Integral invariants for high-symmetry open-shell molecules

A method for calculating the energies of electronic states arising from a degenerate open shellγ ^N in terms of integral invariants H^k(γ,γ) is presented. The calculation proceeds from expansions for the electron repulsion integrals 〈mm/nn〉 on degenerate orbitals ofγ symmetry in terms of H^k(γ,γ). The energies of states for theγ ^N electronic configurations with dimγ≤3 (e^N and t^N configurations) are tabulated. Institute of Catalysis, Siberian Branch, Russian Academy of Sciences. Translated fromZhurnal Strukturnoi Khimii, Vol. 39, No. 2, pp. 183–195, March–April, 1998. This work was supported by RFFR grants No. 96-03-01167 and 96-03-34035.     

Nonstationary perturbation theory for open-shell molecules

Nonstationary perturbation theory equations have been obtained for open-shell molecules. The equations were formulated in terms of a density matrix in the MO-LCAO method. The first variant is coupled perturbation theory in the framework of the restricted Hartree-Fock method for open shells, and the second variant is variational perturbation theory for ground and excited electronic states of molecules, in which the perturbed wave function of the system is constructed in the form of a superposition of the ground and singly excited configurations composed of the Hartree-Fock orbitals of the open shell. A calculation of the Cauchy moments of the dynamic dipole polarizability of several molecules of conjugated open-shell hydrocarbons, viz., doublet states of odd alternant hydrocarbons, as well as triplet excited states and doublet states of radical ions of even alternant hydrocarbons, has been carried out in the framework of both methods.Translated from Teoreticheskaya i Éksperimental'naya Khimiya, Vol. 21, No. 1, pp. 18–27, January–February, 1985.     

Analysis of wave functions for open-shell molecules

During the past decade we have looked at several ways to track the distribution of unpaired electrons during chemical reactions and in different spin states. These methods were inspired by our previous work on singlet di-radicals where the spin density is zero yet there are clearly singly occupied orbitals. More recently we have been concerned with analysis of wave functions for single molecule magnets. This review discusses the mathematical framework by which open-shell systems can be described, in addition to methods that extract the effectively unpaired electron density, the spin state of atoms in a molecule, and other useful properties from a molecular wave function. Some of the difficulties associated with using broken spin Slater determinants to evaluate the exchange coupling parameters in the Heisenberg Hamiltonian are also mentioned.     

CEPA calculations on open-shell molecules

Ab initio calculations at SCF and CEPA levels using large Gaussian basis sets have been performed for the two lowest electronic states,X ² Σ⁺ andA ² Π, of HeAr⁺. Spin-orbit coupling (SOC) effects have been added using a semiempirical treatment. The resulting potential curves for the three statesX,A ₁, andA ₂ have been used to evaluate molecular constants such as vibrational intervals ΔG(v + 1/2) and rotational constantsB _v as well as — by means of a Dunham expansion — equilibrium constants such asR _e , ω _e ,B _e etc. Comparison with the experimental data from UV emission spectroscopy shows that the calculated potential curves are slightly too shallow and have too large equilibrium distances:D _e = 242 cm^?1 andR _e = 2.66 Å compared to the experimental values of 262 cm^?1 and 2.585 Å, respectively, for theX ²Σ⁺ ground state. However, the ab initio calculations yield more bound vibrational levels than observed experimentally and allow for a more complete Dunham analysis, in particular for theA ₂ state. The experimental value of 154 cm^?1 for the dissociation energyD _e of this state is certainly too low; our best estimate is 180±5 cm^?1. For theA ₁ state our calculations are predictions since this state has not yet been observed experimentally.     

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     

Integral invariants in chemical kinetics

Integral invariants of classical mechanical systems are used for the mathematical treatment of equilibrium systems of chemical reaction kinetics. Some conserved quantities and Hamilton equations in chemistry are shown.

---

. .

     

Electron localization and delocalization in open-shell molecules

Localization and delocalization indices derived in the framework of the quantum Atoms in Molecules theory have recently been used to analyze the electron-pair structure of closed-shell molecules. Here we report calculations of localization and delocalization indices for open-shell molecules at the Hartree-Fock (HF) level. Several simple doublet and triplet radical molecules are studied. In general, interatomic delocalization between bonded atoms is heavily dependent on the order and polarity of the bond. Unpaired electrons also have a significant effect on the interatomic delocalization indices. Indeed, for many radicals, the analysis of the spin components reveals that the interatomic delocalization is very different for alpha and beta spin electrons in many cases. In general, at the HF level, the results can be rationalized in terms of orbital contributions. However, the definition of localization and delocalization indices is completely general, and they could be calculated at any level of theory, provided that the one- and two-electron densities are available.     

Correlated electron propagator theory and applications for open-shell atoms and molecules

A unitary group reformulation of electron propagator theory is used to derive the second-order and 2p – h Tamm–Dancoff self-energy approximations for open-shell applications. A highest weight representation of the reference state is chosen in order to facilitate matrix element evaluation, but two special cases precluded by this choice are also discussed. Detailed numerical calculations are described for the lithium atom, oxygen molecule, and the amidogen radical.     

CEPA calculations on open-shell molecules. II. Singlet-triplet energy splitting in π2 configurations of diatomic molecules

The CEPA-PNO method is used for calculating the energy difference ΔE _ST between the³∑⁻ and the¹Δ states of diatomic molecules in electronic π² configurations. An analysis of the contribution of electron correlation to ΔE _ST is performed in terms of physically understandable effects such as direct correlation, dynamic spin polarization, semiinternal and internal excitations. It is shown that these effects are of completely different importance for the molecules treated in this study: For C₂ the direct correlation between the two singly occupied π-orbitals is the dominant correlation contribution to ΔE _ST; for O₂, S₂, SO the internal excitation π _u ² → π _g ² is predominant, whereas for NH and PH there is a close competition between the direct correlation and the spin polarization of the underlying σ-orbitals. The basis set dependence of these effects is investigated, in particular for NH. Our final results reproduce experimental values of ΔE _ST within 0.05–0.10 eV.     

CEPA calculations on open-shell molecules. VI. The first ionization potential of HCO

According to an investigation by Dyke et al. (1980) HCO seems to be one of the rare cases in which the correlation energy of the neutral molecule is smaller (in absolute value) than that of the first monopositive ion, such that the ΔSCF method yields a value for the first vertical ionization potential (IP _v ) which is larger than the experimental value. In order to understand this observation we have performed a series of SCF and CEPA calculations on HCO and HCO⁺ using as many as ten different orbital basis sets. The best ΔSCF result for IP _v is 9.24 eV, i.e. slightly smaller than the “experimental electronic” IP _v of 9.38 eV. Inclusion of electron correlation lowers IP _v as long as small basis sets are used, the convergence with increasing basis size is very slow. Extrapolation to a complete basis leads to a CEPA estimate of 9.26±0.10 eV for IP _v (and 8.05 ±0.10 eV for the adiabatic IP) which shows that the correlation contribution to IP _v is indeed very small. The reason for this is that the gain in correlation energy in HCO due to the presence of the unpaired electron is compensated by a loss of core correlation energy since the low-lying antibonding in-plane C-O-π- orbital is only partially available for excitation in HCO, but fully available in HCO⁺.     

CEPA calculations on open-shell molecules. V. The vibration frequencies of SF and SCl

Ab initio calculations for the ² ground states of SF and SCl have been performed on Hartree-Fock level and with inclusion of valence shell correlation effects by means of the CI and CEPA approaches. The calculated properties are: Equilibrium distances, vibration frequencies, and dipole moment curves in the vicinity of the respective equilibrium geometries. Our best estimates for the 0 1 infrared absorption frequencies _o for SF and SCl are 786 cm^–1 and 520 cm^–1, respectively, both with an uncertainty of about 10 cm^–1. This confirms a recent experimental value obtained by Willner for SF (791 cm^–1), but indicates that for SCl both experimental values reported previously in the literature (617 cm^–1 and 574 cm^–1) are wrong. The S—F and S—Cl bonds in SF and SCl are very similar to the ones in SF₂ and SCl₂, being essentially single p-bonds in either case. In the analogous oxygen-halogen molecules the situation is different, the O—F and O—Cl bonds in the diatomic radicals OF and OCl have partial double bond character and are much stronger than those in OF₂ and OCl₂ or in HOF and HOCl.     

Predicting properties of molecules using graph invariants

Topological indices (TIs) have been used to study structure-activity relationships (SAR) with respect to the physical, chemical, and biological properties of congeneric sets of molecules. Since there are many TIs and many are correlated, it is important that we identify redundancies and extract useful information from TIs into a smaller number of parameters. Moreover, it is important to determine if TIs, or parameters derived from TIs, can be used for global SAR models of diverse sets of chemicals. We calculated seventy-one TIs for three groups of molecules of increasing complexity and diversity: (a) 74 alkanes, (b) 29 alkylbenzenes, and (c) 37 polycyclic aromatic hydrocarbons (PAHs). Principal components analysis (PCA) revealed that a few principal components (PCs) could extract most of the information encoded by the seventy-one TIs. The structural basis of the first few PCs could be derived from their pattern of correlation with individual TIs. For the three sets of molecules, viz. alkanes, alkylbenzenes and PAHs, PCs were able to predict the boiling points reasonably well. Also, for the combined set of 140 chemicals consisting of the alkanes, alkylbenzenes and PAHs, the derived PCs were not as effective in predicting properties as in the case of individual classes of compounds.     

Ab initio program for treatment of related molecules. I. Integral part

Anab initio integral program is described. It utilizes the local symmetries to avoid the redundant computation of integrals over spatially equivalent subsets of the basis. The integrals are grouped in a particular way to facilitate their transfer. The program is very suitable for the treatment of related systems with model geometries. The computing times of different programs are compared and the efficiency of the presented one is demonstrated.     

Molecular design of ambipolar redox-active open-shell molecules: Principles and implementations

The mini-review highlights key structural features and provides guidelines for the molecular design of ambipolar redox-active frameworks, which are of interest for the fabrication of all-organic ‘poleless’ batteries and semiconductors. Open-shell molecules that can form at least three relatively stable redox states (i.e. cationic, neutral, and anionic) are discussed. The efficiency of the suggested strategies is supported by the representative examples taken from the recent publications.     

The spin-unrestricted molecular Kohn-Sham solution and the analogue of Koopmans's theorem for open-shell molecules

Spin-unrestricted Kohn-Sham (KS) solutions are constructed from accurate ab initio spin densities for the prototype doublet molecules NO(2), ClO(2), and NF(2) with the iterative local updating procedure of van Leeuwen and Baerends (LB). A qualitative justification of the LB procedure is given with a "strong" form of the Hohenberg-Kohn theorem. The calculated energies epsilon(isigma) of the occupied KS spin orbitals provide numerical support to the analogue of Koopmans' theorem in spin-density functional theory. In particular, the energies -epsilon(ibeta) of the minor spin (beta) valence orbitals of the considered doublet molecules correspond fairly well to the experimental vertical ionization potentials (VIPs) I(i) (1) to the triplet cationic states. The energy -epsilon(Halpha) of the highest occupied (spin-unpaired) alpha orbital is equal to the first VIP I(H) (0) to the singlet cationic state. In turn, the energies -epsilon(ialpha) of the major spin (alpha) valence orbitals of the closed subshells correspond to a fifty-fifty average of the experimental VIPs I(i) (1) and I(i) (0) to the triplet and singlet states. For the Li atom we find that the exact spin densities are represented by a spin-polarized Kohn-Sham system which is not in its ground state, i.e., the orbital energy of the lowest unoccupied beta spin orbital is lower than that of the highest occupied alpha spin orbital ("a hole below the Fermi level"). The addition of a magnetic field in the -z direction will shift the beta levels up so as to restore the Aufbau principle. This is an example of the nonuniqueness of the mapping of the spin density on the KS spin-dependent potentials discussed recently in the literature. The KS potentials may no longer go to zero at infinity, and it is in general the differences nu(ssigma)( infinity )-epsilon(isigma) that can be interpreted as (averages of) ionization energies. In total, the present results suggest the spin-unrestricted KS theory as a natural one-electron independent-particle model for interpretation and assignment of the experimental photoelectron spectra of open-shell molecules.     

Circular dichroism of some high-symmetry chiral molecules: B3LYP and SAOP calculations

Computational modeling of optical activity, circular dichroism (CD) and optical rotatory dispersion, is rapidly becoming a useful supplement to experimental studies of absolute configuration. Here, we investigate the predictions of two alternative formulations of the rotational strength based on time-dependent density functional theory (TD-DFT), for a series of high symmetry chiral systems. We employ the TD-DFT method as realized in Gaussian 03 suite with the hybrid functional B3LYP and as incorporated in the Amsterdam density functional (ADF) suite with PBE and SAOP functionals. The high-symmetry systems described here are somewhat larger than those used to evaluate the influence of basis sets and density functional choice, and for such large systems the very extensive basis sets recommended by most investigators may not be suitable for routine use. We observe that useful results for these systems can be obtained in modest bases, and in particular that diffuse functions may not be required for informative use of the ADF implementation. The statistical average of orbital potentials (SAOP) model developed by Baerends is essential to the success of the ADF implementation. In some cases chirality is defined by features of the molecular structure remote from the chromophore. This is a severe test of the TD-DFT theory, since high-lying excitations define the most prominent features of the CD spectra, and complicates the use of computations to guide the assignment of absolute configuration. Experimental investigation of the high symmetry systems described here is desirable.     

Efficient and accurate local single reference correlation methods for high-spin open-shell molecules using pair natural orbitals

A production level implementation of the high-spin open-shell (spin unrestricted) single reference coupled pair, quadratic configuration interaction and coupled cluster methods with up to doubly excited determinants in the framework of the local pair natural orbital (LPNO) concept is reported. This work is an extension of the closed-shell LPNO methods developed earlier [F. Neese, F. Wennmohs, and A. Hansen, J. Chem. Phys. 130, 114108 (2009); F. Neese, A. Hansen, and D. G. Liakos, J. Chem. Phys. 131, 064103 (2009)]. The internal space is spanned by localized orbitals, while the external space for each electron pair is represented by a truncated PNO expansion. The laborious integral transformation associated with the large number of PNOs becomes feasible through the extensive use of density fitting (resolution of the identity (RI)) techniques. Technical complications arising for the open-shell case and the use of quasi-restricted orbitals for the construction of the reference determinant are discussed in detail. As in the closed-shell case, only three cutoff parameters control the average number of PNOs per electron pair, the size of the significant pair list, and the number of contributing auxiliary basis functions per PNO. The chosen threshold default values ensure robustness and the results of the parent canonical methods are reproduced to high accuracy. Comprehensive numerical tests on absolute and relative energies as well as timings consistently show that the outstanding performance of the LPNO methods carries over to the open-shell case with minor modifications. Finally, hyperfine couplings calculated with the variational LPNO-CEPA∕1 method, for which a well-defined expectation value type density exists, indicate the great potential of the LPNO approach for the efficient calculation of molecular properties.