Identification of deadwood in configuration spaces through general direct configuration interaction

 In order to identify ineffective and, hence, superfluous configurations in algorithmically generated configuration spaces, a direct configuration interaction (CI) method has been developed for determining completely general configurational expansions based on arbitrary determinantal configuration lists. While based on the determinantal ordering scheme of Knowles and Handy, our direct CI algorithm differs from previous ones by the use of the Slater–Condon expressions in direct conjunction with single and double replacements. A full, as well as a completely general selected, CI program has been implemented. With it, full configuration spaces of Ne, C₂, CO and H₂O with up to about 40 million determinants have been investigated. It has been found that, in all cases, fewer than 1% of the configurations in a natural-orbital-based configuration expansion reproduce the exact results within chemical accuracy. Received: 19 December 2000 / Accepted: 9 May 2001 / Published online: 11 October 2001     

Systematic Study of Selected Diagonalization Methods for Configuration Interaction Matrices

Several modifications to the Davidson algorithm are systematically explored to establish their performance for an assortment of configuration interaction (CI) computations. The combination of a generalized Davidson method, a periodic two‐vector subspace collapse, and a blocked Davidson approach for multiple roots is determined to retain the convergence characteristics of the full subspace method. This approach permits the efficient computation of wave functions for large‐scale CI matrices by eliminating the need to ever store more than three expansion vectors ( b _i) and associated matrix‐vector products ( σ _i), thereby dramatically reducing the I/O requirements relative to the full subspace scheme. The minimal‐storage, single‐vector method of Olsen is found to be a reasonable alternative for obtaining energies of well‐behaved systems to within μE_h accuracy, although it typically requires around 50% more iterations and at times is too inefficient to yield high accuracy (ca. 10^?10 E_h) for very large CI problems. Several approximations to the diagonal elements of the CI Hamiltonian matrix are found to allow simple on‐the‐fly computation of the preconditioning matrix, to maintain the spin symmetry of the determinant‐based wave function, and to preserve the convergence characteristics of the diagonalization procedure. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 1574–1589, 2001     

Some simplification in spin-free configuration interaction studies

A simplification has been attempted in the procedures for determining the matrix elements of the generators of the unitary group U(n) over a tensor basis spanning the irreducible representation 2 ^N/2–S , 1 ^2S for an N-electron system. It has been shown that these matrix elements require, for their determination, only the corresponding representation matrices of cyclic permutations of the group S _N . A viable algorithm has been obtained for determining these representation matrices.     

A multi-configuration reference CEPA method based on pair natural orbitals

Summary A multi-reference CI scheme is proposed which is aiming at a considerable reduction of the generally very large number of configurations of CI expansions in multi-configuration reference cases. This reduction is achieved by combining the idea of internal contraction, the concept of pair natural orbitals (PNO's) and CEPA (coupled electron pair) type approximations for the contributions of higher than double excitations. This latter estimate leads to size consistent results and also permits to employ reference wavefunctions that contain only the dominantly occupied configurations of the considered system. Applications to two test cases, the lowest states (³ P,¹ D and¹ S) of the carbon atom and the symmetry forbiddenC _2v insertion reaction of Be and H₂, show that our method is able to truncate CI expansions to lengths of no more than 10³–10⁴ without losing more than 1–2% of the correlation energy. The calculated excitation energies and energy barriers agree with the full CI results in the respective basis within about 1 kcal/mol. Thus the MC-CEPA-PNO method presents a very efficient way to obtain chemical accuracy in CI-calculations for molecular systems.Dedicated to Prof. Dr. W. Kutzelnigg on the occasion of his 60th birthday     

Raman spectra and structure of γ- Ln2S3 vacancy crystals

Polarized Raman spectra were recorded for oriented γ- La₂S₃ crystals;unpolarized spectra were obtained for y- Ce₂S₃, for (La- Ce)₂S₃ solid solutions, and for XLn₂S₄ compounds (X is metal). The observed crystalline modes are assigned according to symmetry types. It was found that vacancies are distributed in the cation sublattice statistically;conditions for ordering of metal atoms are examined. La ₂S₃ shows resonance scattering at λ = 647.09 nm;this is likely to be due to disorder in the crystal lattice. Translated fromZhumal Struktumoi Khimii, Vol. 38, No. 4, pp. 605–615, July–August, 1997.     

A comparative multi-reference configuration interaction study of the low-lying states of two thione isomers of thiophenol

Multi-reference configuration interaction, MR-CI (including extensivity corrections, named +Q), calculations were performed on the S₀–S₃ states of cyclohexa-2,4-diene-1-thione (thione 24 ) and cyclohexa-2,5-diene-1-thione (thione 25 ), which are thione isomers of thiophenol. Several types of uncontracted MR-CIS and MR-CISD wavefunctions were employed, comprising MR-CI expansions as large as ~365 × 10⁶ configuration state functions. The nature of the studied excited states was characterized. Vertical excitation energies (ΔE) and oscillator strengths (f) were computed. The most intense transitions (S₀ → S₂ for 24 and S₀ → S₃ for 25 ) did not change with the wavefunction, although a variation as large as ~1 eV was obtained for the S₃ state of 24 , at the highest (MR-CI+Q) level. On the other hand, ΔE changed at most by ~0.56 eV for 25 as the wavefunction changes, at the same level. The S₁ state of both thiones was found to have nπ* character and is in the visible region. For 24 , S₂ and S₃ are ππ* and nπ* states, respectively, while for 25 the reverse order is obtained. S₂ and S₃ are in the range ~3.5 to 5.2 eV, again at the highest level. It is the first time that the excited states of the title molecules are studied. The computed results agree with the experimental onset of photoreactions of thiones 24 and 25 found by Reva et al (Phys. Chem. Chem. Phys., 2015 , 17, 4888).     

Crystal and molecular structure of (1S,2S,4aS, 8aS)-N-(N-allyldiaminomethanethione)-1-(2-hydroxy-2,5,5,8a-tetramethyldecahydronaphthalenyl) acetamide

Crystal and molecular structure of a new homodrimanic compound (1S,2S,4aS, 8aS)-N-(N-allyldiaminomethanethione)-1-(2-hydroxy-2,5,5,8a-tetramethyldecahydronaphthalenyl) acetamide has been determined by X-ray diffraction analysis. The crystal is monoclinic, unit cell parameters are: a = 9.577(2) Å, b = 7.414(1) Å, c = 16.856(3) Å; β = 94.83(3)°, space group P2₁, Z = 2, of composition C₂₀H₃₅N₃O₂S. Two cyclohexan fragments have ordinary structure and chair-configuration typical of this compound class in homodrimanic skeleton. Ethanol molecule is located in the outer sphere. The withdrawal of carbon atoms from planar fragments of cyclohexan rings varies within the limits from 0.722(5) Å to − 0.634(5) Å. A dihedral angle between the mean-square planes of the latter equals 16.0(2)°, torsion angle (5)-(5)-(10)-(16) 171.0(1)° indicates their trans-joint. In the side non linear chain allyl group is connected to terminal nitrogen atom of thiosemicarbazide molecule. Intermolecular hydrogen bonds between carbonyl atom of acetamide fragment, ethanol molecule, and donor-acceptor groups of thiosemicarbazide moiety play the main part in crystal structure organization. Original Russian Text Copyright ? 2005 by E. P. Styngach, S. T. Malinovskii, L. P. Bets, L. A. Vlad, M. Gdanets, and F. Z. Makaev __________ Translated from Zhurnal Strukturnoi Khimii, Vol. 46, No. 4, pp.785–789, July–August, 2005.     

Crystal structure of polymorphous modifications of Tm2S3 and related compounds

The role of symmetry in crystal structure formation is revealed for the polymorphous modifications of Tm₂S₃ and related compounds Tm₁₅S{ib22}, Tm₅S₇, and Tm₈S₁₁ using the procedure of selecting and analyzing closepacked planes with cationic, anionic, and combined motifs. The laws of mutual agreement between the cation and anion matrices determining the size of the unit cells are examined, and relationships between the specific cation volumes and symmetry are traced. Translated fromZhumal Strukturnoi Khimii, Vol. 39, No. 4, pp. 669–675, July–August, 1998.     

nBody democratic recoupling and role of Schur Fn. products on Sn, a NMR dual-group view pertinent to coherence-transfer

For automorphic spin symmetries over isotopomer networks defining both NMR coherence transfer and cage-cluster ro-vibrational (RV) weighting phenomena, the role of Schur functions (SF) and their SF products (SFP) (restricted to S_n) is re-evaluated, beyond that typical of atomic physics. It is seen now as being equally applicable to molecular physics. Our special focus here is on the SU2 × S_n dual group for its importance in replacing (Jucys) recoupling schemata by generalised S_n democracy invariants, which ensure the retention of simple reducibility (SR) of carrier spaces for superboson mappings, under U × P(Γ) actions over Liouville space, i.e., as in Physica A 198 (1993) 245. SFP mapping onto S_n and simple Young rule SF mappings onto {[λ]}(S_n) sets are developed in the high n-index, weak-branching limit and utilise known similarities in the algorithms for the Young and Littlewood–Richardson rules. Over discrete k rank, from k=n (down), the Σ_ν{/T ^k(ν)/x007D;k-set orders are related to the ⊵ ordered bipartite χ _12n ^[λ] (S _2n characters. This revised version was published online in July 2006 with corrections to the Cover Date.     

The electronic states of 1,2,5-oxadiazole studied by VUV absorption spectroscopy and CI, CCSD(T) and DFT methods

The 1,2,5-oxadiazole VUV absorption spectrum in the range 5–11.5 eV, shows broad bands centred near 6.2, 7.1, 8.3, 8.8, 10.6 and 11.3 eV. Rydberg states associated with three ionisation energies (IE) were identified in the complex fine structure above 8.7 eV. Electronic vertical excitation energies for singlet and triplet valence, and Rydberg states were computed using ab initio multi-reference multi-root CI methods. There is generally a good correlation between the envelope of the theoretical intensities and the experimental spectrum. The nature of the more intense calculated Rydberg states, and positions of the main valence and Rydberg bands are discussed. The lowest triplet, singlet and Rydberg 3s excited states have equilibrium structures that are non-planar with C_S symmetry, in a chair-like orientation where the O and H atoms lie out of the NCCN plane. This finding is consistent with the doubling of the low energy UV spectral lines [B.J. Forrest, A.W. Richardson, Can. J. Chem., 50 (1972) 2088].The nearly degenerate IE of the UV-photoelectron spectrum (UV–PES, Palmer et al. 1977) makes analysis of the VUV spectrum difficult, leading to the necessity for reinvestigation. Vertical studies (IE_V) using CI, Tamm–Dancoff (TDA) and Green’s Function (GF) methods all gave similar results, with near degeneracy of the first 3IE_V confirming the earlier study. Studies of the adiabatic IE (IE_A) using CCSD(T) and B3LYP methods, showed the energy sequence ²A₂ < ²B₁ < ²B₂, but these states are all saddle points, in contrast to the 4th state (²A₁) which is a minimum. In contrast, MP2 study of the ²B₂ state showed a minimum, with only two saddle points.Complete minima were found after minor twisting of the structures. The lowest energy cationic state is ²A^″ (C_S), which closely resembles the ²B₂ state. The O–N–C–C skeleton is twisted by 8°. The corresponding ²A^′ state (C_S) is effectively identical to the ²B₁ state. Attempts to find minima for other symmetry states were unsuccessful.     

Multicolour models of natural embedding for SU(2≥m≥4)×S 10↓D 5 NMR spin symmetry: Determinacy of nuclear spin weights for (1,12)-car-11B-boranes

The inherent (in)determinacy implicit in the SU(m≥3)×S _n↓G natural embedding aspects of (NMR) spin symmetry of clusters is investigated, as part of a multicolour modelling scheme, where the SU2-branching level meets the initial n(S _n)=/G/ condition. We focus on correlative mappings derived from [λ]_SA (self-associate) irreps for natural group embeddings and compare these with certain Yamanouchi-Gel'fand chain properties of S ₁₀ Mathematical decompositions of M^λ simple S _n-modules with (2≥p≥4)-branchings of λ⊵,λ_SA (for λ⊢N partitions of _n) provide the initial insight into the monocluster spin (NP) physics of [²H_]10, [¹¹B]₁₀ (S ₁₀↓D ₅), as aspects of (1,12)-(HC)₂(¹¹B)₁₀ or (HC)₂(^{2 11}B₁₀ carborane cage isotopomers. The questions raised are significant for their impact on CNP nuclear spin weighting of ro-vibrational spectra. The methods used are those of combinatorics-via-group actions, as physical S _n-encodings applied to nuclear spin algebras. This revised version was published online in July 2006 with corrections to the Cover Date.     

Perspective on “An extended Hückel theory. I. Hydrocarbons”

 The size-consistent self-consistent matrix dressing method has been applied on an open-shell single-configuration reference state. Once the reference state is converged, several low-lying roots can be obtained for the dressed configuration interaction (CI) matrices of appropriate symmetry. The CI matrices were built with a complete-active-space singles and doubles CI method in order to deal properly with multiconfiguration excited states. The vertical ionization and ionization–excitation transitions are obtained from the difference to the closed shell ground-state energy of the neutral molecule. The method has been applied to NH⁺ ₃ and N⁺ ₂ using atomic natural orbital basis sets and state-average adapted molecular orbitals. Two ²A₁ states, very similar and showing great mixing of the (2a _l ⁻¹) and (3a _l ⁻²5a _l ¹) determinants, can be assigned to the broad asymmetric band at 27.6 ± 2 eV in the photoelectron spectrum of NH₃. The possible contribution of a ²Π _g (3σ _g ⁻²1π _g ¹) state to the A shake-up peak of N₂ at 24.6 eV is also discussed. Other states, doublets and quadruplets, are reported for both systems up to 30 eV for NH₃ and 37 eV for N₂. Received: 16 September 1999 / Accepted: 3 February 2000 / Published online: 2 May 2000     

Ab initio configuration interaction study on electronically excited 4-dimethylamino-4′-cyanostilbene

Ab initio configuration interaction calculations have been performed to examine the electronic structures of both trans-4-dimethylamino-4′-cyanostilbene (DCS) and four types of perpendicularly twisted DCSs, trans-DCS is predominantly excited into the S₁ state out of low-lying excited states. The S₁ state is an intramolecular charge-transfer (ICT) state in which the dipole moment is about twice as large as that in S₀. The excited DCS at the 4-dimethylanilino twisted conformation, which becomes S₁ in polar solvents, has a very much larger dipole moment than that in S₁ to trans-DCS. This means that the geometrical structure of the twisted ICT (TICT) is the 4-dimethylanilino twisted form, not the dimethylamino twisted one which is well know from the TICT structure of 4-dimethylaminobenzonitrile. Received: 16 December 1998 / Accepted: 19 March 1999 / Published online: 9 September 1999     

Maxwell–Cartesian spherical harmonics in multipole potentials and atomic orbitals

 The nature of the Maxwell–Cartesian spherical harmonics S ⁽ⁿ⁾ _K and their relation to tesseral harmonics Y _nm is examined with the help of “tricorn arrays” that display the components of a totally symmetric Cartesian tensor of any rank in a systematic way. The arrays show the symmetries of the Maxwell–Cartesian harmonic tensors with respect to permutation of axes, the traceless properties of the tensors, the linearly independent subsets, the nonorthogonal subsets, and the subsets whose linear combinations produce the tesseral harmonics. The two families of harmonics are related by their connection with the gradients of 1/r, and explicit formulas for the transformation coefficients are derived. The rotational transformation of S ⁽ⁿ⁾ _K functions is described by a relatively simple Cartesian tensor method. The utility of the Maxwell–Cartesian harmonics in the theory of multipole potentials, where these functions originated in the work of Maxwell, is illustrated with some newer applications which employ a detracer exchange theorem and make use of the partial linear independence of the functions. The properties of atomic orbitals whose angular part is described by Maxwell–Cartesian harmonics are explored, including their angular momenta, adherence to an Uns?ld-type spherical symmetry relation, and potential for eliminating an angular momentum “contamination” problem in Cartesian Gaussian basis sets. Received: 9 July 2001 / Accepted: 7 September 2001 / Published online: 19 December 2001     

Free electron model in cluster structure theory. Electronic structures of [Mo6S8(CN)6]6?, [Mo6Se8(CN)6]6?, [Re6S8(CN)6]4?, and Rh6(CO)16 clusters

The results of quantum chemical calculations of the electronic structure and geometry of octahedral clusters [Mo₆S₈(CN)₆]⁶⁻, [Mo₆Se₈(CN)₆]⁶⁻, [Re₆S₈(CN)₆]⁴⁻, and Rh₆(CO)₁₆ by the ab initio SCF (RHF) and DFT (B3LYP) methods with various basis sets are presented. The electronic states of the clusters under study in ideal spherically symmetric potential were classified in the orbital quantum number l (1s, 1p, 1d, 1f, 1g, 1h, 1i), l = 0–6. In real crystal field with O_h symmetry these states are split. The calculated new electronic states were matched to the irreducible representations of the point symmetry group O_h. The polarizabilities of the compounds considered are 55–65 ų. A new model for the electronic structure of octahedral clusters containing M₆ groups was proposed. The model is based on the idea of free electrons moving in spherically symmetric potential field. Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 12, pp. 2617–2624, December, 2005.     

Evolutionary algorithm based configuration interaction approach

A new evolutionary algorithm for stochastic configuration interaction (CI) method designed as an affordable approximation to full configuration interaction (FCI) has been described here. The key components of the algorithm are initiation, propagation, and termination steps taking inspiration from the genetic algorithm. The propagation step is performed with cloning (retention of a Slater determinant without change), mutation (single excitation/de‐excitation), and crossover (exchange of α and β strings between two Slater determinants) and termination is selection of few Slater determinants based on certain fitness function (measure of importance of a determinant in the CI space) and rejection of the rest. We find that the absolute value of the CI coefficients is a suitable fitness function when combined with a fixed selection scheme. We have tested its accuracy in 1D Hubbard problem and ground state potential energy surface (PES) has also been constructed for symmetric bond breaking of water molecule, where the errors are found to be around 10 mE_h with low non‐parallelity error, when retaining only a small fraction of the total number of Slater determinants in the final population. This shows that this method has the ability to capture both static and dynamic correlation. Performance and convergence properties of the algorithm are also tested for N₂ triple bond breaking problem. The algorithm opens up a promising way for stochastic sampling of the important determinants in the full Hilbert space.     

Theory of cluster self-organization for crystal-forming systems: A model of transition from disodered to ordered systems

The current state of ideas concerning the self-organization of crystal-forming systems where long-range order spontaneously appears in the arrangement of nanolevel structural units of any nature (micro- and macromolecules or atomic clusters) that initially existed in a dynamic state as a chaotic mixture is considered. Three partially overlapping stages of self-organization of a system accepted in physical models of “order-disorder” kinetic transitions are matched to those used in supramolecular chemistry. An algorithmically constructed model of transition from disordered to hierarchically ordered systems is considered. The geometrical and topological modeling of density fluctuations of n-atomic species (clusters) An in a crystal-forming medium is carried out. A specific set of An clusters with block-diagonal connectivity matrices is recognized. These types of clusters (S ₃⁰), having “sectional” or “hierarchic” partition, are defined as precursors of crystal structures that are capable of evolving most rapidly to give rise to a long-range order in structures. For an S ₃⁰ ring cluster shaped as a triangle, geometrical and topological modeling is carried out for all of the eight topologically and symmetrically possible types of S ₃¹ primary chains built of S ₃⁰ using theory of one-dimensional symmetry groups. Thirty three structural variants of morphologically and topologically different types of S ₃² micronets described by two-dimensional groups of symmetry are considered. Algorithms are presented for combinatorial and topological analysis to search for precursor clusters and restore a three-dimensional net of covalent and noncovalent bonds in a crystal structure by the matrix (cluster) self-assembly mechanism. The model advanced is universal. Examples of self-assembly of a series of cluster-assembled structures of AB₂ alloys of the unique Friauf-Laves family (which counts in 1400 of binary and ternary compounds) are given: for MgCu₂ (cF24) (with its superstructures of ZrCu ₅ and MgSnCu₄ types), MgZn₂ (hP12), and MgNi₂ (hP24) (from AB₂ or A₂B + B₃ three-atom clusters); for ZrZn₂₂ icosahedral structures (from a suprapolyhedral cluster built of a ZrZn₁₆ Friauf polyhedron and two ZnZn₁₂ icosahedra); NaCd₂ (from one A cluster with 61 atoms and two B clusters with 63 atoms); and for a bimolecular compound C₇₈H₃₀ (which is formed of fullerene C₆₀ and a C₂₈H₃₀ molecule). The scenario of formation of self-curving nets with icosahedral symmetry is considered: to form a B₁₂ icosahedron from two isomers with n = 3, a C₂₀ dodecahedron from two isomers with n = 5, and C₆₀ fullerene from pentagonal clusters with n = 5.     

Magnetic and thermodynamic properties of Heisenberg chains and n -nuclear cyclic clusters: S i = 5/2, S i = 3, and S i = 7/2 systems

Based on the previously developed method considering spin and point symmetry, numerical calculations of spin-level spectra have been performed for n-nuclear cyclic clusters with S _i = 5/2 (n ≤ 9), S _i = 3 (n ≤ 8), and S _i = 7/2 (n ≤ 8). The theoretical curves of the magnetic susceptibility, the magnetic contribution to the heat capacity, and the internal energy as a function of temperature have been obtained. The theoretical curves of the magnetic susceptibility and magnetic contribution to the heat capacity have been extrapolated to n → ∞ with a controlled accuracy. These curves are compared to available data. Original Russian Text ? Yu.V. Rakitin, O.R. Starodub, V.M. Rakitina, V.T. Kalinnikov, V.M. Novotortsev, 2007, published in Zhurnal Neorganicheskoi Khimii, 2007, Vol. 52, No. 3, pp. 448–455.     

Algorithm for variational calculations of molecular symmetry in solving anharmonic vibrational problems

A universal program for variational calculations of molecular symmetry in solving anharmonic vibrational problems, realized by the author, is described. The program uses the group-theoretical method. Symmetrized basis wave functions are constructed with the aid of the generalized KJebsch-Gordan series suggested by the author. The method of constructing symmetrized basis wave functions and the program for adequate calculations of molecular symmetry were verified for many molecules of different symmetry groups: O_h, O, T_d, T_h, T, D_∞h, C_t8v, D_nd, D_nh, D_n, C_nv, C_nh, S_2n, C_n, C_i, C_s, and C₁ where 2 ≤n ≤6. It was confirmed that the program provides correct results and high-speed operation. Translated fromZhurnal Strukturnoi Khimii, Vol. 38, No. 6, pp. 1146–1153, November–December, 1997.