occur because the MCSCF energy is not used. The theory is illustrated by applications to the bond energies and potential energy curves of H2, N2, F2, CaO, Cr2, and NiCl and the electronic excitation energies of Be, C, N, N(+), O, O(+), Sc(+), Mn, Co, Mo, Ru, N2, HCHO, C4H6, c-C5H6, and pyrazine. The method presented has a computational cost and scaling similar to MCSCF, but a quantitative accuracy, even with the present first approximations to the new types of density functionals, that is comparable to much more expensive multireference perturbation theory methods.

Synergy between pair coupled cluster doubles and pair density functional theory

“…As suggested in Ref. [12], a simple way (n, ∞) and the DFT correlation energy density. The f (r) used here is an adaptation for pCCD from the one developed for CAS-DFT by Miehlich et al [15], analogous to the adaptation for PHF in our PHF+f DFT method [26].…”

“…The simplest is to use small active spaces and reparametrized functionals [1][2][3], but this is not always satisfactory. Alternatively, the (interelectron) Coulomb operator may be divided into complementary MR and DFT parts to create MR+DFT hybrids without double counting [9][10][11][12], at the cost of introducing inexact DFT exchange (which is associated with self-interaction error [32][33][34]). A different strategy that avoids the use of DFT exchange is to use a local scaling factor f (r) of the correlation energy density [15][16][17].…”

“…In this case, however, f (r) does not eliminate the double counting exactly; f (r) is approximated at the LDA level. Again, despite these imperfections, these approaches have been used before and the outcomes have been encouraging [11,12,17].…”

Density matrix renormalization group with efficient dynamical electron correlation through range separation

“…Alternative approaches for a simultaneous treatment of static and dynamic correlation in a hybrid DMRG approach that avoid a range-separation ansatz exist. In future work, we are considering the pair-density functional theory which was recently put forward for MC-SCF methods 50 as well as the site occupation density functional theory proposed by Fromager 51 . It should finally be emphasized that hybrids between DFT and wave function methods are also expected to have less dramatic dependence on the one-electron basis set than standard wave function methods.…”

“…In turn, the coupling of DFT to a CAS-type wave function method provides the required flexibility for cases where static correlation becomes important and where DFT is likely to fail [41][42][43][44] . Although hybrid approaches between wave function theory (WFT) and DFT have not been considered for DMRG yet, they have already been studied with other wave func-tion methods [45][46][47][48][49][50][51] . While approximate DFT functionals will introduce errors on the absolute scale, relative energies for hybrid WFT-DFT approaches can be obtained with good accuracy [52][53][54][55][56] .…”