Combining the Complete Active Space Self-Consistent Field Method and the Full Configuration Interaction Quantum Monte Carlo within a Super-CI Framework, with Application to Challenging Metal-Porphyrins

A novel stochastic Complete Active Space Self-Consistent Field (CASSCF) method has been developed and implemented in the Molcas software package. A two-step procedure is used, in which the CAS configuration interaction secular equations are solved stochastically with the Full Configuration Interaction Quantum Monte Carlo (FCIQMC) approach, while orbital rotations are performed using an approximated form of the Super-CI method. This new method does not suffer from the strong combinatorial limitations of standard MCSCF implementations using direct schemes and can handle active spaces well in excess of those accessible to traditional CASSCF approaches. The density matrix formulation of the Super-CI method makes this step independent of the size of the CI expansion, depending exclusively on one- and two-body density matrices with indices restricted to the relatively small number of active orbitals. No <i>sigma</i> vectors need to be stored in memory for the FCIQMC eigensolvera substantial gain in comparison to implementations using the Davidson method, which require three or more vectors of the size of the CI expansion. Further, no orbital Hessian is computed, circumventing limitations on basis set expansions. Like the parent FCIQMC method, the present technique is scalable on massively parallel architectures. We present in this report the method and its application to the free-base porphyrin, Mg­(II) porphyrin, and Fe­(II) porphyrin. In the present study, active spaces up to 32 electrons and 29 orbitals in orbital expansions containing up to 916 contracted functions are treated with modest computational resources. Results are quite promising even without accounting for the correlation outside the active space. The systems here presented clearly demonstrate that large CASSCF calculations are possible via FCIQMC-CASSCF without limitations on basis set size

Second-Order Perturbation Theory for Generalized Active Space Self-Consistent-Field Wave Functions

A multireference second-order perturbation theory approach based on the generalized active space self-consistent-field (GASSCF) wave function is presented. Compared with the complete active space (CAS) and restricted active space (RAS) wave functions, GAS wave functions are more flexible and can employ larger active spaces and/or different truncations of the configuration interaction expansion. With GASSCF, one can explore chemical systems that are not affordable with either CASSCF or RASSCF. Perturbation theory to second order on top of GAS wave functions (GASPT2) has been implemented to recover the remaining electron correlation. The method has been benchmarked by computing the chromium dimer ground-state potential energy curve. These calculations show that GASPT2 gives results similar to CASPT2 even with a configuration interaction expansion much smaller than the corresponding CAS expansion

SplitGAS Method for Strong Correlation and the Challenging Case of Cr₂

A new multiconfigurational quantum chemical method, SplitGAS, is presented. The configuration interaction expansion, generated from a generalized active space, GAS, wave function is split in two parts, a principal part containing the most relevant configurations and an extended part containing less relevant, but not negligible, configurations. The partition is based on an orbital criterion. The SplitGAS method has been employed to study the HF, N₂, and Cr₂ molecules. The results on these systems, especially on the challenging, multiconfigurational Cr₂ molecule, are satisfactory. While SplitGAS is comparable with the GASSCF method in terms of memory requirements, it performs better than the complete active space method followed by second-order perturbation theory, CASPT2, in terms of equilibrium bond length, dissociation energy, and vibrational properties

Systematic Expansion of Active Spaces beyond the CASSCF Limit: A GASSCF/SplitGAS Benchmark Study

The applicability and accuracy of the generalized active space self-consistent field, (GASSCF), and (SplitGAS) methods are presented. The GASSCF method enables the exploration of larger active spaces than with the conventional complete active space SCF, (CASSCF), by fragmentation of a large space into subspaces and by controlling the interspace excitations. In the SplitGAS method, the GAS configuration interaction, CI, expansion is further partitioned in two parts: the principal, which includes the most important configuration state functions, and an extended, containing less relevant but not negligible ones. An effective Hamiltonian is then generated, with the extended part acting as a perturbation to the principal space. Excitation energies of ozone, furan, pyrrole, nickel dioxide, and copper tetrachloride dianion are reported. Various partitioning schemes of the GASSCF and SplitGAS CI expansions are considered and compared with the complete active space followed by second-order perturbation theory, (CASPT2), and multireference CI method, (MRCI), or available experimental data. General guidelines for the optimum applicability of these methods are discussed together with their current limitations

Correction to Multiconfiguration Pair-Density Functional Theory: Barrier Heights and Main Group and Transition Metal Energetics

Correction to Multiconfiguration Pair-Density Functional Theory: Barrier Heights and Main Group and Transition Metal Energetic

Multiconfiguration Pair-Density Functional Theory: Barrier Heights and Main Group and Transition Metal Energetics

Kohn–Sham density functional theory, resting on the representation of the electronic density and kinetic energy by a single Slater determinant, has revolutionized chemistry, but for open-shell systems, the Kohn–Sham Slater determinant has the wrong symmetry properties as compared to an accurate wave function. We have recently proposed a theory, called multiconfiguration pair-density functional theory (MC-PDFT), in which the electronic kinetic energy and classical Coulomb energy are calculated from a multiconfiguration wave function with the correct symmetry properties, and the rest of the energy is calculated from a density functional, called the on-top density functional, that depends on the density and the on-top pair density calculated from this wave function. We also proposed a simple way to approximate the on-top density functional by translation of Kohn–Sham exchange-correlation functionals. The method is much less expensive than other post-SCF methods for calculating the dynamical correlation energy starting with a multiconfiguration self-consistent-field wave function as the reference wave function, and initial tests of the theory were quite encouraging. Here, we provide a broader test of the theory by applying it to bond energies of main-group molecules and transition metal complexes, barrier heights and reaction energies for diverse chemical reactions, proton affinities, and the water dimerization energy. Averaged over 56 data points, the mean unsigned error is 3.2 kcal/mol for MC-PDFT, as compared to 6.9 kcal/mol for Kohn–Sham theory with a comparable density functional. MC-PDFT is more accurate on average than complete active space second-order perturbation theory (CASPT2) for main-group small-molecule bond energies, alkyl bond dissociation energies, transition-metal–ligand bond energies, proton affinities, and the water dimerization energy

Correction to Multiconfiguration Pair-Density Functional Theory

Correction to Multiconfiguration Pair-Density Functional Theor

Multiconfiguration Pair-Density Functional Theory

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, <i>c</i>-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

Oxidative Stretching of Metal–Metal Bonds to Their Limits

Oxidation of quadruply bonded Cr₂(dpa)₄, Mo₂(dpa)₄, MoW­(dpa)₄, and W₂(dpa)₄ (dpa = 2,2′-dipyridylamido) with 2 equiv of silver­(I) triflate or ferrocenium triflate results in the formation of the two-electron-oxidized products [Cr₂(dpa)₄]²⁺ (<b>1</b>), [Mo₂(dpa)₄]²⁺ (<b>2</b>), [MoW­(dpa)₄]²⁺ (<b>3</b>), and [W₂(dpa)₄]²⁺ (<b>4</b>). Additional two-electron oxidation and oxygen atom transfer by <i>m</i>-chloroperoxybenzoic acid results in the formation of the corresponding metal–oxo compounds [Mo₂O­(dpa)₄]²⁺ (<b>5</b>), [WMoO­(dpa)₄]²⁺ (<b>6</b>), and [W₂O­(dpa)₄]²⁺ (<b>7</b>), which feature an unusual linear M···MO structure. Crystallographic studies of the two-electron-oxidized products <b>2</b>, <b>3</b>, and <b>4</b>, which have the appropriate number of orbitals and electrons to form metal–metal triple bonds, show bond distances much longer (by >0.5 Å) than those in established triply bonded compounds, but these compounds are nonetheless diamagnetic. In contrast, the Cr–Cr bond is completely severed in <b>1</b>, and the resulting two isolated Cr³⁺ magnetic centers couple antiferromagnetically with <i>J</i>/<i>k</i>_B= −108(3) K [−75(2) cm^–1], as determined by modeling of the temperature dependence of the magnetic susceptibility. Density functional theory (DFT) and multiconfigurational methods (CASSCF/CASPT2) provide support for “stretched” and weak metal–metal triple bonds in <b>2</b>, <b>3</b>, and <b>4</b>. The metal–metal distances in the metal–oxo compounds <b>5</b>, <b>6</b>, and <b>7</b> are elongated beyond the single-bond covalent radii of the metal atoms. DFT and CASSCF/CASPT2 calculations suggest that the metal atoms have minimal interaction; the electronic structure of these complexes is used to rationalize their multielectron redox reactivity

Computational Insights into Uranium Complexes Supported by Redox-Active α-Diimine Ligands

The electronic structures of two uranium compounds supported by redox-active α-diimine ligands, (^MesDAB^Me)₂U­(THF) (<b>1</b>) and Cp₂U­(^MesDAB^Me) (<b>2</b>) (^MesDAB^Me = [ArNC­(Me)­C­(Me)NAr]; Ar = 2,4,6-trimethylphenyl (Mes)), have been investigated using both density functional theory and multiconfigurational self-consistent field methods. Results from these studies have established that both uranium centers are tetravalent, that the ligands are reduced by two electrons, and that the ground states of these molecules are triplets. Energetically low-lying singlet states are accessible, and some transitions to these states are visible in the electronic absorption spectrum