Multiconfigurational Wave Functions Research Articles

Atomic data for doubly-excited states [formula omitted] of He-like ions and [formula omitted] of Li-like ions with [formula omitted] and [formula omitted

The wavelengths and radiative transition probabilities for transitions 2lnl′→1snl″, 2lnl′→1s2l″, 1s2lnl′→1s2nl″, 1s2lnl′→1s22l″, and the autoionization decay probabilities for doubly-excited states 2lnl′, 1s2lnl′ were calculated in He-like and Li-like ions with atomic numbers Z=6–36 for the principal quantum number n=2,3 and the orbital quantum one l=0–2. The calculations were carried out by means of the MZ code based on the Z-expansion method. Relativistic corrections were taken into account within the framework of the Breit operator. The main difference with previous calculations by MZ code consists in accounting for the first order corrections in powers of 1/Z, corresponding to the screening effects, in calculations of autoionization rates. New data for comparatively large rates are of about 20%–50% less as compared to previous ones and are in an agreement within 10% with the results of calculations made by the methods based on multi-configuration wave functions with non-relativistic and relativistic orbitals. Some refinements and corrections concerning the energies and radiative transition probabilities were also introduced in the MZ code. In this paper the main formulas used in a modified MZ code and the data needed for the description of dielectronic satellites with n=2,3 are given.

Role of the Imide Axial Ligand in the Spin and Oxidation State of Manganese Corrole and Corrolazine Complexes.

Electronic structure calculations have been performed on four different Mn corrole and corrolazine complexes to clarify the role of the imide axial ligand on the relative stability of the different spin states and the stabilization of the high-valent Mn ion in these complexes. Multiconfigurational perturbation theory energy calculations on the DFT-optimized geometries show that all complexes have a singlet ground state except the complex with the strongest electron-withdrawing substituent on the imide axial ligand, which is found to have a triplet ground state. The analysis of the σ and π interaction between the metal and imide ligand shows that this spin crossover is caused by a subtle interplay of geometrical factors (Mn-N distance and coordination angle) and the electron-withdrawing character of the substituent on the imide, which reduces the electron donation to the metal center. The analysis of the multiconfigurational wave functions reveals that the formally Mn(V) ion is stabilized by an important electron transfer from both the equatorial corrole/corrolazine ligand and the axial imide. The macrocycle donates roughly half an electron, being somewhere between the closed-shell trianionic and the dianionic radical form. The imide ligand transfers 2.5 electrons to the metal center, resulting in an effective d-electron count close to five in all complexes.

Multiconfiguration Pair-Density Functional Theory Is as Accurate as CASPT2 for Electronic Excitation.

A correct description of electronically excited states is critical to the interpretation of visible-ultraviolet spectra, photochemical reactions, and excited-state charge-transfer processes in chemical systems. We have recently proposed a theory called multiconfiguration pair-density functional theory (MC-PDFT), which is based on a combination of multiconfiguration wave function theory and a new kind of density functional called an on-top density functional. Here, we show that MC-PDFT with a first-generation on-top density functional performs as well as CASPT2 for an organic chemistry database including valence, Rydberg, and charge-transfer excitations. The results are very encouraging for practical applications.

Recasting wave functions into valence bond structures: A simple projection method to describe excited states.

A method is proposed to obtain coefficients and weights of valence bond (VB) determinants from multi configurational wave functions. This reading of the wave functions can apply to ground states as well as excited states. The method is based on projection operators. Both energetic and overlap-based criteria are used to assess the quality of the resulting VB wave function. The approach gives a simple access to a VB rewriting for low-lying states, and it is applied to the allyl cation, to the allyl radical and to the ethene (notably to the V-state). For these states, large overlap between VB and multi reference wave functions are easily obtained. The approach proves to be useful to propose an interpretation of the nature of the V-state of ethene.

Molcas 8: New capabilities for multiconfigurational quantum chemical calculations across the periodic table.

In this report, we summarize and describe the recent unique updates and additions to the Molcas quantum chemistry program suite as contained in release version 8. These updates include natural and spin orbitals for studies of magnetic properties, local and linear scaling methods for the Douglas-Kroll-Hess transformation, the generalized active space concept in MCSCF methods, a combination of multiconfigurational wave functions with density functional theory in the MC-PDFT method, additional methods for computation of magnetic properties, methods for diabatization, analytical gradients of state average complete active space SCF in association with density fitting, methods for constrained fragment optimization, large-scale parallel multireference configuration interaction including analytic gradients via the interface to the Columbus package, and approximations of the CASPT2 method to be used for computations of large systems. In addition, the report includes the description of a computational machinery for nonlinear optical spectroscopy through an interface to the QM/MM package Cobramm. Further, a module to run molecular dynamics simulations is added, two surface hopping algorithms are included to enable nonadiabatic calculations, and the DQ method for diabatization is added. Finally, we report on the subject of improvements with respects to alternative file options and parallelization.

Stochastic Multiconfigurational Self-Consistent Field Theory.

The multiconfigurational self-consistent field theory is considered the standard starting point for almost all multireference approaches required for strongly correlated molecular problems. The limitation of the approach is generally given by the number of strongly correlated orbitals in the molecule, since its cost will grow exponentially with this number. We present a new multiconfigurational self-consistent field approach, wherein linear determinant coefficients of a multiconfigurational wave function are optimized via the stochastic full configuration interaction quantum Monte Carlo technique at greatly reduced computational cost, with nonlinear orbital rotation parameters updated variationally based on this sampled wave function. This extends this approach to strongly correlated systems with far larger active spaces than it is possible to treat via conventional means. By comparison with this traditional approach, we demonstrate that the introduction of stochastic noise in both the determinant amplitudes and the gradient and Hessian of the orbital rotations does not preclude robust and reliable convergence of the orbital optimization. It can even improve the ability to avoid convergence to local minima in the orbital space, and therefore aid in finding variationally lower-energy solutions. We consider the effect on the convergence of the orbitals as the number of walkers and the sampling time within the active space increases, as well as the effect on the final energy and error. The scope of the new protocol is demonstrated with a study of the increasingly strongly correlated electronic structure in a series of polycyclic aromatic hydrocarbons, up to the large coronene molecule in a complete active space of 24 π electrons in 24 orbitals, requiring only modest computational resources

Electronic structures and geometries of the XF3 (X = Cl, Br, I, At) fluorides.

The potential energy surfaces of the group 17 XF3 (X = Cl, Br, I, At) fluorides have been investigated for the first time with multiconfigurational wave function theory approaches. In agreement with experiment, bent T-shaped C(2v) structures are computed for ClF3, BrF3, and IF3, while we predict that an average D(3h) structure would be experimentally observed for AtF3. Electron correlation and scalar relativistic effects strongly reduce the energy difference between the D(3h) geometry and the C(2v) one, along the XF3 series, and in the X = At case, spin-orbit coupling also slightly reduces this energy difference. AtF3 is a borderline system where the D(3h) structure becomes a minimum, i.e., the pseudo-Jahn-Teller effect is inhibited since electron correlation and scalar-relativistic effects create small energy barriers leading to the global C(2v) minima, although both types of effects interfere.

Energies, fine structures, and transitions of the core-excited sextet states 6Se,o(n) and 6Pe,o(n) (n=1–5) of B-like ions

A comprehensive theoretical study of atomic characteristics of energy levels and transitions for the core-excited 6Se,o(n) and 6Pe,o(n) (n=1–5) states of the boron isoelectronic sequence (Z=6–14) are investigated by the Rayleigh–Ritz variation method and multi-configuration interaction wavefunctions. The relativistic corrections and mass polarization effects are included by first-order perturbation theory. The configuration structures of the high-lying sextet series 6Se,o(n) and 6Pe,o(n) (n=1–5) of the B-like ions are assigned. The transition rates and wavelengths for the electric dipole transitions 6Se,o(n)—6Po,e(n) (n=1–5) of the B-like ions are calculated and compared with currently available theoretical and experimental data. Furthermore, the radiative transition rates and wavelengths for the important dipole transitions are discussed with the increase of nuclear charge number Z. The calculations will provide useful data for identification of spectral lines arising from the solar atmosphere and the experimental study in future work.

Multiconfiguration Pair-Density Functional Theory Outperforms Kohn-Sham Density Functional Theory and Multireference Perturbation Theory for Ground-State and Excited-State Charge Transfer.

The correct description of charge transfer in ground and excited states is very important for molecular interactions, photochemistry, electrochemistry, and charge transport, but it is very challenging for Kohn-Sham (KS) density functional theory (DFT). KS-DFT exchange-correlation functionals without nonlocal exchange fail to describe both ground- and excited-state charge transfer properly. We have recently proposed a theory called multiconfiguration pair-density functional theory (MC-PDFT), which is based on a combination of multiconfiguration wave function theory with a new type of density functional called an on-top density functional. Here we have used MC-PDFT to study challenging ground- and excited-state charge-transfer processes by using on-top density functionals obtained by translating KS exchange-correlation functionals. For ground-state charge transfer, MC-PDFT performs better than either the PBE exchange-correlation functional or CASPT2 wave function theory. For excited-state charge transfer, MC-PDFT (unlike KS-DFT) shows qualitatively correct behavior at long-range with great improvement in predicted excitation energies.

Selection of active spaces for multiconfigurational wavefunctions.

The efficient and accurate description of the electronic structure of strongly correlated systems is still a largely unsolved problem. The usual procedures start with a multiconfigurational (usually a Complete Active Space, CAS) wavefunction which accounts for static correlation and add dynamical correlation by perturbation theory, configuration interaction, or coupled cluster expansion. This procedure requires the correct selection of the active space. Intuitive methods are unreliable for complex systems. The inexpensive black-box unrestricted natural orbital (UNO) criterion postulates that the Unrestricted Hartree-Fock (UHF) charge natural orbitals with fractional occupancy (e.g., between 0.02 and 1.98) constitute the active space. UNOs generally approximate the CAS orbitals so well that the orbital optimization in CAS Self-Consistent Field (CASSCF) may be omitted, resulting in the inexpensive UNO-CAS method. A rigorous testing of the UNO criterion requires comparison with approximate full configuration interaction wavefunctions. This became feasible with the advent of Density Matrix Renormalization Group (DMRG) methods which can approximate highly correlated wavefunctions at affordable cost. We have compared active orbital occupancies in UNO-CAS and CASSCF calculations with DMRG in a number of strongly correlated molecules: compounds of electronegative atoms (F2, ozone, and NO2), polyenes, aromatic molecules (naphthalene, azulene, anthracene, and nitrobenzene), radicals (phenoxy and benzyl), diradicals (o-, m-, and p-benzyne), and transition metal compounds (nickel-acetylene and Cr2). The UNO criterion works well in these cases. Other symmetry breaking solutions, with the possible exception of spatial symmetry, do not appear to be essential to generate the correct active space. In the case of multiple UHF solutions, the natural orbitals of the average UHF density should be used. The problems of the UNO criterion and their potential solutions are discussed: finding the UHF solutions, discontinuities on potential energy surfaces, and inclusion of dynamical electron correlation and generalization to excited states.

Cerium oxidation state and covalent 4f-orbital contributions in the ground state of bis(η8-pentalene)cerium

The sandwich complex bis(η8-pentalene)cerium Ce(C8H6)2 was investigated by relativistic quantum chemical ab initio methods and the nature of the interaction between the Ce 4f and ligand π orbitals was analyzed. It is shown for a multi-configurational ground state wavefunction of the complete active space variety that by means of an orbital rotation in the space of the strongly interacting orbitals it is possible to generate nearly pure Ce 4f and ligand π orbitals without changing the total energy. These orbitals are occupied by one electron each, leading to an open-shell 4f1π1 ground state configuration. The admixture of the energetically higher closed-shell 4f0π2 configuration leads to a singlet ground state, making the complex another example for a molecular analog of a Ce(III)-based lattice Kondo system. A further analysis using local spin and occupation number fluctuations in a two-electron two-orbital description reveals that the Ce 4f-ligand π interaction corresponds to a significantly stretched covalent bond: orbital and configuration mixing coefficients as well as local spin and occupation number fluctuations are very similar to those of hydrogen dimer H2 at an interatomic distance of about two times the equilibrium distance.

The Electronic Structure of the Al3(-) Anion: Is it Aromatic?

Multiconfigurational high-level electronic structure calculations show that the Al3(-) ring-like cluster anion has three close low-lying electronic states of different spin, all of them having strong multiconfigurational character. The aromaticity of the cluster has, therefore, been studied by means of total electron delocalization and normalized multicenter electron delocalization indices evaluated from the multiconfigurational wave functions of each state. The lowest-lying singlet and triplet states are found to be highly aromatic, whereas the next lowest-lying state, the quintet state, has much less, though non-negligible, aromatic character.

Configuration interaction singles natural orbitals: an orbital basis for an efficient and size intensive multireference description of electronic excited states.

Multireference quantum chemical methods, such as the complete active space self-consistent field (CASSCF) method, have long been the state of the art for computing regions of potential energy surfaces (PESs) where complex, multiconfigurational wavefunctions are required, such as near conical intersections. Herein, we present a computationally efficient alternative to the widely used CASSCF method based on a complete active space configuration interaction (CASCI) expansion built from the state-averaged natural orbitals of configuration interaction singles calculations (CISNOs). This CISNO-CASCI approach is shown to predict vertical excitation energies of molecules with closed-shell ground states similar to those predicted by state averaged (SA)-CASSCF in many cases and to provide an excellent reference for a perturbative treatment of dynamic electron correlation. Absolute energies computed at the CISNO-CASCI level are found to be variationally superior, on average, to other CASCI methods. Unlike SA-CASSCF, CISNO-CASCI provides vertical excitation energies which are both size intensive and size consistent, thus suggesting that CISNO-CASCI would be preferable to SA-CASSCF for the study of systems with multiple excitable centers. The fact that SA-CASSCF and some other CASCI methods do not provide a size intensive/consistent description of excited states is attributed to changes in the orbitals that occur upon introduction of non-interacting subsystems. Finally, CISNO-CASCI is found to provide a suitable description of the PES surrounding a biradicaloid conical intersection in ethylene.

Probing deactivation pathways of DNA nucleobases by two-dimensional electronic spectroscopy: first principles simulations.

The SOS//QM/MM [Rivalta et al., Int. J. Quant. Chem., 2014, 114, 85] method consists of an arsenal of computational tools allowing accurate simulation of one-dimensional (1D) and bi-dimensional (2D) electronic spectra of monomeric and dimeric systems with unprecedented details and accuracy. Prominent features like doubly excited local and excimer states, accessible in multi-photon processes, as well as charge-transfer states arise naturally through the fully quantum-mechanical description of the aggregates. In this contribution the SOS//QM/MM approach is extended to simulate time-resolved 2D spectra that can be used to characterize ultrafast excited state relaxation dynamics with atomistic details. We demonstrate how critical structures on the excited state potential energy surface, obtained through state-of-the-art quantum chemical computations, can be used as snapshots of the excited state relaxation dynamics to generate spectral fingerprints for different de-excitation channels. The approach is based on high-level multi-configurational wavefunction methods combined with non-linear response theory and incorporates the effects of the solvent/environment through hybrid quantum mechanics/molecular mechanics (QM/MM) techniques. Specifically, the protocol makes use of the second-order Perturbation Theory (CASPT2) on top of Complete Active Space Self Consistent Field (CASSCF) strategy to compute the high-lying excited states that can be accessed in different 2D experimental setups. As an example, the photophysics of the stacked adenine-adenine dimer in a double-stranded DNA is modeled through 2D near-ultraviolet (NUV) spectroscopy.

Unbiased reduced density matrices and electronic properties from full configuration interaction quantum Monte Carlo.

Properties that are necessarily formulated within pure (symmetric) expectation values are difficult to calculate for projector quantum Monte Carlo approaches, but are critical in order to compute many of the important observable properties of electronic systems. Here, we investigate an approach for the sampling of unbiased reduced density matrices within the full configuration interaction quantum Monte Carlo dynamic, which requires only small computational overheads. This is achieved via an independent replica population of walkers in the dynamic, sampled alongside the original population. The resulting reduced density matrices are free from systematic error (beyond those present via constraints on the dynamic itself) and can be used to compute a variety of expectation values and properties, with rapid convergence to an exact limit. A quasi-variational energy estimate derived from these density matrices is proposed as an accurate alternative to the projected estimator for multiconfigurational wavefunctions, while its variational property could potentially lend itself to accurate extrapolation approaches in larger systems.

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.

Ab initio study of the RbSr electronic structure: potential energy curves, transition dipole moments, and permanent electric dipole moments.

Excited states and the ground state of the diatomic molecule RbSr were calculated by post Hartree-Fock molecular orbital theory up to 22000 cm(-1). We applied a multireference configuration interaction calculation based on multiconfigurational self-consistent field wave functions. Both methods made use of effective core potentials and core polarization potentials. Potential energy curves, transition dipole moments, and permanent electric dipole moments were determined for RbSr and could be compared with other recent calculations. We found a good agreement with experimental spectra, which have been obtained recently by helium nanodroplet isolation spectroscopy. For the lowest two asymptotes (Rb (5s (2)S) + Sr (5s4d (3)P°) and Rb (5p (2)P°) + Sr (5s(2) (1)S)), which exhibit a significant spin-orbit coupling, we included relativistic effects by two approaches, one applying the Breit-Pauli Hamiltonian to the multireference configuration interaction wave functions, the other combining a spin-orbit Hamiltonian and multireference configuration interaction potential energy curves. Using the results for the relativistic potential energy curves that correspond to the Rb (5s (2)S) + Sr (5s4d (3)P°) asymptote, we have simulated dispersed fluorescence spectra as they were recently measured in our lab. The comparison with experimental data allows to benchmark both methods and demonstrate that spin-orbit coupling has to be included for the lowest states of RbSr.

Ab Initio Calculation of Molecular Diffraction.

We discuss the application of ab initio X-ray diffraction (AIXRD) to the interpretation of time-resolved and static X-ray diffraction. In our approach, elastic X-ray scattering is calculated directly from the ab initio multiconfigurational wave function via a Fourier transform of the electron density, using the first Born approximation for elastic scattering. Significant gains in efficiency can be obtained by performing the required Fourier transforms analytically, making it possible to combine the calculation of ab initio X-ray diffraction with expensive quantum dynamics simulations. We show that time-resolved X-ray diffraction can detect not only changes in molecular geometry but also changes in the electronic state of a molecule. Calculations for cis-, trans-, and cyclo-butadiene, as well as benzene and 1,3-cyclohexadiene are included.

Computational determination of the dominant triplet population mechanism in photoexcited benzophenone.

In benzophenone, intersystem crossing occurs efficiently between the S1(nπ*) state and the T1 state of dominant nπ* character, leading to excited triplet states after photoexcitation. The transition mechanism between S1(nπ*) and T1 is still a matter of debate, despite several experimental studies. Quantum mechanical calculations have been performed in order to assess the relative efficiencies of previously proposed mechanisms, in particular, the direct S1→ T1 and indirect S1→ T2(ππ*) → T1 ones. Multiconfigurational wave function based methods are used to discuss the nature of the relevant states and also to determine minimum energy paths and conical intersections. It is found that the T1 state has a mixed nπ*/ππ* character and that the T2(ππ*) state acts as an intermediate state between the S1 and T1 states. This result is in line with recent experiments, which suggested a two-step kinetic model to populate the phosphorescent state after photoexcitation [Aloïse et al., J. Phys. Chem. A, 2008, 112, 224-231].

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, 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.