Dipole Moment Calculations Using Multiconfiguration Pair-Density Functional Theory and Hybrid Multiconfiguration Pair-Density Functional Theory

Abstract

The dipole moment is the molecular property that most directly indicates molecular polarity. The accuracy of computed dipole moments depends strongly on the quality of the calculated electron density, and the breakdown of single-reference methods for strongly correlated systems can lead to poor predictions of the dipole moments in those cases. Here, we derive the analytical expression for obtaining the electric dipole moment by multiconfiguration pair density functional theory (MC-PDFT), and we assess the accuracy of MC-PDFT for predicting dipole moments at equilibrium and nonequilibrium geometries. We show that MC-PDFT dipole moment curves have reasonable behavior even for stretched geometries, and they significantly improve upon the CASSCF results by capturing more electron correlation. The analysis of a dataset consisting of 18 first-row transition metal diatomics and 6 main-group polyatomic molecules with multireference character suggests that MC-PDFT and its hybrid extension (HMC-PDFT) perform comparably to CASPT2 and MRCISD+Q methods and have a mean unsigned deviation of 0.2-0.3 D with respect to the best available dipole moment reference values. We explored the dependence of the predicted dipole moments upon the choice of the on-top density functional and active space, and we recommend the tPBE and hybrid tPBE0 on-top choices for the functionals combined with the moderate correlated participating orbital scheme for selecting the active space. With these choices, the mean unsigned deviations (in debyes) of the calculated equilibrium dipole moments from the best estimates are 0.77 for CASSCF, 0.29 for MC-PDFT, 0.24 for HMC-PDFT, 0.28 for CASPT2, and 0.25 for MRCISD+Q. These results are encouraging because the computational cost of MC-PDFT or HMC-PDFT is largely reduced compared to the CASPT2 and MRCISD+Q methods.

Electronic structure of strongly correlated systems: recent developments in multiconfiguration pair-density functional theory and multiconfiguration nonclassical-energy functional theory

“…To evaluate MC-PDFT dipole moments, we performed analytic differentiation of the MC-PDFT energy using Lagrange's method of undetermined multipliers. 158 This approach shares the same set of Lagrange multipliers as discussed in the previous section. Since the HMC-PDFT energy is a linear combination of the CASSCF and MC-PDFT energies, the HMC-PDFT analytic dipole moment comes with no additional cost compared to MC-PDFT.…”

“…To assess the performance of MC-PDFT for predicting dipole moments, we first explored dipole moments of diatomic molecules as functions of geometry. 158 Using a CASSCF calculation with a full-valence active space for the first step, we compared MC-PDFT with the tPBE on-top functional and HMC-PDFT with the tPBE0 functional to MRCI with single and double excitations and a quadruples correction (MRCISD + Q). For HF, CO, and NO, we found that the MC-PDFT and HMC-PDFT dipole moments were much closer to MRCI + Q than to CASSCF.…”

“…We also tested MC-PDFT and HMC-PDFT for 18 diatomics containing a transition metal and H, C, N, O, or S. 158 We used the moderate correlated participating orbitals 43 ( mod -CPO) active space prescription. The mean unsigned error of the various methods is as follows: 0.55 D for CASSCF, 0.29 D for MC-PDFT with tPBE, 0.24 D for HMC-PDFT with tPBE0, and 0.28 and 0.25 D respectively for the much more expensive CASPT2 and MRCISD + Q calculations.…”

Organic Emitters Showing Excited-States Energy Inversion: An Assessment of MC-PDFT and Correlation Energy Functionals Beyond TD-DFT

“…Note that the on-top second-order reduced density matrix represents the probability that two opposite-spin electrons are found at point r and integrates to the total number of interacting pairs. Multiconfiguration Pair-Density Functional Theory (MC-PDFT [29][30][31][32]) can be thus viewed as a post-MCSCF method that evaluates the energy of any state with on-top pair-density function theory. Basically, for a MCSCF wavefunction, |Ψ MCSCF = ∑ µ C µ |Ψ µ , one can obtain the total electronic energy as:…”

Dipole Moments and Transition Dipole Moments Calculated by Pair-Density Functional Theory with State Interaction

“…One powerful multireference method is the multiconfiguration pair-density functional theory (MC-PDFT), an electronic structure theory that evaluates the electronic energy from the multiconfigurational wave function using the kinetic energy, density, on-top pair density, and an on-top density functional. Recently, we derived 13 the analytic dipole moment expression for CAS-PDFT, which is MC-PDFT based on a CASSCF reference state. However, that treatment did not yield transition dipole moments.…”