Multiconfigurational Self-consistent Field Calculations Research Articles

High-Resolution Spectroscopy on the Laser-Cooling CandidateLa−

The bound-bound transition from the 5d^{2}6s^{2} ^{3}F_{2}^{e} ground state to the 5d6s^{2}6p ^{3}D_{1}^{o} excited state in negative lanthanum has been proposed as a candidate for laser cooling, which has not yet been achieved for negative ions. Anion laser cooling holds the potential to allow the production of ultracold ensembles of any negatively charged species. We have studied the aforementioned transition in a beam of negative La ions by high-resolution laser spectroscopy. The center-of-gravity frequency was measured to be 96.592 80(10)THz. Seven of the nine expected hyperfine structure transitions were resolved. The observed peaks were unambiguously assigned to the predicted hyperfine transitions by a fit, confirmed by multiconfigurational self-consistent field calculations. From the determined hyperfine structure we conclude that La^{-} is a promising laser cooling candidate. Using this transition, only three laser beams would be required to repump all hyperfine levels of the ground state.

Auger decay and subsequent fragmentation pathways of ethylene followingK-shell ionization

The fragmentation pathways and dynamics of ethylene molecules after core ionization are explored using coincident measurements of the Auger electron and fragment ions by employing the cold target recoil-ion momentum spectroscopy method. The influence of several factors on the dynamics and kinematics of the dissociation is studied. These include propensity rules, ionization mechanisms, symmetry of the orbitals from which the Auger electrons originate, multiple scattering, conical intersections, interference, and possible core-hole localization for the double ionization of this polyatomic molecule. Energy correlation maps allow probing the multidimensional potential energy surfaces and, in combination with our multiconfiguration self-consistent field calculations, identifying the populated electronic states of the dissociating dication. The measured angular distributions of the Auger electrons in the molecular frame further support and augment these assignments. The deprotonation and molecular hydrogen ion elimination channels show a nearly isotropic Auger electron angular distribution with a small elongation along the direction perpendicular to the molecular axis. For the symmetric breakup the angular distributions show a clear influence of multiple scattering on the outgoing electrons. The lowest kinetic energy release feature of the symmetric breakup channel displays a fingerprint of entangled Auger and photoelectron motion in the angular emission pattern identifying this transition as an excellent candidate to probe core-hole localization at a conical intersection of a polyatomic molecule.

Polarizable QM/MM Multiconfiguration Self-Consistent Field Approach with State-Specific Corrections: Environment Effects on Cytosine Absorption Spectrum.

We present the formulation and implementation of a polarizable quantum mechanics/molecular mechanics (QM/MM) strategy to describe environment effects in multiconfiguration self-consistent field calculations. The strategy is applied to the calculation of the vertical absorption spectrum of cytosine in water. In our approach, mutual polarization of the solute and the solvent is solved self-consistently at the complete-active-space self-consistent-field (CASSCF) level, and the resulting set of charges and dipoles is used to calculate vertical excitation energies using the complete-active-space second-order perturbative (CASPT2) approach and its multistate (MS-CASPT2) variant. In order to treat multiple excited states, we converge the solvent polarization with respect to the state-averaged density of the solute. In order to obtain the final energies, however, we introduce a state-specific correction, where the solvent polarization is recomputed with the density of each state, and demonstrate that this correction brings the excitation energies closer to the values obtained with state-optimized orbitals. Comparison with PCM and nonpolarizable QM/MM calculations shows the importance of specific solute-solvent interactions and environment polarization in describing experiments. Overall, the calculated excitations for the π → π* states in water show good agreement with the experimental spectrum, whereas the n → π* appear at energies above 6 eV, approximately 1 eV higher than in the gas phase. Beyond solvents, the new method will allow studying the impact of heterogeneous biological environments in multiple excited states, as well as the treatment of multichromophoric systems where charge transfer and exciton states play important roles.

Electronic spectrum of the UO and UO(+) molecules.

Electronic theory calculations are applied to the study of the UO molecule and the UO(+) ion. Relativistic effective core potentials are used along with the accompanying valence spin-orbit operators. Polarized double-ς and triple-ς basis sets are used. Molecular orbitals are obtained from state-averaged multiconfiguration self-consistent field calculations and then used in multireference spin-orbit configuration interaction calculations with a number of millions of terms. The ground state of UO has open shells of 5f(3)7s(1), angular momentum Ω = 4, and a spin-orbit-induced avoided crossing near the equilibrium internuclear distance. Many UO excited states are studied with rotational constants, intensities, and experimental comparisons. The ground state of UO(+) is of 5f(3) nature with Ω = 9/2. Many UO(+) excited states are also studied. The open-shell nature of both UO and UO(+) leads to many low-lying excited states.

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.

A time-dependent density-functional theory and complete active space self-consistent field method study of vibronic absorption and emission spectra of coumarin.

Time-dependent density-functional theory (TD-DFT) and complete active space multiconfiguration self-consistent field (CASSCF) calculations have been used to determine equilibrium structures and vibrational frequencies of the ground state and several singlet low-lying excited states of coumarin. Vertical and adiabatic transition energies of S1, S2, and S3 have been estimated by TD-B3LYP and CASSCF/PT2. Calculations predict that the dipole-allowed S1 and S3 states have a character of (1)(ππ*), while the dipole-forbidden (1)(nπ*) state is responsible for S2. The vibronic absorption and emission spectra of coumarin have been simulated by TD-B3LYP and CASSCF calculations within the Franck-Condon approximation, respectively. The simulated vibronic spectra show good agreement with the experimental observations available, which allow us to reasonably interpret vibronic features in the S0→S1 and S0→S3 absorption and the S0←S1 emission spectra. Based on the calculated results, activity, intensity, and density of the vibronic transitions and their contribution to the experimental spectrum profile have been discussed.

Theoretical Analyses on Phosphorescent Processes in Pt(thpy)2 and Its Derivatives

Theoretical estimation of the peak wavelengths of phosphorescence was performed at the MCSCF+SOCI/SBKJC+p level of theory for several typical platinum complexes in the research field of organic-light-emitting-diodes (OLEDs), where MCSCF+SOCI is the abbreviation of multiconfiguration self-consistent field calculations followed by second-order configuration interaction calculations. The spin–orbit coupling (SOC) integrals among low-lying electronic states of different spin multiplicities were explicitly calculated within the Zeff approximation. By using these computational methods, the experimental results for peak wavelengths of phosphorescence were reasonably explained for cis-bis[2-(2′-thienyl)pyridinato-N,C3′]platinum(II) and its derivatives. The replacement of one of the 2-(2′-thienyl)pyridinate (thpy) ligands by a 2,4-pentanedionate (acac) ligand causes a blue shift of the phosphorescent peak by about 10 nm. The use of a 1,3-bis(phenyl)propane-1,3-dionate (bpp), 1,3-bis(n-methoxyphenyl)propane-1,3-dio...

Reducing the propensity for unphysical wavefunction symmetry breaking in multireference calculations of the excited states of semiconductor clusters

Unphysical spatial symmetry breaking in multiconfigurational self-consistent field calculations can lead to undesirable artifacts in the potential energy surfaces and electronic properties of molecules. Herein, we report several examples of such symmetry breaking in calculations of the excited states of small semiconductor clusters and related molecules at the state-averaged complete active space self-consistent field (SA-CASSCF) level of theory. A multireference approach is proposed to reduce its incidence: the singly excited active space complete active space configuration interaction (SEAS-CASCI) method. In SEAS-CASCI, the orbitals are determined by variationally minimizing an energy expression that does not depend on the off-diagonal Hamiltonian matrix elements which drive symmetry breaking at the SA-CASSCF level of theory. By application to several highly symmetric molecules, SEAS-CASCI is demonstrated to reduce the propensity for unphysical spatial symmetry breaking and eliminate resulting errors in the potential energy surfaces and molecular properties relative to the SA-CASSCF description. The SEAS method is also found to eliminate unphysical wavefunction distortion in asymmetric molecules. Finally, SEAS-CASCI is demonstrated to accurately describe the biradicaloid region of the potential energy surface of ethylene.

Multiconfigurational Self-Consistent Field Calculations of the Magnetically Induced Current Density Using Gauge-Including Atomic Orbitals.

Nondynamical electron correlation based on a genuine multiconfigurational theory is of considerable importance for a balanced ab initio calculation of aromatic and antiaromatic molecules either with open-shell character or quasi-degeneracy in the electronic states. Among the various aromaticity indices, the calculation of magnetically induced ring current densities (MICD) has emerged as a strong contender, providing both a qualitative and a quantitative description of the effect. We report here the first implementation of MICD at the multiconfigurational self-consistent field (MCSCF) level of theory together with example calculations. This extension makes the method applicable to systems that cannot be appropriately handled with earlier implementations based on a single-reference starting function. We present the formulation of the MCSCF MICD theory along with applications to a prototypical antiaromatic (cyclobutadiene) and an aromatic (benzyne) system, both systems that require a multiconfigurational description. We compare the MCSCF results to those obtained using Hartree-Fock and Kohn-Sham density functional theory and discuss the effects of static correlation on the aromaticity.

Cyclic and Linear NiO2: A Multireference Configuration Interaction Study

Linear (ONiO) and triangular (Ni(O2))isomers of NiO2 are investigated by multiconfiguration self-consistent field (MCSCF) and multireference configuration interaction (MRCI) calculations. For ONiO, the ground electronic term is a 1Σg+ term. The lower-lying excited terms are 3Πg, 1Πg, and 5Πu at relative energies of 0.55, 0.95, and 1.20eV, respectively. For Ni(O2), the ground electronic term is a1A1 term with an energy of 1.53 eV with respect to the ONiO ground state. Lower-lying excited terms are 5B2, 5A1, and 3B2 at 0.58, 0.62, and 0.73 eV with respect to the 1A1 state, respectively.A transition structure between the ground states of both isomers has been located with an energy of 2.76 eV above the ONiO ground state. For the fragmentation Ni(O2) → Ni + O2 the (electronic) reaction energy is estimated to 1.15 eV. The wave function based results demonstrate the failure of previous density functional investigations and emphasize the importance of a multireference treatment.

Ab initio calculations of the lowest electronic states in the CuNO system

The lowest singlet and triplet electronic levels of the A' and A" symmetry species of the neutral copper-nitrosyl (CuNO) system are calculated by ab initio methods at the multi-reference configuration interaction (MRCI) level of theory with single and double excitations, and at the coupled cluster level of theory with both perturbational (CCSD(T)) and full inclusion of triple excitations (CCSDT). Experimental data are difficult to obtain, hence the importance of carrying out calculations as accurate as possible to address the structure and dynamics of this system. This paper aims at validating a theoretical protocol to develop global potential energy surfaces for transition metal nitrosyl complexes. For the MRCI calculations, the comparison of level energies at linear structures and their values from C(2v) and C(s) symmetry restricted calculations has allowed to obtain clear settings regarding atomic basis sizes, active orbital spaces and roots obtained at the multi-configurational self-consistent field (MCSCF) level of theory. It is shown that a complete active space involving 18 valence electrons, 11 molecular orbitals and the prior determination of 12 roots in the MCSCF calculation is needed for overall qualitatively correct results from the MRCI calculations. Atomic basis sets of the valence triple-zeta type are sufficient. The present calculations yield a bound singlet A' ground state for CuNO. The CCSD(T) calculations give a quantitatively more reliable account of electronic correlation close to equilibrium, while the MRCI energies allow to ensure the qualitative assessment needed for global potential energy surfaces. Relativistic coupled cluster calculations using the Douglas-Kroll-Hess Hamiltonian yield a dissociation energy of CuNO into Cu and NO to be (59 ± 5) kJ mol(-1) ((4940 ± 400) hc cm(-1)). Favorable comparison is made with some of previous theoretical results and a few known experimental data.

Photoexcited tautomerization of vinyl alcohol to acetylaldehydevia a conical intersection from contracted Schrödinger theory

The photoexcited tautomerization of vinyl alcohol to acetylaldehyde via a conical intersection is analyzed through the direct calculation of two-electron reduced density matrices (2-RDMs) from solutions to the anti-Hermitian contracted Schrödinger equation (ACSE). The study utilizes the recent generalization of the ACSE method for the treatment of excited states [G. Gidofalvi and D. A. Mazziotti, Phys. Rev. A, 2009, 80, 022507]. We computed absolute energies of the critical points as well as various intermediate points along the ground- and excited-state potential energy surface of vinyl alcohol and acetylaldehyde. The ACSE, MCSCF, second-order multireference many-body perturbation theory (MRMP2), and various coupled cluster methods all demonstrate the existence of a family of pathways from vinyl alcohol to acetylaldehyde via a conical intersection that are monotonically decreasing in energy. The conical intersection, proposed for the first time in this paper, is both structurally and energetically similar to the ground-state transition state. We observe a relationship between conical intersections and transition states both in this paper and in our previous work on bicyclobutane's ring conical intersection [J. W. Snyder, Jr. and D. A. Mazziotti, J. Chem. Phys., 2011, 135, 024107]. To treat multireference correlation, we seeded the ACSE with an initial 2-RDM guess from a multiconfiguration self-consistent field (MCSCF) calculation. The ACSE recovers more single-reference correlation energy than MRMP2 and more multireference correlation energy than comparable single-reference wave function methods. The 2-RDMs from the ACSE nearly satisfy all necessary N-representability conditions.

Conical Intersection of the Ground and First Excited States of Water: Energies and Reduced Density Matrices from the Anti-Hermitian Contracted Schrödinger Equation

A conical intersection between the ground and first-excited states of water is computed through the direct calculation of two-electron reduced density matrices (2-RDMs) from solutions of the anti-Hermitian contracted Schrödinger equation (ACSE). This study is an extension of a previous study in which the ACSE was used to compute the energies around a conical intersection in the triplet excited states of methylene [Snyder, J. W., Jr.; Rothman, A. E.; Foley, J. J.; Mazziotti, D. A. J. Chem. Phys. 2010, 132, 154109]. We compute absolute energies of the 1(1)A' and 2(1)A' states of water (H(2)O) and the location of the conical intersection. The ACSE energies are compared to those from ab initio wave function methods. To treat multireference correlation, we seed the ACSE with an initial 2-RDM from a multiconfiguration self-consistent field (MCSCF) calculation. Unlike the situation for methylene, the two states in the vicinity of the conical intersection of water both have the same spatial symmetry. Hence, the study demonstrates the ability of the ACSE to resolve states of the same spatial symmetry that are nearly degenerate in energy. The 2-RDMs from the ACSE nearly satisfy necessary N-representability conditions. Comparison of the results from double-ζ and augmented double-ζ basis sets demonstrates the importance of augmented (or diffuse) functions for determining the location of the conical intersection.

Photoexcited conversion of gauche-1,3-butadiene to bicyclobutane via a conical intersection: Energies and reduced density matrices from the anti-Hermitian contracted Schrödinger equation

The photoexcited reaction pathway from gauche-1,3-butadiene to bicyclobutane via a conical intersection is analyzed through the direct calculation of two-electron reduced density matrices (2-RDMs) from solutions to the anti-Hermitian contracted Schrödinger equation (ACSE). The study utilizes the recent generalization of the ACSE method for the treatment of excited states [G. Gidofalvi and D. A. Mazziotti, Phys. Rev. A 80, 022507 (2009)]. We computed absolute energies of the critical points as well as various intermediate points along the ground-and excited-state potential energy surface of gauche-1,3-butadiene and bicyclobutane. To treat multi-reference correlation, we seeded the ACSE with an initial 2-RDM from a multi-configuration self-consistent field (MCSCF) calculation. The ACSE, MCSCF, and second-order multi-reference many-body perturbation theory (MRPT2) all demonstrate that there exists a family of pathways from gauche-1,3-butadiene to bicyclobutane via a conical intersection that are monotonically decreasing in energy, confirming a conjecture by Sicilia et al. [J. Phys. Chem. A 111, 2182 (2007)]. The ACSE recovers more single-reference correlation energy than MRPT2 and more multi-reference correlation energy than comparable single-reference wave function methods. The 2-RDMs from the ACSE nearly satisfy all necessary N-representability conditions.

Configuration selection within vibrational multiconfiguration self-consistent field theory: Application to bridged lithium compounds

A configuration selection scheme has been used to speed up vibrational multiconfiguration self-consistent field calculations. Deviations with respect to reference calculations were found to be negligible while yielding an acceleration of about two orders of magnitude. Its application to bridged lithium compounds (Li(2)H(2), Li(2)F(2), Li(2)O(2), and Li(3)F(3)) based on high-level coupled-cluster potential energy surfaces provides accurate vibrational transitions for all fundamental modes. The explicit inclusion of 4-mode couplings was found to be important for Li(2)H(2).

Basis set construction for molecular electronic structure theory: Natural orbital and Gauss–Slater basis for smooth pseudopotentials

A simple yet general method for constructing basis sets for molecular electronic structure calculations is presented. These basis sets consist of atomic natural orbitals from a multiconfigurational self-consistent field calculation supplemented with primitive functions, chosen such that the asymptotics are appropriate for the potential of the system. Primitives are optimized for the homonuclear diatomic molecule to produce a balanced basis set. Two general features that facilitate this basis construction are demonstrated. First, weak coupling exists between the optimal exponents of primitives with different angular momenta. Second, the optimal primitive exponents for a chosen system depend weakly on the particular level of theory employed for optimization. The explicit case considered here is a basis set appropriate for the Burkatzki-Filippi-Dolg pseudopotentials. Since these pseudopotentials are finite at nuclei and have a Coulomb tail, the recently proposed Gauss-Slater functions are the appropriate primitives. Double- and triple-zeta bases are developed for elements hydrogen through argon. These new bases offer significant gains over the corresponding Burkatzki-Filippi-Dolg bases at various levels of theory. Using a Gaussian expansion of the basis functions, these bases can be employed in any electronic structure method. Quantum Monte Carlo provides an added benefit: expansions are unnecessary since the integrals are evaluated numerically.

Performance of dynamically weighted multiconfiguration self-consistent field and spin-orbit coupling calculations of diatomic molecules of Group 14 elements

The efficacy of several multiconfiguration self-consistent field (MCSCF) methods in the subsequent spin-orbit coupling calculations was studied. Three MCSCF schemes to generate molecular orbitals were analyzed: state-specific, state-averaged, and dynamically weighted MCSCF. With Sn(2)(+) as the representative case, we show that the state-specific MCSCF orbitals lead to discontinuities in potential energy curves when avoided crossings of electronic states occur; this problem can be solved using the state-averaged or dynamically weighted MCSCF orbitals. The latter two schemes are found to give similar results when dynamic electron correlation is considered, which we calculated at the level of multiconfigurational quasidegenerate perturbation theory (MCQDPT). We employed the recently developed Douglas-Kroll spin-orbit adapted model core potential, ZFK3-DK3, and the dynamically weighted MCSCF scheme to calculate the spectroscopic constants of the mono-hydrides and compared them to the results obtained using the older set of potentials, MCP-TZP. We also showed that the MCQDPT tends to underestimate the dissociation energies of the hydrides and discussed to what extent coupled-cluster theory can be used to improve results.

Pathway analysis of super-exchange electronic couplings in electron transfer reactions using a multi-configuration self-consistent field method

We present a novel pathway analysis of super-exchange electronic couplings in electron transfer reactions using localized molecular orbitals from multi-configuration self-consistent field (MCSCF) calculations. In our analysis, the electronic coupling and the tunneling pathways can be calculated in terms of the configuration interaction (CI) Hamiltonian matrix obtained from the localized MCSCF wave function. Making use of the occupation restricted multiple active spaces (ORMAS) method can effectively produce the donor, acceptor, and intermediate configuration state functions (CSFs) and CIs among these CSFs. In order to express the electronic coupling as a sum of individual tunneling pathways contributions, we employed two perturbative methods: Löwdin projection-iteration method and higher-order super-exchange method. We applied them to anion couplings of butane-1,4-diyl and pentane-1,5-diyl. The results were (1) the electronic couplings calculated from the two perturbative methods were in reasonable agreement with those from a non-perturbative method (one-half value of the energy difference between the ground and first excited states), (2) the main tunneling pathways consisted of a small number of lower-order super-exchange pathways where bonding, anti-bonding, or extra-valence-shell orbitals were used once or twice, and (3) the interference among a huge number of higher-order super-exchange pathways significantly contributed to the overall electronic coupling, whereas each of them contributed only fractionally. Our method can adequately take into account both effects of non-dynamical electron correlation and orbital relaxation. Comparing with the analyses based on the Koopmans' theorem (ignoring both effects) and the ORMAS-CIs from frozen localized reference orbitals (ignoring the effect of orbital relaxation), we discuss these effects.

Conical intersections in triplet excited states of methylene from the anti-Hermitian contracted Schrödinger equation

A conical intersection in triplet excited states of methylene is computed through the direct calculation of two-electron reduced density matrices (2-RDMs) from solutions of the anti-Hermitian contracted Schrodinger equation (ACSE). The study synthesizes recent extensions of the ACSE method for the treatment of excited states [G. Gidofalvi and D. A. Mazziotti, Phys. Rev. A 80, 022507 (2009)] and arbitrary-spin states [A. E. Rothman, J. J. Foley, and D. A. Mazziotti, Phys. Rev. A 80, 052508 (2009)]. We compute absolute energies of the 1 (3)B(1), 1 (3)A(2), and 2 (3)B(1) states of methylene (CH(2)) and the location of the conical intersection along the 1 (3)A(2)-2 (3)B(1) potential-energy surfaces. To treat multireference correlation, we seed the ACSE with an initial 2-RDM from a multiconfiguration self-consistent field (MCSCF) calculation. The ACSE produces energies that significantly improve upon those from MCSCF and second-order multireference many-body perturbation theory, and the 2-RDMs from the ACSE nearly satisfy necessary N-representability conditions. Comparison of the results from augmented double-zeta and triple-zeta basis sets demonstrates the importance of augmented (or diffuse) functions for determining the location of the conical intersection.

Theoretical and Experimental Spectroscopy of the S2 State of CHF and CDF: Dynamically Weighted Multireference Configuration Interaction Calculations for High-Lying Electronic States

Dynamically adjusting the weights in state-averaged multiconfigurational self-consistent field (SA-MCSCF) calculations using an energy-dependent functional allows the electronic wave function to smoothly evolve across the potential energy surface (PES) and correctly preserves differing asymptotic electronic-state degeneracy patterns. We have developed a generalized dynamic weighting (GDW) method to treat high-lying electronic states. To test the method, a global PES was constructed for the S2 (B̃) state of CHF (CDF), which lies nearly 31000 cm−1 above the minimum of the ground state. The GDW method was used to produce SA-MCSCF reference states for subsequent multireference configuration interaction (MRCI) calculations, whose Davidson-corrected energies were extrapolated to the complete basis set limit. Quantum mechanical vibrational energy calculations for CDF were performed using the fitted PES, and the predicted energy levels are in excellent agreement with an extensive set of experimentally determined (optical−optical double resonance) levels, with a mean unsigned error of only 12 cm−1.