Symmetry Adapted Research Articles

Efficacy of the symmetry-adapted basis for ab initio nucleon-nucleus interactions for light- and intermediate-mass nuclei

We study the efficacy of a new ab initio framework that combines the symmetry-adapted (SA) no-core shell-model approach with the resonating group method (RGM) for unified descriptions of nuclear structure and reactions. We obtain ab initio neutron-nucleus interactions for 4He, 16O, and 20Ne targets, starting with realistic nucleon-nucleon potentials. We discuss the effect of increasing model space sizes and symmetry-based selections on the SA-RGM norm and direct potential kernels, as well as on phase shifts, which are the input to calculations of cross sections. We demonstrate the efficacy of the SA basis and its scalability with particle numbers and model space dimensions, with a view toward ab initio descriptions of nucleon scattering and capture reactions up through the medium-mass region.

Nuclear Dynamics and Reactions in the Ab Initio Symmetry-Adapted Framework

We review the ab initio symmetry-adapted (SA) framework for determining the structure of stable and unstable nuclei, along with related electroweak, decay, and reaction processes. This framework utilizes the dominant symmetry of nuclear dynamics, the shape-related symplectic [Formula: see text] symmetry, which has been shown to emerge from first principles and to expose dominant degrees of freedom that are collective in nature, even in the lightest species or seemingly spherical states. This feature is illustrated for a broad scope of nuclei ranging from helium to titanium isotopes, enabled by recent developments of the ab initio SA no-core shell model expanded to the continuum through the use of the SA basis and that of the resonating group method. The review focuses on energies, electromagnetic transitions, quadrupole and magnetic moments, radii, form factors, and response function moments for ground-state rotational bands and giant resonances. The method also determines the structure of reaction fragments that is used to calculate decay widths and α-capture reactions for simulated X-ray burst abundance patterns, as well as nucleon–nucleus interactions for cross sections and other reaction observables.

Neural network based quasi-diabatic Hamiltonians with symmetry adaptation and a correct description of conical intersections.

In a previous paper, we have demonstrated that artificial neural networks (NNs) can be used to generate quasidiabatic Hamiltonians (Hd) that are capable of representing adiabatic energies, energy gradients, and derivative couplings. In this work, two additional issues are addressed. First, symmetry-adapted functions such as permutation invariant polynomials are introduced to account for complete nuclear permutation inversion symmetry. Second, a partially diagonalized representation is introduced to facilitate a better description of near degeneracy points. The diabatization of 1, 21A states of NH3 is used as an example. The NN fitting results are compared to that of a previous fitting with symmetry adapted polynomials.

Full-valence density matrix renormalisation group calculations on meta-benzyne based on unrestricted natural orbitals. Revisit of seamless continuation from broken-symmetry to symmetry-adapted models for diradicals

ABSTRACTIn this work, we show that the natural orbitals of unrestricted hybrid density functional theory (UHDFT) can be used as the active space orbitals to perform multireference (MR) calculations, for example, the density matrix renormalisation group (DMRG) and Mukherjee-type (Mk) MR coupled-cluster (CC) method. By including a sufficiently large number of these natural orbitals, full-valence (FV) active space can be identified without recourse of the expensive self-consistent procedures for DMRG-SCF. Several useful chemical indices are derived based on the occupation numbers of the natural orbitals for seamless continuation from broken-symmetry (BS) to symmetry-adapted (SA) methods. These procedures are used on 1,3-didehydrobenzene (meta-benzyne) to calculate its singlet (S)-triplet (T) gap. We compare our results to available experiments and computational results obtained by several other groups. We see our procedures as a seamless bridge between single-reference BS methods, such as UHDFT, and the SA MR methods, such as FV DMRG and MkMRCC.

Quantum phase crossovers with finite atom number in the Dicke model

Two-level atoms interacting with a one-mode cavity field at zero temperature have order parameters which reflect the presence of a quantum phase transition at a critical value of the atom–cavity coupling strength. Two popular examples are the number of photons inside the cavity and the number of excited atoms. Coherent states provide a mean field description, which becomes exact in the thermodynamic limit. Employing symmetry-adapted (SA) SU(2) coherent states the quantum crossover, precursor of the critical behavior, can be described for a finite number of atoms. A variation after projection treatment, involving a numerical minimization of the SA energy surface, associates the quantum crossover with a discontinuity in the order parameters, which originates from competition between two local minima in the SA energy surface. Although this discontinuity is not present in finite systems, it provides a good description of 1/N effects in the observables.

Spatial dependence of hyperpoarizabilty of van der Waals supermolecule H2-Ne. Its influence on spectral properties of hyper-Rayleigh collisionally scattered light

A series of reports has been published previously for several last years on the nonlinear hyper-Rayleigh (HR) collisional light scattering in molecular media composed of interacting atomic and atomic-diatomic compounds. The presented study continues and extends these theoretical and numerical analyses. Especially, a careful insight into the spectral properties of the most recent case considered - the HR scattering in H2-Ne gas mixtures -is given by calculating and examining a complete set of spectral line contributions related to symmetry adapted (SA) components of the collisional first hyperpolarizability tensor ΔβΛL(K). The influence of the ΔβΛL(K)(R) functional behavior on the HR spectral line shapes is found and characterized in details. A possibility of estimating numerical values of the SA contributions is discussed. Moreover, the long distance functional dependencies of Δβ(R), obtained by means of the numerical quantum chemistry methods, are confronted with those derived on the basis of an analytical approach within the molecular interaction models proposed at the early stage of development of the collisional nonlinear spectroscopy by Kielich and coworkers.

Asymptotic multipolar expansion of collision-induced properties

The collision-induced (CI) dipole moment, CI polarizability, and CI hyperpolarizability are considered for three H(2)-rare gas (Rg) pairs (Rg = He, Ne, Ar). In this study, the symmetry-adapted (SA) components, the projection of the CI dipole, polarizability, and hyperpolarizability on an appropriately tailored set of spherical harmonics are calculated. A set of equations for the respective SA components is derived. The Cartesian components of the CI properties calculated by quantum chemistry methods for three intermolecular geometries are used in our calculations as input data. The analytical, multipolar long-range behavior of the CI properties studied is considered within a multipole-induced multipole model. Taking the SA components at large distances, the ab initio SA numerical results and the model semianalytical data were compared. In general, a good agreement has been found. The results of our study are expected to be of value in spectral line shape analysis and in modeling of processes in the Earth's and planetary atmospheres.

Symmetry breaking in spin-restricted Hartree–Fock solutions: the case of the C2 molecule and the N2+ and F2+ cations

We examine a spin-preserving stability of restricted Hartree-Fock (RHF) and open-shell RHF (ROHF) solutions for homonuclear diatomic species, namely the molecule C(2) and the N(2)(+) and F(2)(+)cations, in the entire relevant range of internuclear separations. In the presence of respective singlet, doublet, or triplet instabilities we explore the implied broken-symmetry (BS) solutions and check their stability. We also address the occurrence of vanishing roots rendered by the stability problem in the case of BS solutions. Since for homonuclear diatomic species the space symmetry of the nuclear framework cannot be broken, the only possible space symmetry breaking involves either the electron charge distribution or the rotational symmetry about the internuclear axis. Such BS solutions arise primarily due to the symmetry breaking of the valence, mostly frontier, molecular orbitals, which approach atomic-type orbitals in the dissociation limit. The resulting BS RHF or ROHF solutions yield much more realistic potential energy curves (PECs) than do the symmetry-adapted (SA) solutions. For the sake of comparison, we also generated corresponding PECs using the density functional theory (DFT). Finally, we examine the role of BS molecular orbitals in post-HF correlated approaches to the many-electron problem, specifically in the computation of equilibrium properties using the coupled-cluster method with singles and doubles (CCSD) and its perturbatively corrected version for triples, the CCSD(T) method.

Independent particle model of spontaneous symmetry breaking in planar π-electron systems

The singlet stability of symmetry adapted (SA), restricted Hartree-Fock (RHF) solutions, and the implied symmetry breaking for several planar, π-electron systems, is investigated using the semiempirical Pariser-Parr-Pople Hamiltonian in the whole range of the coupling constant. We focus here on highly symmetric cyclic polyenes C10H10 and C14H14 and their various distorted analogues of lower symmetry, in particular on the perimeter models of naphthalene and anthracene (p-naphthalene and p-anthracene) modeling the so-called [n]-annulenes. Relying on earlier results for general systems with conjugated double-bonds, we explore the character and properties of both the SA and broken-symmetry (BS) RHF solutions for these systems and relate their behavior to those of highly symmetric cyclic polyenes and corresponding polyacenes. In this way we are able to provide a better understanding of the spontaneous symmetry breaking in these systems at the Hartree-Fock level of approximation.

A complete list of symmetry adapted expressions to the fourth power for compact bending potentials in molecules with and symmetry from a general symbolic algebra program

An algorithm is proposed to perform the reduction of direct product representations of finite groups and, in particular, to perform the complete symmetry adaption of power representations, which may be used to obtain compact analytical representations of potential energy surfaces following an idea described in Marquardt and Quack [J. Chem. Phys. 109, 10628 (1998)]. The algorithm is general in the sense that it can be applied to any finite group being characterized by its set of irreducible representations. It is automatic in the sense that, in case the reduction yields multiple degenerate irreducible subspaces of the same species, all degenerate irreducible subspaces are obtained with a coherent phase relation. The algorithm is based on the standard reduction rule of traditional representation theory. A symbolic algebra computer program based on MAPLE is presented and applied here to obtain the complete list of symmetry adapted expressions of bond angle coordinates up to the fourth power in all irreducible representations of the C3v and Td point groups.

Theory of chemical bonds in metalloenzymes V: Hybrid‐DFT studies of the inorganic [8Fe–7S] core

AbstractThe electronic and spin structures of the [8Fe–7S] inorganic model complex (1) have been investigated using the density functional theory (DFT) UB3LYP and UBLYP methods. The experimental geometry of 1 was used to calculate the sets of exchange interactions (J values) between Fe‐spin centers within spin Hamiltonian models. The experimental temperature‐dependent paramagnetism is well reproduced on the basis of the calculated J values. The calculated low‐lying energy levels are also consistent with those of the experiments. This indicates that the ground spin state of 1 is singlet (S = 0) for the weak antiferromagnetic coupling between two [4Fe–3S] cubane units, which take the Fe(III)Fe(II)3 formal charge and S = 7/2 spin states. The first‐order density matrix of the most stable broken‐symmetry (BS) solution has been diagonalized to elucidate the natural molecular orbitals (MOs) and their occupation number of 1, which are hardly obtained by the symmetry‐adapted (SA) CASSCF approach. Several chemical indices such as the effective bond orders before and after spin projection are also calculated by using the occupation numbers to provide the SA MO picture of labile chemical bonds of 1. It is found that the BS MO calculation followed by spin‐projected SA analysis is an effective and practical alternative to the CASSCF MO approach for the multinuclear transition metal clusters. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006

Theoretical calculations of effective exchange integrals by spin projected and unprojected broken-symmetry methods. I. Cluster models of K2NiF4-type solids

Previously, various symmetry-adapted (SA) and broken-symmetry (BS) computations have been performed for strongly correlated transition metal species so as to examine magnetic properties in simple cluster models such as binuclear transition metal oxides. Though SA computations such as the complete active space configuration interaction and CASSCF are desirable for estimating physical constants, these computations are heavy for larger cluster models with strongly correlated electrons. K2NiF4 is known to be the two-dimensional perovskite-type antiferromagnet and to have the electronic configuration similar to that of La2CuO4. Here, we have examined the utility and applicability of the BS spin-polarized hybrid-density functional theory (HUDFT) for cluster models of K2NiF4. As the result, HUDFT calculation such as UB2LYP has provided the reasonable effective exchange integral (Jab) followed by our approximately spin projected scheme, in comparison to the experimental one. It was also found that the square planar tetranuclear model has provided the most reasonable Jab value by HUDFT. In addition, external effects such as putting point charges around cluster models and changing distances between nickel and fluorine have been also examined. The natural orbital analysis by HUDFT has been carried out to obtain natural orbitals and their occupation numbers. Charge density, spin density, and chemical indices expressed by the occupation numbers have been also obtained to elucidate the nature of the chemical bonds in the K2NiF4-type solids.

Systematic Comparisons between Broken Symmetry and Symmetry-Adapted Approaches to Transition States by Chemical Indices: A Case Study of the Diels−Alder Reactions

Systematic comparisons between broken symmetry (BS) and symmetry-adapted (SA) approaches to transition states were carried out using chemical indices, which are defined using the occupation numbers of natural orbitals (NOs) by both BS and SA methods. For the former, unrestricted Hartree−Fock, UMP2, UQCISD(T), and UCCSD(T), together with hybrid density functional theory (DFT) methods such as UB2LYP and UB3LYP, were employed, while CASSCF, UNO CASCI, UNO CASCI MP2 (≡ CASMP2(CI)), and MRMP2 were used for the latter approach. To begin with, we examined the functional behaviors of chemical indices during the C−C bond dissociation process of ethylene or the least motion association reaction of triplet methylenes. The bonding characteristics revealed by the chemical indices are largely dependent on the computational methods. Next, the energy difference between concerted and nonconcerted transition structures of the Diels−Alder reaction of butadiene and ethylene was calculated by both BS and SA methods. The origin of excellent energetics by the hybrid DFT methods for this reaction was revealed in terms of the chemical indices. These indices are also useful to understand the nature of the chemical bonds at TSs and the role of unstable intermediates in stepwise mechanisms. Furthermore, the ability of phenyl groups to stabilize nonsynchronous TS for the Diels−Alder and the Cope rearrangement reactions was examined by the NO analyses of the BS hybrid DFT solutions. The difference of the chemical indices for substituted systems provides useful information for the theoretical consideration of the reaction mechanism and stereospecificity of pericyclic reactions. Finally, implications of the computational results are discussed in relation to appropriate selection of effective hybrid DFT methods for large systems with moderate radical characters, for which the SA computations are difficult.

Solution of the time-dependent Schrödinger equation for highly symmetric potentials

The method of symmetry adapted wavepackets (SAWP) to solve the time-dependent Schrödinger equation for a highly symmetric potential energy surface is introduced. The angular dependence of a quantum-mechanical wavepacket is expanded in spherical harmonics where the number of close-coupled equations for the corresponding radial functions can be efficiently reduced by symmetry adaption of the rotational basis using the SAWP approach. Various techniques to generate symmetry adapted spherical harmonics (SASHs) for the point groups of highest symmetry (octahedral, icosahedral) are discussed. The standard projection operator technique involves the use of Wigner rotation matrices. Two methods to circumvent numerical instabilities occurring for large azimuthal quantum numbers are suggested. The first is based on a numerical scheme which employs Gaussian integrations yielding exact and stable results. The second is a recursive algorithm to generate higher order SASHs accurately and efficiently from lower order ones. The paper gives a complete set of “seed functions” generated by projection techniques which can be used to obtain SASHs for all irreducible representations of the octahedral and icosahedral point groups recursively.

From nuclei to molecules: a symmetry adapted algebraic model of vibrational excitations

Algebraic techniques orginally used in nuclear physics can be extended to molecular systems by combining Lie-algebraic and point-group techniques. The symmetry adaptation gives rise to a direct connection between the algebraic and traditional configuration-space descriptions. We illustrate the model with an application to .

Point group symmetry adaptation in clifford algebra unitary group approach

AbstractThe symmetry adaptation procedure of Chen et al. [Sciencia Sinica 23, 1116 (1980)], which can account for the invariance properties of the Hamiltonian with respect to any finite point group G, is both modified and adapted to the Clifford algebra unitary group approach (CAUGA). From orthogonal symmetry adapted Mo's, one first constructs a pure configuration many‐electron basis adapted to the chain U(ni) ⊃ G ⊃ G(s) in terms of the U(ni) Gel'fand–Tsetlin (GT) basis, where ni is the dimension of the irrep defining a given pure configuration, and G(s) designates the canonical chain supplying a unique labeling. The pure configuration basis is then coupled to the desired G‐adapted states using the point group Clebsch–Gordan coefficients and the U(n1) ⊂ U(n1 + n2) ⊂ …︁ ⊂ U(n) basis by using the permutation group outer‐product reduction coefficients. This basis can be expressed in terms of the U(n) GT basis by using the U(n) subduction coefficients (SAC'S). The SDC'S are particularly simple for the highest weight states (Hess's) of various subproblems, which can be in turn represented through the U(2n) two‐box Weyl tableaux of CAUGA. The non‐HWS's are obtained by applying the U(ni) lowering generators to the HWS's. In this way we can directly obtain the spin and point group adapted CAUGA basis. The procedure is illustrated on a nontrivial example.

Adaptation à la symétrie, équivalents opératoriels et résonance magnétique

Basic quantities for symmetry adaptation are discussed in connection with molecular and solid state physics. This gives rise to a formalism whose central elements are operator equivalents adapted to a point group. Such symmetry adapted operator equivalents are defined in terms of Schwinger operators so that they cover the off-diagonal and diagonal cases. Special emphasis is put on the applications of the formalism to magnetic resonance. More specifically, it is shown how to apply the formalism to the construction, the study of the transformation properties, and the determination of the eigenstates of a generalized spin Hamiltonian. Numerous examples are given, as well as key tables relative to the chain [Formula: see text], to simplify the application of the formalism to electron paramagnetic resonance.

ChemInform Abstract: SYMMETRY ADAPTED FUNCTIONS AND NORMALIZED SPHERICAL HARMONIC (NSH) HAMILTONIANS FOR THE POINT GROUPS O(H), T(D), D(4H), D(2D), C(4V), D(2H), C(2V)

Chemischer InformationsdienstVolume 5, Issue 21 Physical Inorganic Chemistry ChemInform Abstract: SYMMETRY ADAPTED FUNCTIONS AND NORMALIZED SPHERICAL HARMONIC (NSH) HAMILTONIANS FOR THE POINT GROUPS O(H), T(D), D(4H), D(2D), C(4V), D(2H), C(2V) J. C. HEMPEL, J. C. HEMPELSearch for more papers by this authorJ. C. DONINI, J. C. DONINISearch for more papers by this authorB. R. HOLLEBONE, B. R. HOLLEBONESearch for more papers by this authorA. B. P. LEVER, A. B. P. LEVERSearch for more papers by this author J. C. HEMPEL, J. C. HEMPELSearch for more papers by this authorJ. C. DONINI, J. C. DONINISearch for more papers by this authorB. R. HOLLEBONE, B. R. HOLLEBONESearch for more papers by this authorA. B. P. LEVER, A. B. P. LEVERSearch for more papers by this author First published: May 28, 1974 https://doi.org/10.1002/chin.197421003AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat No abstract is available for this article. Volume5, Issue21May 28, 1974 RelatedInformation

Computer Generation of Molecular Symmetry Orbitals

We show how molecular symmetry adapted linear combinations of atomic orbitals can be produced by carrying out standard manipulations on the matrices that represent values for each of two rotationally invariant operators evaluated over the atomic orbitals of interest. Attention is given to obtaining final orbitals that have not only the desired transformation properties but that are also ``beautiful'' in the sense that they have simple atomic orbital coefficients like the symmetry orbitals that are traditionally discussed in the literature and generated by inspection. Our implementation of this method in Fortran on a CDC 3600 requires a time for symmetry adaptation that increases as the symmetry of the molecule becomes greater. In several calculations that we have done on small (5–12 atoms), reasonably symmetrical (D2h–Oh) molecules, the adaptation required times comparable to those required for calculating the overlap and kinetic energy matrix elements used in the symmetry adaptation.