Group theoretical classification of states of a molecular system at definite states of its constituent parts

A method is developed for determining the allowed states of a many-particle system at definite states of the subsystems (impurity ion complexes in crystals, nucleon clusters in nuclei, polyatomic molecules). The advantage of the method proposed here over those previously developed is that the present one is independent of the number of particles in the subsystems. Depending on the value of spin S_a the subsystems are considered either as bosons or fermions. The representation character expression whose expansion gives all the terms ^2s+1Γ^(α) with a definite value of total spin of the system is written as the product of permutation and orbital factors. The orbital factor is defined in terms of characters of the local point group representations. Some alternative ways of calculating the permutation factor are discussed: a) with the use of tables of reduction of the unitary transformation group U₂S_a+1 in a three-dimensional rotation group; b) through the plethysm coefficients; and c) in terms of 3nj-symbols of three-dimensional rotation group. Some examples of application of the method to various systems are presented.     

Subduction coefficients for 〈2N/2–S, 12S〉 ↓ 〈2, 1〉 ⊗ 〈2, 1〉 of U(n) ↓ U(n1) ⊗ U(n2)

The determination of the subduction coefficients for states of the unitary group U(n) under the restrictions U(n) ↓ U(n₁) ? U(n₂) have been considered for the spin free states of many electron systems. Using the transformation properties of the tensor basis spanning the irreducible representation 〈2^N/2–S, 1^2S〉 of U(n) under the permutations of electron coordinates, a simple programmable procedure has been developed for the determination of these coefficients. The procedure has been illustrated using a simple example.     

Application of the unitary group approach to evaluate spin density for configuration interaction calculations in a basis of S2 eigenfunctions

We present an implementation of the spin‐dependent unitary group approach to calculate spin densities for configuration interaction calculations in a basis of spin symmetry‐adapted functions. Using S² eigenfunctions helps to reduce the size of configuration space and is beneficial in studies of the systems where selection of states of specific spin symmetry is crucial. To achieve this, we combine the method to calculate U(n) generator matrix elements developed by Downward and Robb (Theor. Chim. Acta 1977, 46, 129) with the approach of Battle and Gould to calculate U(2n) generator matrix elements (Chem. Phys. Lett. 1993, 201, 284). We also compare and contrast the spin density formulated in terms of the spin‐independent unitary generators arising from the group theory formalism and equivalent formulation of the spin density representation in terms of the one‐ and two‐electron charge densities.     

Unitary group approach to general system partitioning. I. Calculation of U(n = n1 + n2): U(n1) × u(n2) reduced matrix elements and reduced wigner coefficients

This paper is the first in a series of two directed toward a unitary calculus for group-function-type approaches to the many-electron correlation problem. In this paper we present a complete derivation of the matrix elements of the U(n = n₁ + n₂) generators, for the representations approapriate to many-electron systems, in a basis symmetry adapted to the subgroup U(n₁) × U(n₂). Explicit formulae for the fundamental U(n):U(n₁) × U(n₂) reduced Wigner coefficients, which are needed for the general multishell problem, are also obtained. The symmetry properties of the reduced Wigner coefficients and reduced matrix elements are investigated, and a suitable phase convention is given.     

Theoretical investigation of LaC3n+ (n = 0, 1, 2) clusters by density functional theory

LaC₃ⁿ⁺ (n=0, 1, 2) clusters have been studied using B3LYP (Becke 3-parameter–Lee-Yang-Parr) density functional method. The basis set is Dunning/Huzinaga valence double zeta for carbon and [2s2p2d] for lanthanum, denoted LANL1DZ. Four isomers are presented for each cluster; two of them are edge binding isomers with C_2v symmetry, the other two are linear chains with C_∞v symmetry. Meanwhile, two spin states for each isomer, that is, singlet and triplet for LaC₃⁺, doublet and quartet for LaC₃ and LaC₃²⁺, respectively, are also considered. Geometries, vibrational frequencies, infrared intensities, and other quantities are reported and discussed. The results indicate that at some spin states; the C_2v symmetry isomers are the dominant structures, while for the other spin states, linear isomers are energetically favored. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 66 : 301–307, 1998     

Magnetic and thermodynamic properties of ferrimagnetic cyclic clusters and infinite chains with alternating spins of the [s-S]n Type: s = 1/2 and S = 3/2–7/2

Based on the method considering spin and spatial symmetry, numerical calculations of spin-level spectra have been performed for cyclic clusters and infinite chains with alternating spins of the [s-S]_n type, where s = 1/2 and S = 3/2–7/2. The temperature dependences of internal energy and magnetic susceptibility for ferrimagnetic systems of this type have been determined. The curves of the magnetic susceptibility as a function of temperature have been extrapolated to n → ∞. The theoretical curves are compared with experimental data.     

Unitary group approach to spin-adapted open-shell coupled cluster theory

We show that the irreducible tensor operators of the unitary group provide a natural operator basis for the exponential Ansatz which preserves the spin symmetry of the reference state, requires a minimal number of independent cluster amplitudes for each substitution order, and guarantees the invariance of the correlation energy under unitary transformations of core, open-shell, and virtual orbitals. When acting on the closed-shell reference state with n_c doubly occupied and n_v unoccupied (virtual) orbitals, the irreducible tensor operators of the group U(n_c) ? U(n_V) generate all Gelfand-Tsetlin (GT) states corresponding to appropriate irreducible representation of U(n_c + n_v). The tensor operators generating the M-tuply excited states are easily constructed by symmetrizing products of M unitary group generators with the Wigner operators of the symmetric group S_M. This provides an alternative to the Nagel-Moshinsky construction of the GT basis. Since the corresponding cluster amplitudes, which are also U(n_c) ? U(n_s) tensors, can be shown to be connected, the irreducible tensor operators of U(n_c) ? U(n_v) represent a convenient basis for a spin-adapted full coupled cluster calculation for closed-shell systems. For a high-spin reference determinant with n, singly occupied open-shell orbitals, the corresponding representation of U(n), n=n_c + n_v + n_s is not simply reducible under the group U(n_c) ? U(n_s) ? U(n_v). The multiplicity problem is resolved using the group chain U(n) ? U(n_c + n_v) ? U(n_s) ? U(n_c) ?U(n_s)? U(n_v) ? U(n_v). The labeling of the resulting configuration-state functions (which, in general, are not GT states when n_c > 1) by the irreducible representations of the intermediate group U(n_c + n_v) ?U(n_s) turns out to be equivalent to the classification based on the order of interaction with the reference state. The irreducible tensor operators defined by the above chain and corresponding to single, double, and triple substitutions from the first-, second-, and third-order interacting spaces are explicitly constructed from the U(n) generators. The connectedness of the corresponding cluster amplitudes and, consequently, the size extensivity of the resulting spin-adapted open-shell coupled cluster theory are proved using group theoretical arguments. The perturbation expansion of the resulting coupled cluster equations leads to an explicitly connected form of the spin-restricted open-shell many-body perturbation theory. Approximation schemes leading to manageable computational procedures are proposed and their relation to perturbation theory is discussed. © 1995 John Wiley & Sons, Inc.     

Molecular point group adaptation of spin–free configurations

Symmetry adaptation of spin–free multishell electron configurations in molecules to general non-Abelian point groups has been carried out. Using the basis spanning the irreducible representation [2^N/2?S, 1^2S, 0^n?N/2?S] of the unitary group U(n) as primitives, the Wigner operators for point groups were applied to generate the required basis. In the process it was found that the segments of the Weyl tableaux could be handled individually. Using this and the matric algebra of permutation group a viable procedure has been developed for point groups adaptation. A program based on the procedure has been generated and implemented.     

Far‐Infrared Spectra of Yttrium‐Doped Gold Clusters AunY (n=1–9)

The geometric, spectroscopic, and electronic properties of neutral yttrium‐doped gold clusters Au_nY (n=1–9) are studied by far‐infrared multiple photon dissociation (FIR‐MPD) spectroscopy and quantum chemical calculations. Comparison of the observed and calculated vibrational spectra allows the structures of the isomers present in the molecular beam to be determined. Most of the isomers for which the IR spectra agree best with experiment are calculated to be the energetically most stable ones. Attachment of xenon to the Au_nY cluster can cause changes in the IR spectra, which involve band shifts and band splittings. In some cases symmetry changes, as a result of the attachment of xenon atoms, were also observed. All the Au_nY clusters considered prefer a low spin state. In contrast to pure gold clusters, which exhibit exclusively planar lowest‐energy structures for small sizes, several of the studied species are three‐dimensional. This is particularly the case for Au₄Y and Au₉Y, while for some other sizes (n=5, 8) the 3D structures have an energy similar to that of their 2D counterparts. Several of the lowest‐energy structures are quasi‐2D, that is, slightly distorted from planar shapes. For all the studied species the Y atom prefers high coordination, which is different from other metal dopants in gold clusters.     

Density functional study of structural and electronic properties of maximum‐spin n+1Aun−1Ag clusters

The structures and relative stabilities of high‐spin ⁿ⁺¹Au_n?1Ag and ⁿAu_n?1Ag⁺ (n = 2–8) clusters have been studied with density functional calculation. We predicted the existence of a number of previously unknown isomers. Our results revealed that all structures of high‐spin neutral or cationic Au_n?1Ag clusters can be understood as a substitution of an Au atom by an Ag atom in the high‐spin neutral or cationic Au_n clusters. The properties of mixed gold–silver clusters are strongly sized and structural dependence. The high‐spin bimetallic clusters tend to be holding three‐dimensional geometry rather than planar form represented in their low‐spin situations. Silver atom prefers to occupy those peripheral positions until to n = 8 for high‐spin clusters, which is different from its position occupied by light atom in the low‐spin situations. Our theoretical calculations indicated that in various high‐spin Au_n?1Ag neutral and cationic species, ⁵Au₃Ag, ³AuAg and ⁵Au₄Ag⁺ hold high stability, which can be explained by valence bond theory. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

Calculation of SN 1 + N 2 ⊃ SN1 ⊗ sn 2 and U(n1 + n2) ⊃ u (n1) ⊗ u (n2) subduction coefficients by using spin graph

By use of the graphical method of spin algebra, simple, and closed expressions for S_N1+_N2 ? S_N1 ? S_N2 and U(n₁ + n₂) ? U(n₁) ? U(n₂), subduction coefficients are derived.     

Explicit representation of Gel'fand–Tsetlin states in clifford algebra unitary group approach

A explicit expression for the unitary group Clebsch–Gordan coefficients, which couple two fully antisymmetric single-column states into the two-column Gel'fand–Tsetlin states, is given in terms of isoscalar factors for the canonical subgroup chain U(n) ? U(n – 1) ? …? ? U(1). The isoscalar factors are expressed through the step numbers labeling canonical basis states and enable a straightforward construction of Gel'fand–Tsetlin states in the Clifford algebra unitary group approach, without the use of the tables for the symmetric group outer-product reduction coefficients.     

Structures,Unimolecular Fragmentations,and Reactivities of the Self‐Assembled Multimetallic/Peptide Complexes [Mnn(GlyGly‐H)2n−1]+ and [Mnn+1(GlyGly‐H)2n]2+

Complexes of Mn²⁺ with deprotonated GlyGly are investigated by sustained off‐resonance irradiation collision‐induced dissociation (SORI‐CID), infrared multiple‐photon dissociation spectroscopy, ion–molecule reactions, and computational methods. Singly [Mn_n(GlyGly‐H)_2n?1]⁺ and doubly [Mn_n+1(GlyGly‐H)_2n]²⁺ charged clusters are formed from aqueous solutions of MnCl₂ and GlyGly by electrospray ionization. The most intense ion produced was the singly charged [M₂(GlyGly‐H)₃]⁺ cluster. Singly charged clusters show extensive fragmentations of small neutral molecules such as water and carbon dioxide as well as dissociation pathways related to the loss of NH₂CHCO and GlyGly. For the doubly charged clusters, however, loss of GlyGly is observed as the main dissociation pathway. Structure elucidation of [Mn₃(GlyGly‐H)₄]²⁺ clusters has also been done by IRMPD spectroscopy as well as DFT calculations. It is shown that the lowest energy structure of the [Mn₃(GlyGly‐H)₄]²⁺ cluster is deprotonated at all carboxylic acid groups and metal ions are coordinated with carbonyl oxygen atoms, and that all amine nitrogen atoms are hydrogen bonded to the amide hydrogen. A comparison of the calculated high‐spin (sextet) and low‐spin (quartet) state structures of [Mn₃(GlyGly‐H)₄]²⁺ is provided. IRMPD spectroscopic results are in agreement with the lowest energy high‐spin structure computed. Also, the gas‐phase reactivity of these complexes towards neutral CO and water was investigated. The parent complexes did not add any water or CO, presumably due to saturation at the metal cation. However, once some of the ligand was removed via CO₂ laser IRMPD, water was seen to add to the complex. These results are consistent with high‐spin Mn²⁺ complexes.     

Electronic structure and stability of BP clusters: theoretical calculations for (BP)n (n = 2–4)

The geometry, electronic configurations, harmonic vibrational frequencies, and stability of the structural isomers of boron phosphide clusters have been investigated using density functional theory (DFT). CCSD(T) calculations show that the lowest‐energy structures are cyclic (IIt, IVs) with D_nh symmetry for dimers and trimers. The caged structure for B₄P₄ lie higher in energy than the monocyclic structure with D_2d symmetry (VIs). The B–P bond dominates the structures for many isomers, so that one preferred dissociation channel is loss of the BP monomer. The hybridization and chemical bonding in the different structures are also discussed. Comparisons with boron nitride clusters, the ground state structures of B_nP_n (n = 2, 3) clusters are analogous to those of their corresponding B_nN_n (n = 2, 3) counterparts. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Spin-free quantum chemistry. XXIII. The generator-state approach

In the unitary-group formulation of quantum chemistry, the spin-projected, configuration-state spaces of quantum chemistry are realized by the irreducible representation spaces (IRS ) of the freeon unitary group U_(n), where n is the number of freeon orbitals. The Pauli-allowed IRS are labeled by the partitions [λ] = [2^(N/2)?s, 1^2S], where N and S are the particle number and the spin, respectively. The generator-state approach (GSA ) to the unitary-group formulation consists of (1) the construction of the overcomplete, nonorthonormal generator basis for each IRS ; (2) the Lie-algebraic computation of matrix elements over generator states; (3) the Moshinsky–Nagel construction of the complete, orthonormal Gel'fand basis in terms of the generator basis; and (4) the computation of matrix elements over Gel'fand states in terms of matrix elements over generator states.     

Structures and isomerization of neutral and zwitterion serine–water clusters: Computational study

Calculations are presented for the structure and the isomerization reaction of various conformers of the bare serine, neutral serine–(H₂O)_n and serine zwitterion–(H₂O)_n (n = 1, 2) clusters. The effects of binding water molecules on the relative stability and the isomerization processes are examined. Hydrogen bonding between serine and the water molecule(s) may significantly affect the relative stability of conformers of the neutral serine–(H₂O)_n (n = 1, 2) clusters. The sidechain (OH group) in serine is found to have a profound effect on the structure and isomerization of serine–(H₂O)_n (n = 1, 2) clusters. Conformers with the hydrogen bonding between water and the hydroxyl group of serine are predicted. A detailed analysis is presented of the isomerization (proton transfer) pathways between the neutral serine–(H₂O)₂ and serine zwitterion–(H₂O)₂ clusters by carrying out the intrinsic reaction coordinate analysis. At least two water molecules need to bind to produce the stable serine zwitterion–water cluster in the gas phase. The isomerization for the serine–(H₂O)₂ cluster proceeds by the concerted double and triple proton transfer mechanism occurring via the binding water molecules, or via the hydroxyl group. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Optimization of bimetallic Cu–Au and Ag–Au clusters by using a modified adaptive immune optimization algorithm

A modified adaptive immune optimization algorithm (AIOA) is designed for optimization of Cu–Au and Ag–Au bimetallic clusters with Gupta potential. Compared with homoatom clusters, there are homotopic isomers in bimetallic cluster, so atom exchange operation is presented in the modified AIOA. The efficiency of the algorithm is tested by optimization of Cu_nAu_38‐n (0 ≤ n ≤ 38). Results show that all the structures with the putative global minimal energies are successfully located. In the optimization of Ag_nAu_55‐n (0 ≤ n ≤ 55) bimetallic clusters, all the structures with the reported minimal energies are obtained, and 36 structures with even lower potential energies are found. On the other hand, with the optimized structures of Cu_nAu_55‐n, it is shown that all 55‐atom Cu–Au bimetallic clusters are Mackay icosahedra except for Au₅₅, which is a face‐centered cubic (fcc)‐like structure; Cu₅₅, Cu₁₂Au₄₃, and Cu₁Au₅₄ have two‐shell Mackay icosahedral geometries with I_h point group symmetry. © 2009 Wiley Periodicals, Inc. J Comput Chem 2009     

Global Minimum‐Energy Structure and Spectroscopic Properties of I2.−⋅n H2O Clusters: A Monte Carlo Simulated Annealing Study

The vibrational (IR and Raman) and photoelectron spectral properties of hydrated iodine‐dimer radical‐anion clusters, I₂^.? ? n H₂O (n=1–10), are presented. Several initial guess structures are considered for each size of cluster to locate the global minimum‐energy structure by applying a Monte Carlo simulated annealing procedure including spin–orbit interaction. In the Raman spectrum, hydration reduces the intensity of the I? I stretching band but enhances the intensity of the O? H stretching band of water. Raman spectra of more highly hydrated clusters appear to be simpler than the corresponding IR spectra. Vibrational bands due to simultaneous stretching vibrations of O? H bonds in a cyclic water network are observed for I₂^.? ? n H₂O clusters with n≥3. The vertical detachment energy (VDE) profile shows stepwise saturation that indicates closing of the geometrical shell in the hydrated clusters on addition of every four water molecules. The calculated VDE of finite‐size small hydrated clusters is extrapolated to evaluate the bulk VDE value of I₂^.? in aqueous solution as 7.6 eV at the CCSD(T) level of theory. Structure and spectroscopic properties of these hydrated clusters are compared with those of hydrated clusters of Cl₂^.? and Br₂^.?.     

A method for the construction of orthogonal spin eigenfunctions

A method for the construction of the essentially idempotent and Hermitian diagonal elements of the matric algebra of the permutation group S_n is proposed. For the irreducible representation [λ] = [λ₁, λ₂] characterising a spin state S of an n-electron system, it is found that this method generates the complete set of spin projections from the appropriate primitive spin functions. The method is applied to a 7-electron system in the spin state S = M_S = 1/2 and the results are listed in the Appendix.     

Finding all‐nonmetal transition‐metal‐like superatom and its magnetic building block

In novel superatom chemistry, it is very attractive that all‐metal clusters can mimic the behaviors of nonmetal atoms and simple nonmetal molecules. Wizardly all‐metal halogen‐like superatom Al₁₃ with 2P⁵ sub shell (corresponding to the 3p⁵ of chlorine) is the most typical example. In contrast, how to mimic the behaviors of magnetic transition‐metal atom using all‐nonmetal cluster is an intriguing challenge for superatom chemistry. In response to this based on human intuition, using quantum chemistry methods and extending jellium model from metal cluster to all‐nonmetal cluster, we have found out that all‐nonmetal octahedral B₆ cluster with characteristic jellium electron configuration 1S²1P⁶2S²1D⁸ in the triplet ground state can mimic the behaviors of transition‐metal Ni atom with electron configuration 3s²3p⁶4s²3d⁸ in electronic configuration, physics and chemistry. Interestingly, the characteristic order of 1S1P2S1D for the B₆ nonmetal cluster with short B‐B lengths is different from that of the traditional jellium model—1S1P1D2S for metal clusters with long M‐M lengths, which exhibits a novel size effect of nonmetal cluster on jellium orbital ordering. Based on the jellium electron configuration, the B₆ with the spin moment value of 2μ_B is a new all‐nonmetal transition‐metal nickel‐like superatom exhibiting a new kind of all‐nonmetal magnetic superatom. Finding the application of the all‐nonmetal magnetic superatom, we encapsulate the magnetic superatom B₆ inside fully hydrogenated fullerene forming a clathrate B₆@C₆₀H₆₀ with the spin moment value of 2μ_B. As the C₆₀H₆₀ cage as a polymerization unit can conserve the spin moment of endohedral B₆, the clathrate B₆@C₆₀H₆₀ is a new all‐nonmetal magnetic superatom building block. Naturally, magnetic superatom structures of the B₆ and B₆@C₆₀H₆₀ may be metastable.