Ab initio quasirelativistic calculations on electronic transitions in ICl by the multireference many‐body perturbation theory

A quasirelativistic perturbative method of ab initio calculations on ground and excited molecular electronic states and transition properties within the relativistic effective core potential approximation is presented and discussed. The method is based on the construction of a state‐selective many‐electron effective Hamiltonian in the model space spanned by an appropriate set of Slater determinants by means of the second‐order many‐body multireference perturbation theory. The neglect of effective spin–orbit interactions outside of the model space allows the exploitation of relatively high nonrelativistic symmetry during the evaluation of perturbative corrections and therefore dramatic reduction of the cost of computations without any contraction of the model‐space functions. One‐electron transition properties are evaluated via the perturbative construction of spin‐free transition density matrices. Illustrative calculations on the X0⁺ ? A1, B0⁺, and (ii)1 transitions in the ICl molecule are reported. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Theoretical study on the size consistency of the second and third order energies of the multireference perturbation theory

The size consistency of the second and third order energies of the multireference perturbation theory(Chen F, Davidson E, Iwata S. Int J Quant Chem, 2002, 86: 256) is investigated theoretically with a su-per-molecular model composed of N-hydrogen molecules separated by a large distance. It is found that the two perturbation series corresponding to two Hamiltonian partitions are not size consistent at the second and third order. However, two size consistent forms are suggested for two Hamiltonian parti-tions at the second order, if some approximations to the denominators of the original second order energies are assumed.     

Quasi-degenerate perturbation theory with general multiconfiguration self-consistent field reference functions

The quasi-degenerate perturbation theory (QDPT) with complete active space (CAS) self-consistent field (SCF) reference functions is extended to the general multiconfiguration (MC) SCF references functions case. A computational scheme that utilizes both diagrammatic and sum-over-states approaches is presented. The second-order effective Hamiltonian is computed for the external intermediate configurations (including virtual or/and core orbitals) by the diagrammatic approach and for internal intermediate configurations (including only active orbitals) by the configuration interaction matrix-based sum-over-states approach. The method is tested on the calculations of excitation energies of H(2)O, potential energy curves of LiF, and valence excitation energies of H(2)CO. The results show that the present method yields very close results to the corresponding CAS-SCF reference QDPT results and the available experimental values. The deviations from CAS-SCF reference QDPT values are less than 0.1 eV on the average for the excitation energies of H(2)O and less than 1 kcal/mol for the potential energy curves of LiF. In the calculation of the valence excited energies of H(2)CO, the maximum deviation from available experimental values is 0.28 eV.     

A new size extensive multireference perturbation theory

A new multireference perturbation series is derived based on the Rayleigh–Schrödinger perturbation theory. It is orbitally invariant. Its computational cost is comparable to the single reference Møller–Plesset perturbation theory. It is demonstrated numerically that the present multireference second‐ and third‐order energies are size extensive by two types of supermolecules composed of H₂ and BH monomers. Spectroscopic constants of as well as the ground state energies of H₂O, NH₂, and CH₂ at three bond lengths have been calculated with the second multireference perturbation theory. The dissociation behaviors of CH₄ and HF have also been investigated. Comparisons with other approximate theoretical models as well as the experimental data have been carried out to show their relative performances. © 2013 Wiley Periodicals, Inc.     

Multireference perturbation theory with optimized partitioning. II. Applications to molecular systems

The second‐order multireference perturbation theory using an optimized partitioning, denoted as MROPT(2), is applied to calculations of various molecular properties—excitation energies, spectroscopic parameters, and potential energy curves—for five molecules: ethylene, butadiene, benzene, N₂, and O₂. The calculated results are compared with those obtained with second‐ and third‐order multireference perturbation theory using the traditional partitioning techniques. We also give results from computations using the multireference configuration interaction (MRCI) method. The presented results show very close resemblance between the new method and MRCI with renormalized Davidson correction. The accuracy of the new method is good and is comparable to that of second‐order multireference perturbation theory using Møller‐Plesset partitioning. © 2003 Wiley Periodicals, Inc. J Comput Chem 24: 1390–1400, 2003     

Multireference perturbation theory with improved virtual orbitals for radicals: More degeneracies,more problems

IVO-SSMRPT is an affordable and accurate type of state-specific multireference perturbation (SSMRPT) theory that adds dynamic correlation energy to improved virtual orbital (IVO) complete active space configuration interaction (CASCI) wave functions using a single-root parametrization of multi-root Hilbert-space ansatz. We applied it to many chemically important di- and tri-radicals to analyze the geometries and electronic properties of spectroscopic interest for both closed- and open-shell singlet- and nonsinglet ground as well as excited states. We observed that IVO-SSMRPT identifies optimized geometries, splitting between multiplets and frequencies for several radicals that are similar to those displayed by current generation state-of-the-art methods but with admiringly decreased computational effort. This study illustrates the importance of having an accurate treatment of both nondynamical and dynamical correlation effects when examining multiradical species. Chemically and spectroscopically relevant answers can be obtained using our computationally tractable method. Our method will be a serviceable avenue for portraying open-shell interactions in other radicals.     

Diagrammatic formulation of the second‐order many‐body multipartitioning perturbation theory

The second‐order multireference perturbation theory employing multiple partitioning of the many‐electron Hamiltonian into a zero‐order part and a perturbation is formulated in terms of many‐body diagrams. The essential difference from the standard diagrammatic technique of Hose and Kaldor concerns the rules of evaluation of energy denominators which take into account the dependence of the Hamiltonian partitioning on the bra and ket determinantal vectors of a given matrix element, as well as the presence of several two‐particle terms in zero‐order operators. The novel formulation naturally gives rise to a “sum‐over‐orbital” procedure of correlation calculations on molecular electronic states, particularly efficient in treating the problems with large number of correlated electrons and extensive one‐electron bases. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 73: 395–401, 1999     

Multiconfigurational SCF approach for high-symmetry molecules and its applications to fullerene trianion

A symmetry-adapted multiconfiguration self-consistent field (MC SCF) approach aimed at calculations of high-symmetry molecules is proposed. The self-consistency procedure applicable to the molecular terms of any symmetry and multiplicity is developed. It holds the symmetry transformation properties of varied molecular orbitals, thus taking advantage of the relationships within the set of two-electron integrals through molecular invariants. For orbital optimization, a unified coupling operator is constructed on the basis of the pseudosecular method providing for efficient convergence to energy minimum. Based on the group-theory technique, computer codes have been developed for straightforward determination of the invariant expansions for two-electron integrals and configuration interaction (CI) matrix elements. Calculated in this way, the expansion coefficients are presented for the three-electron states that originate from joint t_1u and t_1g shells of an icosahedral fullerene C₆₀, the case important for the calculations of anion C₆₀³⁻ representing the charge state of the fullerene molecule in the superconducting ionic solids K₃C₆₀ or Rb₃C₆₀. The results of MC SCF calculations for lowest quasi-π-electronic states of C₆₀³⁻ are discussed. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 68: 293–304, 1998     

A multireference perturbation theory study on the vertical electronic spectrum of thiophene

The vertical electronic spectrum of the thiophene molecule is investigated by means of second and third order multireference perturbation theory (NEVPT). Single-state and quasi-degenerate NEVPT calculations of more than 25 singlet excited states have been performed. The study is addressed to the theoretical characterization of the four lowest-energy valence states, as well as the 3s, 3p and 3d Rydberg states. In addition, the excitation energies of two and valence states are also reported. For almost all the excited states, coupled cluster calculations (CCSD and CCSDR(3)) have been also carried out, using the same geometry and basis set used for the NEVPT ones, in order to make the comparison between the results of the two methods meaningful. A remarkable accordance between the NEVPT and CC excitation energies is found. The present results, over all, confirm the experimental assignments but, above all, represent an important contribution to the assignments of some low-energy states, valence and Rydberg, for which a firm interpretation is not available in the literature.     

Generalized perturbation theory: Quality of the first-order wave function

A recently proposed version of generalized perturbation theory, in which the whole energy correction is taken care of in second order, is investigated with respect to the quality of its first-order wave function. It is demonstrated that the overlap of the wave function generated in this procedure with the exact solution is in most cases much closer to unity than those of the Rayleigh-Schroedinger or Brillouin-Wigner perturbation theories. Certain approximations, by means of which realistic systems become amenable to investigation within the presently discussed framework, are studied.Based on a section of a thesis to be submitted by N. M. to the senate of the Technion — Israel Institute of Technology, in partial fulfilment of the requirements for the D.Sc. degree, and presented in the Second International Congress of Quantum Chemistry, New Orleans, April 1976.     

Unitary group adapted state specific multireference perturbation theory: Formulation and pilot applications

We present here a comprehensive account of the formulation and pilot applications of the second‐order perturbative analogue of the recently proposed unitary group adapted state‐specific multireference coupled cluster theory (UGA‐SSMRCC), which we call as the UGA‐SSMRPT2. We also discuss the essential similarities and differences between the UGA‐SSMRPT2 and the allied SA‐SSMRPT2. Our theory, like its parent UGA‐SSMRCC formalism, is size‐extensive. However, because of the noninvariance of the theory with respect to the transformation among the active orbitals, it requires the use of localized orbitals to ensure size‐consistency. We have demonstrated the performance of the formalism with a set of pilot applications, exploring (a) the accuracy of the potential energy surface (PES) of a set of small prototypical difficult molecules in their various low‐lying states, using natural, pseudocanonical and localized orbitals and compared the respective nonparallelity errors (NPE) and the mean average deviations (MAD) vis‐a‐vis the full CI results with the same basis; (b) the efficacy of localized active orbitals to ensure and demonstrate manifest size‐consistency with respect to fragmentation. We found that natural orbitals lead to the best overall PES, as evidenced by the NPE and MAD values. The MRMP2 results for individual states and of the MCQDPT2 for multiple states displaying avoided curve crossings are uniformly poorer as compared with the UGA‐SSMRPT2 results. The striking aspect of the size‐consistency check is the complete insensitivity of the sum of fragment energies with given fragment spin‐multiplicities, which are obtained as the asymptotic limit of super‐molecules with different coupled spins. © 2015 Wiley Periodicals, Inc.     

Theoretical study of molecular conduction: I. Effective Green's function based on perturbation theory

There have been many experimental and theoretical studies on molecular conduction, as it is a fundamental parameter in the study of molecular‐scale electronics. We have investigated the features of molecular conduction using a Green's function method, which has often been used to solve problems in quantum transport and is also effective in elucidating electron transport in molecules. We have obtained the novel effective Green's functions, including the first‐order energy corrections, by accommodating the self‐energy of the electrodes as perturbation terms. Although these approximate Green's functions only provide information on the first‐order energy corrections, they can involve the elementary properties of molecular conduction. We propose a scheme for the analysis of the relations between molecular orbitals and their roles in molecular conduction and present analytical calculations for normal and cyclic polyenes. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Semiclassical self-consistent field perturbation theory for the hydrogen atom in a magnetic field

A recently developed perturbation theory for solving self-consistent field equations is applied to the hydrogen atom in a strong magnetic field. This system has been extensively studied using other methods and is therefore a good test case for the new method. The perturbation theory yields summable large-order expansions. The accuracy of the self-consistent field approximation varies according to field strength and quantum state but is often higher than the accuracy from adiabatic approximations. A new derivation is presented for the asymptotic adiabatic approximation, the most useful of the adiabatic approaches. This derivation uses semiclassical perturbation theory without invoking an adiabatic hypothesis. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 183–192, 1998     

New perspectives in multireference perturbation theory: the n-electron valence state approach

The n-electron valence state perturbation theory (NEVPT) is a form of multireference perturbation theory which is based on a zero order reference wavefunction of CAS-CI type (complete active space configuration interaction) and which is characterized by the utilization of correction functions (zero order wavefunctions external to the CAS) of multireference nature, obtained through the diagonalization of a suitable two-electron model Hamiltonian (Dyall’s Hamiltonian) in some well defined determinant spaces. A review of the NEVPT approach is presented, starting from the original second order state-specific formulation, going through the quasidegenerate multi-state extension and arriving at the recent implementations of the third order in the energy and of the internally contracted configuration interaction. The chief properties of NEVPT—size consistence and absence of intruder states—are analyzed. Finally, an application concerning the calculation of the vertical spectrum of the biologically important free base porphin molecule, is presented.     

Testing theory beyond molecular structure: Electron density distributions of complex molecules

The electron density distribution (EDD) of a molecular system can be determined experimentally from elaborate X‐ray diffraction measurements or calculated with quantum mechanical methods: This provides a unique opportunity for mutual validation of the experimental and theoretical methods—a validation that goes far beyond comparison of molecular structures. Two examples of complex molecular systems of biologic relevance are presented. The first is the cocrystallized complex of betaine, imidazole, and picric acid, 1, which is a 75‐atom molecular complex serving as a model for the active site in the serine proteases class of enzymes, the so‐called catalytic triad. For 1 the experimental charge density was determined by combined modeling of single crystal synchrotron X‐ray and neutron diffraction data measured at 28(1) K, and it is compared with ab initio theoretical calculations at the B3LYP/6‐311G(d,p) level of theory. Overall, the agreement is good, but in one strong N? H? O hydrogen bond clear differences are observed. The second example concerns the EDD of the mixed valence trinuclear oxo‐centered iron carboxylate, [Fe₃O(OOCC(CH₃)₃)₆(NC₅H₅)₃], 2. This molecule contains 133 atoms (542 electrons) including three open‐shell iron atoms, and the experimental investigation is based on synchrotron X‐ray diffraction data. Calculations in the experimental geometry at the commonly used UB3LYP/LanL2DZ level of theory are not able to reproduce a number of experimentally observed electron density features. In particular, the sp³‐like distribution on the central oxygen atom and the electron deformations on the iron centers are at variance with experiment. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2004     

Local electric dipole moments: A generalized approach

We present an approach for calculating local electric dipole moments for fragments of molecular or supramolecular systems. This is important for understanding chemical gating and solvent effects in nanoelectronics, atomic force microscopy, and intensities in infrared spectroscopy. Owing to the nonzero partial charge of most fragments, “naively” defined local dipole moments are origin‐dependent. Inspired by previous work based on Bader's atoms‐in‐molecules (AIM) partitioning, we derive a definition of fragment dipole moments which achieves origin‐independence by relying on internal reference points. Instead of bond critical points (BCPs) as in existing approaches, we use as few reference points as possible, which are located between the fragment and the remainder(s) of the system and may be chosen based on chemical intuition. This allows our approach to be used with AIM implementations that circumvent the calculation of critical points for reasons of computational efficiency, for cases where no BCPs are found due to large interfragment distances, and with local partitioning schemes other than AIM which do not provide BCPs. It is applicable to both covalently and noncovalently bound systems. © 2016 Wiley Periodicals, Inc.     

Biphenyl: A stress tensor and vector‐based perspective explored within the quantum theory of atoms in molecules

We use quantum theory of atoms in molecules (QTAIM) and the stress tensor topological approaches to explain the effects of the torsion φ of the C‐C bond linking the two phenyl rings of the biphenyl molecule on a bond‐by‐bond basis using both a scalar and vector‐based analysis. Using the total local energy density H( r _b), we show the favorable conditions for the formation of the controversial H–H bonding interactions for a planar biphenyl geometry. This bond‐by‐bond QTAIM analysis is found to be agreement with an earlier alternative QTAIM atom‐by‐atom approach that indicated that the H–H bonding interaction provided a locally stabilizing effect that is overwhelmed by the destabilizing role of the C‐C bond. This leads to a global destabilization of the planar biphenyl conformation compared with the twisted global minimum. In addition, the H( r _b) analysis showed that only the central torsional C‐C bond indicated a minimum for a torsion φ value coinciding with that of the conventional global energy minimum. The H–H bonding interactions are found to be topologically unstable for any torsion of the central C‐C bond away from the planar biphenyl geometry. Conversely, we demonstrate that for 0.0° < φ < 39.95° there is a resultant increase in the topological stability of the C nuclei comprising the central torsional C‐C bond. Evidence is found of the effect of the H–H bonding interactions on the torsion φ of the central C‐C bond of the biphenyl molecule in the form of the QTAIM response β of the total electronic charge density ρ( r _b). Using a vector‐based treatment of QTAIM we confirm the presence of the sharing of chemical character between adjacent bonds. In addition, we present a QTAIM interpretation of hyperconjugation and conjugation effects, the former was quantified as larger in agreement with molecular orbital (MO) theory. The stress tensor and the QTAIM H atomic basin path set areas are independently found to be new tools relevant for the incommensurate gas to solid phase transition occurring in biphenyl for a value of the torsion reaction coordinate φ ≈ 5°. © 2015 Wiley Periodicals, Inc.     

A two‐scale approach to electron correlation in multiconfigurational perturbation theory

We present a new approach for the calculation of dynamic electron correlation effects in large molecular systems using multiconfigurational second‐order perturbation theory (CASPT2). The method is restricted to cases where partitioning of the molecular system into an active site and an environment is meaningful. Only dynamic correlation effects derived from orbitals extending over the active site are included at the CASPT2 level of theory, whereas the correlation effects of the environment are retrieved at lower computational costs. For sufficiently large systems, the small errors introduced by this approximation are contrasted by the substantial savings in both storage and computational demands compared to the full CASPT2 calculation. Provided that static correlation effects are correctly taken into account for the whole system, the proposed scheme represent a hierarchical approach to the electron correlation problem, where two molecular scales are treated each by means of the most suitable level of theory. © 2014 Wiley Periodicals, Inc.     

Second-order atomic Fukui indices from the electron-pair density in the framework of the atoms in molecules theory

Atomic Fukui indices, which are obtained from the electron density, have been previously shown to be useful in predicting which atoms in a molecule are most likely to suffer nucleophilic, electrophilic, or radicalary attacks. Here, we present a second-order generalization of these indices based on the electron pair density. We show how second-order atomic Fukui indices can be used to analyze the effects of electron loss or gain in several molecules from an electron pair point of view. Further, these indices also highlight which atoms or pairs of atoms are more likely to suffer nucleophilic, electrophilic, or radical attacks. In conclusion, second-order indices can complement first-order ones by affording relevant information on molecular reactivity from an electron pair perspective.     

Second order of perturbation theory correction to the radial distribution function of a liquid with the interaction potential of hard spheres plus a square-well

A liquid with the interaction potential of hard spheres plus a square-well is analyzed using the Monte-Carlo technique. Numerical results for the perturbation theory series over a square-well potential are obtained in the form of the Barker and Henderson discrete representation. Approximating expressions for the correction to a liquid radial distribution function in the second order of perturbation theory are presented. The obtained results allow us to define this correction with a root-mean-square deviation of about 0.007. It is shown that the given approach provides a complete calculation in the second order of perturbation theory, and also the determination of the third order correction to the free energy for a liquid interacting with the potential of the Lennard-Jones type.