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.     

Freeon tensor product states and the unitary group formulation of the many-electron problem

The freeon tensor product basis provides a rapid method for the evaluation of matrix elements in the unitary group formulation of quantum chemistry. The method employs fast transformations between the Gel'fand and freeon tensor product basis.     

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.     

A linear combination of molecular orbital formalism

The assignment of the alternancy quantum number to the N-electron states of neutral alternant hydrocarbons is discussed within the spin-free unitary group formulation. Alternancy is defined with respect to both molecular graphs and molecular orbital eigenvalues. The properties of the molecular orbitals of alternant hydrocarbons result from requiring the assignments in terms of atomic orbital and molecular orbital Gel'fand states to be consistent. State correlation arguments are used to remove the arbitrary phase factor present in previous treatments.     

Matrix element factorization in the unitary group approach for configuration interaction calculations

Techniques of diagrammatic spin algebra are employed to derive segment factorization formulas for spin-adapted matrix elements of one- and two-electron excitation operators. The spin-adapted basis is formed by the Yamanouchi–;Kotani geneological coupling method, and therefore constitutes an irreducible basis of the unitary group U(N), as prescribed by Gel'fand and Tsetlin. Several features distinguish this paper from similar work that has recently been published. First, intermediate steps in the derivation of each segment factor are fully documented. Comprehensive tables list the spin diagrams and phases that contribute to the possible segment factors. Second, a special effort has been made to distinguish between those parts of a segment factor that can be ascribed to a spin diagram and those parts which arise from the orbitals. The results of this paper should thus be useful for those who wish to extend diagrammatic spin algebra to evaluation of matrix elements for states built from nonorthogonal orbitals. Third, a novel graphical method has been introduced to keep track of phase changes that are induced by line up permutations of creation and annihilation operators. This technique may be useful for extension of our analysis to higher excitations. The necessary concepts of second quantization and diagrammatic spin algebra are developed in situ, so the present derivation should be accessible to those who have little prior knowledge of such methods.     

Spin-free quantum theory. XXV. The unitary-group formulation of fermion many-body theory

The fermion unitary group formulation (UGF ) of many-body theory is based on the unitary group U(2n) where n is the number of freeon orbitals. This formulation, which conserves particle-number but not spin, is isomorphic to the particle-number-conserving, second-quantized formulation (SQF ). In UGF we derive the familiar diagrammatic algorithm for matrix elements, M(Y) = (?1)^H+L where H and L denote the numbers of hole lines and loops in the diagram D(Y) of M(Y). The unitary group derivation is considerably simpler than is the conventional, second-quantized derivation that employs time-dependence, Wick's theorem, normal-order, and contractions. In neither fermion UGF nor SQF is spin conserved. We carry out in UGF the spin-projection (symmetry adaptation to SU (2)) of the fermion vectors and obtain with a spin-free Hamiltonian the same matrix elements as with the freeon UGF (part 24 of this series). The fermion unitary group formulation for a spin-free Hamiltonian should be regarded as an alternate path to spin-free quantum chemistry.     

Valence bond approach exploiting Clifford algebra realization of Rumer–Weyl basis

A detailed algorithm is described that enables an implementation of a general valence bond (VB ) method using the Clifford algebra unitary group approach (CAUGA ). In particular, a convenient scheme for the generation and labeling of classical Rumer–Weyl basis (up to a phase) is formulated, and simple rules are given for the evaluation of matrix elements of unitary group generators, and thus of any spin-independent operator, in this basis. The case of both orthogonal and nonrothogonal atomic orbital bases is considered, so that the proposed algorithm can also be exploited in molecular orbital configuration interaction calculations, if desired, enabling a greater flexibility for N-electron basis-set truncation than is possible with the standard Gel'fand–Tsetlin basis. Finally, an exploitation of this formalism for the VB method, based on semiempirical Pariser–Parr–Pople (PPP )-type Hamiltonian and nonorthogonal overlap-enhanced atomic orbital basis, and its computer implementation, enabling us to carry out arbitrarily truncated or full VB calculations, is described in detail.     

Combinatorics of angular momentum recoupling theory: spin networks,their asymptotics and applications

The quantum theory of angular momentum and the associated Racah–Wigner algebra of the Lie group SU(2) have been widely used in many branches of theoretical and applied physics, chemical physics, and mathematical physics. This paper starts with an account of the basics of such a theory, which represents the most exhaustive framework in dealing with interacting many-angular momenta quantum systems. We then outline the essential features of this algebra, that can be encoded, for each fixed number N = (n + 1) of angular momentum variables, into a combinatorial object, the spin network graph, where vertices are associated with finite-dimensional, binary coupled Hilbert spaces while edges correspond to either phase or Racah transforms (implemented by 6j symbols) acting on states in such a way that the quantum transition amplitude between any pair of vertices is provided by a suitable 3nj symbol. Applications of such a combinatorial setting—both in fully quantum and in semiclassical regimes—are briefly discussed providing evidence of a unifying background structure.     

Unitary group approach to the many-electron problem. I. Matrix element evaluation and shift operators

This is the first paper in a series of three directed toward the evaluation of spin-dependent Hamiltonians directly in the spin-orbit basis. In this paper we present a new and complete derivation of the matrix elements of the U(n) generators in the electronic Gel'fand basis. The approach employed differs from previous treatments in that the matrix elements of nonelementary generators are obtained directly. A general matrix element formula is derived which explicitly demonstrates the segment level formalism obtained previously by Shavitt using different methods. A simple relationship between the matrix elements of raising and lowering generators is determined which indicates that in CI calculations, only the matrix elements of raising generators need be calculated. Some results on the matrix elements of products of two generators are also presented.     

Freeon unitary group formulation of Hartree-Fock theory

Summary Hartree—Fock theory was a major topic in Professor Löwdin's famous 1955Physical Review papers. His development was based on fermion orbitals and the Slater determinant. Since that time there has been developed, at the University of Texas, the freeon, unitary-group formulation of quantum chemistry as a viable alternative to the fermionic formulations of nonrelativistic quantum chemistry. We wish to express our appreciation to Professor Löwdin for his strong support of our freeon studies and for many helpful conversations.     

Spin symmetry and gel'fand patterns of higher SU2×ℓ n groups

The nature of? ₅ spin algebra is considered together with? ₅↓? ₅↓C _4v mapping in order to specify the dual spin symmetry for MQ-NMR ofnido-¹¹B₅H₉. The forms of Gel'fand shapes for? ₅ spin symmetry are presented to show how they specify thefull range of multiplicities found in higher-n ? _n -partitions. Tuples, or number-partitions and their? _n ↓G invariance sets provide models for both the five-foldI _i =3/2 component, and the two distinct types of spin-1/2 subsystems. The full spin symmetry is derived in terms of the direct product ((? ₅↓C _4v )?(? ₄↓C _4v ))^1/2?(? ₅↓C _4v )^3/2. The concepts used are implicit in the substructure ofp-tuple model invariances over the subduced symmetry, or derive from the inner tensor product algebras under the? _n group. Both as a check on the combinatorially derived multiplicities of [λ]s and for insight into (non-simply-reducible) substructure of number-partitions, the study of mapping from :hrr'.:-tuples onto the? _n -partitional set for higherI _i is invaluable. The motivation for this work lies in its pertinence to the MQ-NMR spin dynamics of clusterlike molecules. The accessible information content of a spin algebra over either form of spin space is bound up with a suitable symmetry partitioning of the problem, as implied by the use of {T ^kq (v:[λ])} bases within higherq subspaces of the Liouville formalism.     

Unitary symmetry and classification of the states of <Emphasis Type="Italic">n</Emphasis>-spin clusters. Magnetic and thermodynamic parameters

The unitary symmetry and classification of spin clusters by spin momenta S are considered on the basis of reduction of the full linear group to unitary groups U _{2s + 1} and orthogonal rotation group R ₃. Reduction of the permutation group P _n of n spins to the point group of the cluster is applied to the classification of the spatial states of a spin cluster with the use of permutation quantum numbers introduced in this work and the Young diagrams of the permutation group P _n . Examples of the classification of spin systems with spins s = 1/2, 1, 3/2, 2, and 5/2 with U _{2s + 1} × P _n groups (n = 5–15) are reported. This classification is common for all spin clusters and is the same for both cyclic clusters and 3D clusters with symmetry groups of a crystal. On the basis of this classification, the magnetic and thermodynamic parameters of a spin system are calculated as a function of the number of spins and temperature. For s = 1/2 clusters, the analytical formulas are derived for magnetic susceptibility, internal energy, heat capacity, and entropy as a function of quantum numbers for a cluster with any number of spins, and their dependences on temperature and the number of atoms are considered.     

Characters of two-row representations of the symmetric group

Characters of irreducible representations (irreps) of the symmetric group corresponding to the two-row Young diagrams, i.e., describing transformation properties of N-electron eigenfunctions of the total spin operators, have been expressed as explicit functions of the number of electrons N and of the total spin quantum number S. The formulas are useful in various areas of theory of many-electron systems, particularly in designing algorithms for evaluation of spectral density moments. © 1997 John Wiley & Sons, Inc.     

A quantum electronic theory of chemical processes—The inverted energy profile case: CH3+ + H2 reaction

A quantum approach to chemical processes is developed. The chemical interconversion is described as an electronic process. The reaction corresponds to histories involving quantum states belonging to different stationary molecular Hamiltonians. The system may be embedded in a weak (thermal) and/or external electromagnetic field. The electromagnetic transverse fields lead to transition moments yielding finite probability amplitudes for the system to change from one quantum state to another. Bottleneck subspaces (transition states) are defined; they mediate the interconversions in generic unimolecular and bimolecular processes. Active precursor and successor complexes are introduced to help bridge reactant and product electronic states. The stationary states are modeled with Born-Oppenheimer Hamiltonians. At a qualitative level, the theory is general. The rate, measured as a time derivative of product concentration, is expressed in terms of concentrations of active precursor and successor complexes. The kinetic coefficients are given in terms of quantum processes involving electronic bottleneck states. Stationary structures and vibrational zero-point energies characterizing the reactive CH₃++H₂ system are determined at a Hartree-Fock level of theory with 6-31++G** basis set. The vibrational levels are corrected with anharmonicity effects. The saddle point of index one for hydrogen scrambling reactions has been obtained and shown to be related to the CH₅⁺ molecular complex together with the precursor and successor complexes geometries. The unusual properties of the system with respect to standard transition-state theory are fairly well described within this approach, in particular, isotope scrambling as well as photon emission during formation of the carbocation. The theory suggests that these types of reactions, which are found in outer space, may contribute to the scattering of the cosmic microwave background. © 1997 John Wiley & Sons, Inc.     

Theory and spectroscopy of an incarcerated quantum rotor: The infrared spectroscopy,inelastic neutron scattering and nuclear magnetic resonance of H2@C60 at cryogenic temperature

The supramolecular complex, H₂@C₆₀, represents a model of a quantum rotor in a nearly spherical box. In providing a real example of a quantum particle entrapped in a small space, the system cuts to the heart of many important and fundamental quantum mechanical issues. This review compares the predictions of theory of the quantum behaviour of H₂ incarcerated in C₆₀ with the results of infrared spectroscopy, inelastic neutron scattering and nuclear magnetic resonance. For H₂@C₆₀, each of these methods supports the quantization of translational motion of H₂ and the coupling of the translational motion with rotational motion and provides insights to the factors leading to breaking of the degeneracies of states expected for a purely spherical potential. Infrared spectroscopy and inelastic neutron scattering experiments at cryogenic temperatures provide direct evidence of a profound quantum mechanical feature of H₂ predicted by Heisenberg based on the Pauli principle: the existence of two nuclear spin isomers, a nuclear spin singlet (para-H₂) and a nuclear triplet (ortho-H₂). Nuclear magnetic resonance is capable of probing the local lattice environment of H₂@C₆₀ through analysis of the H₂ motional effects on the ortho-H₂ spin dynamics (para-H₂, the nuclear singlet state, is NMR silent). In this review we will show how the information obtained by three different forms of spectroscopy join together with quantum theory to create a complementary and consistent picture which strikingly shows the intrinsically quantum nature of H₂@C₆₀.     

The Adsorption of Ag on (CdTe)<Subscript>13</Subscript> Core-Cage Nanocluster: A Computational Study

Cadmium chalcogenide semiconductor quantum dots, especially doped nanoclusters, have attracted great attention for their effects on photo generated carriers and their lifetime due to introduced trapping states by changing surface unbonded orbitals. Here, we investigate the adsorption of Ag on “magic-sized” cadmium chalcogenide (CdTe)₁₃ core-cage nanoclusters, Cd₁₃Te₁₃Ag, by first-principles density functional theory. All possible adsorption sites, top, bridge, and hollow sites, have been considered. Particular attention is paid to the energy band structures of Cd₁₃Te₁₃Ag. The study demonstrates that the hollow sites, the centers of hexagons, are the favorite Ag adsorption sites. Unlike observed shallow acceptor level of doped QDs, two unusual deep mid-gap states with different spins, spin up and spin down, are observed. These two deep states shift with Ag moving towards the core of cage. The detail properties of adsorption configurations and these two deep states are analyzed. These two deep states should have important role to their optical applications.     

A unitary group formulation of many-body theory: The spin-shift formalism

This paper lays the algebraic foundation of a unitary group approach to many-body theory. We define a set of second quantized spin-shift operators which are used to construct spin-adapted many-electron configuration functions. We investigate the particle-hole transformation, normal ordering, and contraction of spin shifts. This gives an orbital Wick's theorem reflecting the permutational structure of the states spanning the irreducible representations of the spatial unitary group. We study the spinless Hamiltonian, and the relationship between spin shifts and unitary group generators.     

The combinatorics of graphical unitary group approach to many electron correlation

The combinatorics of Gel'fand states which are useful in the graphical unitary group approach to many electron correlation problem and spin free quantum chemistry is considered. Using operator theoretic methods it is shown that the generators of Gel'fand states are S-functions.     

Local physical quantities for spin based on the relativistic quantum field theory in molecular systems

Local physical quantities for spin are investigated on the basis of the four‐ and two‐component relativistic quantum theory. In the quantum field theory, local physical quantities for spin such as the spin angular momentum density, spin torque density, zeta force density, and zeta potential play important roles in spin dynamics. We discuss how to calculate these local physical quantities based on the two‐component relativistic quantum theory. Some different types of relativistic numerical calculations of local physical quantities in Li atom and C₆H₆ are demonstrated and compared. Local physical quantities for each orbital are also discussed, and it is seen that a total local zeta potential is given as a result of some cancellation of large contributions from each orbital. © 2016 Wiley Periodicals, Inc.