The Use of Effective Core Potentials with Multiconfiguration Pair-Density Functional Theory.

The reliable and accurate prediction of chemical properties is a key goal in quantum chemistry. Transition-metal-containing complexes can often pose difficulties to quantum mechanical methods for multiple reasons, including many electron configurations contributing to the overall electronic description of the system and the large number of electrons significantly increasing the amount of computational resources required. Often, multiconfigurational electronic structure methods are employed for such systems, and the cost of these calculations can be reduced by the use of an effective core potential (ECP). In this work, we explore both theoretical considerations and performances of ECPs applied in the context of multiconfiguration pair-density functional theory (MC-PDFT). A mixed-basis set approach is used, using ECP basis sets for transition metals and all-electron basis sets for nonmetal atoms. We illustrate the effects that an ECP has on the key parameters used in the computation of MC-PDFT energies, and we explore how ECPs affect the prediction of physical observables for chemical systems. The dissociation curve for a metal dimer was explored, and ionization energies for transition metal-containing diatomic systems were computed and compared to experimental values. In general, we find that ECP approaches employed with MC-PDFT are able to predict ionization energies with improved accuracy compared to traditional Kohn-Sham density functional theory approaches.

A multiconfiguration pair-density functional theory-based approach to molecular junctions.

Due to their small size and unique properties, single-molecule electronics have long seen research interest from experimentalists and theoreticians alike. From a theoretical standpoint, modeling these systems using electronic structure theory can be difficult due to the importance of electron correlation in the determination of molecular properties, and this electron correlation can be computationally expensive to consider, particularly multiconfigurational correlation energy. In this work, we develop a new approach for the study of single-molecule electronic systems, denoted NEGF-MCPDFT, which combines multiconfiguration pair-density functional theory (MCPDFT) with the non-equilibrium Green's function formalism (NEGF). The use of MCPDFT with NEGF allows for the efficient inclusion of both static and dynamic electron correlations in the description of the junction's electronic structure. Complete active space self-consistent field wave functions are used as references in the MCPDFT calculation, and as with any active space method, effort must be made to determine the proper orbital character to include in the active space. We perform conductance and transmission calculations on a series of alkanes (predominantly single-configurational character) and benzyne (multiconfigurational character), exploring the role that active space selection has on the computed results. For the alkane junctions explored (where dynamic electron correlation dominates), the MCPDFT-NEGF results agree well with the DFT-NEGF results. For the benzyne junction (which has a significant static correlation), we see clear differences in the MCPDFT-NEGF and DFT-NEGF results and evidence that NEGF-MCPDFT is capturing additional electron correlation effects beyond those provided by the Perdew-Burke-Ernzerhof functional.

Analytic gradients for multiconfiguration pair-density functional theory with density fitting: Development and application to geometry optimization in the ground and excited states.

Density fitting reduces the computational cost of both energy and gradient calculations by avoiding the computation and manipulation of four-index electron repulsion integrals. With this algorithm, one can efficiently optimize the geometries of large systems with an accurate multireference treatment. Here, we present the derivation of multiconfiguration pair-density functional theory for energies and analytic gradients with density fitting. Six systems are studied, and the results are compared to those obtained with no approximation to the electron repulsion integrals and to the results obtained by complete active space second-order perturbation theory. With the new approach, there is an increase in the speed of computation with a negligible loss in accuracy. Smaller grid sizes have also been used to reduce the computational cost of multiconfiguration pair-density functional theory with little effect on the optimized geometries and gradient values.

Analytic gradients for state-averaged multiconfiguration pair-density functional theory.

Analytic gradients are important for efficient calculations of stationary points on potential energy surfaces, for interpreting spectroscopic observations, and for efficient direct dynamics simulations. For excited electronic states, as are involved in UV-Vis spectroscopy and photochemistry, analytic gradients are readily available and often affordable for calculations using a state-averaged complete active space self-consistent-field (SA-CASSCF) wave function. However, in most cases, a post-SA-CASSCF step is necessary for quantitative accuracy, and such calculations are often too expensive if carried out by perturbation theory or configuration interaction. In this work, we present the analytic gradients for multiconfiguration pair-density functional theory based on SA-CASSCF wave functions, which is a more affordable alternative. A test set of molecules has been studied with this method, and the stationary geometries and energetics are compared to values in the literature as obtained by other methods. Excited-state geometries computed with state-averaged pair-density functional theory have similar accuracy to those from complete active space perturbation theory at the second-order.

Calculation of Chemical Reaction Barrier Heights by Multiconfiguration Pair-Density Functional Theory with Correlated Participating Orbitals.

The accurate description of reaction barrier heights is challenging for quantum mechanical methods due to the need for a balanced treatment of dynamic and static correlation energies because their importance varies during the course of a chemical reaction. While some regions of potential energy surfaces are well-described by a single-reference wave function or by Kohn-Sham density functional theory, in other cases a multireference treatment is needed. For systems with many active electrons, most accurate multireference methods have prohibitive computational scalings with system size. Multiconfiguration pair-density functional theory, MC-PDFT, is a more affordable multireference approach that computes the total electron correlation energy in a single step by using the multiconfiguration kinetic energy, density, and on-top pair density and an on-top density functional. In this work, we apply MC-PDFT to a benchmark database (DBH24/18) of 24 diverse reaction barrier heights. We explore the role of active space and basis set selection on the performance of MC-PDFT. We find that MC-PDFT is able to calculate reaction barrier heights with a similar accuracy to complete active space second order perturbation theory, CASPT2, but at a lower computational cost, and we find that MC-PDFT is less dependent on basis set selection than CASPT2.

State-interaction pair-density functional theory.

The accurate description of ground- and excited-state potential energy surfaces poses a challenge for many electronic structure methods, especially in regions where strong electronic state interaction occurs. Here we introduce a new methodology, state-interaction pair-density functional theory (SI-PDFT), to target molecular systems exhibiting strong interaction of electronic states. SI-PDFT is an extension of multiconfiguration pair-density functional theory in which a set of N electronic states is generated through the diagonalization of an N × N effective Hamiltonian. We demonstrate the accuracy of the method by performing calculations on the ionic-neutral avoided crossing in lithium fluoride and the 1ππ-1πσ* avoided crossing in the H-O bond photodissociation in phenol. We show that SI-PDFT can be a useful tool in the study of photochemistry and nonadiabatic dynamics.

Efficient algorithm for multiconfiguration pair-density functional theory with application to the heterolytic dissociation energy of ferrocene.

The recently developed multiconfiguration pair-density functional theory (MC-PDFT) combines multiconfiguration wave function theory with a density functional that depends on the on-top pair density of an electronic system. In an MC-PDFT calculation, there are two steps: a conventional multiconfiguration self-consistent-field (MCSCF) calculation and a post-MCSCF evaluation of the energy with an on-top density functional. In this work, we present the details of the MC-PDFT algorithm that avoids steeply scaling steps that are present in other post-self-consistent-field multireference calculations of dynamic correlation energy. We demonstrate the favorable scaling by considering systems of H2 molecules with active spaces of several different sizes. We then apply the MC-PDFT method to calculate the heterolytic dissociation enthalpy of ferrocene. We find that MC-PDFT yields results that are at least as accurate as complete active space second-order perturbation theory and are more stable with respect to basis set, but at a fraction of the cost in both time and memory.

Enhanced computational efficiency in the direct determination of the two-electron reduced density matrix from the anti-Hermitian contracted Schrödinger equation with application to ground and excited states of conjugated π-systems.

Determination of the two-electron reduced density matrix (2-RDM) from the solution of the anti-Hermitian contracted Schrödinger equation (ACSE) yields accurate energies and properties for both ground and excited states. Here, we develop a more efficient method to solving the ACSE that uses second-order information to select a more optimal step towards the solution. Calculations on the ground and excited states of water, hydrogen fluoride, and conjugated π systems show that the improved ACSE algorithm is 10-20 times faster than the previous ACSE algorithm. The ACSE can treat both single- and multi-reference electron correlation with the initial 2-RDM from a complete-active-space self-consistent-field (CASSCF) calculation. Using the improved algorithm, we explore the relationship between truncation of the active space in the CASSCF calculation and the accuracy of the energy and 2-RDM from the ACSE calculation. The accuracy of the ACSE, we find, is less sensitive to the size of the active space than the accuracy of other wavefunction methods, which is useful when large active space calculations are computationally infeasible.

Modulating the electronic structure of chromophores by chemical substituents for efficient energy transfer: application to fluorone.

Strong electron correlation within a quasi-spin model of chromophores was recently shown to enhance exciton energy transfer significantly. Here we investigate how the modulation of the electronic structure of the chromophores by chemical substitution can enhance energy-transfer efficiency. Unlike previous work that does not consider the direct effect of the electronic structure on exciton dynamics, we add chemical substituents to the fluorone dimer to study the effect of electron-donating and electron-withdrawing substituents on exciton energy transfer. The exciton dynamics are studied from the solution of a quantum Liouville equation for an open system whose model Hamiltonian is derived from excited-state electronic structure calculations. Both van der Waals energies and coupling energies, arising from the Hellmann-Feynman force generated upon transferring the dimers from infinity to a finite separation, are built into the model Hamiltonian. Though these two effects are implicitly treated in dipole-based models, their explicit and separate treatment as discussed here is critical to forging the correct connection with the electronic structure calculations. We find that the addition of electron-donating substituents to the fluorone system results in an increase in exciton-transfer rates by factors ranging from 1.3-1.9. The computed oscillator strength is consistent with the recent experimental results on a larger heterodimer system containing fluorone. The oscillator strength increases with the addition of electron-donating substituents. Our results indicate that the study of chromophore networks via electronic structure will help in the future design of efficient synthetic light-harvesting systems.

Effect of molecular-orbital rotations on ground-state energies in the parametric two-electron reduced density matrix method.

Different sets of molecular orbitals and the rotations connecting them are of great significance in molecular electronic structure. Most electron correlation methods depend on a reference wave function that separates the orbitals into occupied and unoccupied spaces. Energies and properties from these methods depend upon rotations between the spaces. Some electronic structure methods, such as modified coupled electron pair approximations and the recently developed parametric two-electron reduced density matrix (2-RDM) methods [D. A. Mazziotti, Phys. Rev. Lett. 101, 253002 (2008)], also depend upon rotations between occupied orbitals and rotations between unoccupied orbitals. In this paper, we explore the sensitivity of the ground-state energies from the parametric 2-RDM method to rotations within the occupied space and within the unoccupied space. We discuss the theoretical origin of the rotational dependence and provide computational examples at both equilibrium and non-equilibrium geometries. We also study the effect of these rotations on the size extensivity of the parametric 2-RDM method. Computations show that the orbital rotations have a small effect upon the parametric 2-RDM energies in comparison to the energy differences observed between methodologies such as coupled cluster and parametric 2-RDM. Furthermore, while the 2-RDM method is rigorously size extensive in a local molecular orbital basis set, calculations reveal negligible deviations in nonlocal molecular orbital basis sets such as those from canonical Hartree-Fock calculations.

Strongly correlated barriers to rotation from parametric two-electron reduced-density-matrix methods in application to the isomerization of diazene.

Recently, parameterization of the two-electron reduced density matrix (2-RDM) has made possible the determination of electronic energies with greater accuracy and lower cost than traditional electron-pair theories including coupled cluster with single and double excitations [D. A. Mazziotti, Phys. Rev. Lett. 101, 253002 (2008)]. We examine the method's performance for strongly correlated barriers to rotation; in particular, we study two distinct pathways in the isomerization of diazene (N(2)H(2)) from cis to trans: (i) a strongly correlated rotational pathway and (ii) a moderately correlated inversion pathway. While single reference wavefunction methods predict that the rotational barrier is higher than the inversional barrier, the parametric 2-RDM method predicts that the rotational barrier is lower than the inversional barrier by 3.1 kcal/mol in the extrapolated basis set limit. The parametric 2-RDM results are in agreement with those from multireference methods including multireference perturbation theory and the solution to the anti-Hermitian contracted Schrödinger equation. We report energies, optimized structures, and natural orbital occupation numbers for three diazene minima and two transition states.

OpenMolcas: From Source Code to Insight.

In this Article we describe the OpenMolcas environment and invite the computational chemistry community to collaborate. The open-source project already includes a large number of new developments realized during the transition from the commercial MOLCAS product to the open-source platform. The paper initially describes the technical details of the new software development platform. This is followed by brief presentations of many new methods, implementations, and features of the OpenMolcas program suite. These developments include novel wave function methods such as stochastic complete active space self-consistent field, density matrix renormalization group (DMRG) methods, and hybrid multiconfigurational wave function and density functional theory models. Some of these implementations include an array of additional options and functionalities. The paper proceeds and describes developments related to explorations of potential energy surfaces. Here we present methods for the optimization of conical intersections, the simulation of adiabatic and nonadiabatic molecular dynamics, and interfaces to tools for semiclassical and quantum mechanical nuclear dynamics. Furthermore, the Article describes features unique to simulations of spectroscopic and magnetic phenomena such as the exact semiclassical description of the interaction between light and matter, various X-ray processes, magnetic circular dichroism, and properties. Finally, the paper describes a number of built-in and add-on features to support the OpenMolcas platform with postcalculation analysis and visualization, a multiscale simulation option using frozen-density embedding theory, and new electronic and muonic basis sets.

Analytic Gradients for Complete Active Space Pair-Density Functional Theory.

Analytic gradient routines are a desirable feature for quantum mechanical methods, allowing for efficient determination of equilibrium and transition state structures and several other molecular properties. In this work, we present analytical gradients for multiconfiguration pair-density functional theory (MC-PDFT) when used with a state-specific complete active space self-consistent field reference wave function. Our approach constructs a Lagrangian that is variational in all wave function parameters. We find that MC-PDFT locates equilibrium geometries for several small- to medium-sized organic molecules that are similar to those located by complete active space second-order perturbation theory but that are obtained with decreased computational cost.