Normalized irreducible tensorial matrices and the Wigner–Eckart theorem for unitary groups: A superposition hamiltonian constructed from octahedral NITM

The concepts of normalized irreducible tensorial matrices (NITM ) are extended to all finite and compact unitary groups by a development that clarifies their relationship to group theory and matrix algebra. NITM for a unitary group G are shown to be elements of a basis obtained by symmetry adapting to G the matrix basis of a matrix space M (α₁ × α₂). Elements [X] ∈ M (α₁ α₂) transform under G_a ∈ G according to [G_a] [X][G^?1_a], where [G_a] and [G^?1_a] belong to irreducible representations of G . The usual properties of NITM and the Wigner–Eckart theorem follow from these results, which are valid for both finite and compact unitary groups. The NITM span M (α₁ × α₂) are orthonormal under the trace and transform irreducibly with respect to G . This NITM basis of M (α₁ × α₂) is said to be simple. A compound NITM basis of a matrix space results when the space is partitioned into two or more subspaces, each spanned by a simple NITM basis. NITM determined from Griffith's V coefficients for the octahedral group are tabulated and used to construct a six-coordinate superposition Hamiltonian.     

Elements of irreducible tensorial matrices generated by finite groups with applications to ligand field Hamiltonians

Using symmetry to determine Hamiltonian matrix elements for quantum systems with finite group symmetry is a special case of obtaining group-generated irreducible tensorial matrices. A group-generated irreducible tensorial matrix transforms irreducibly under the group and is a linear combination of group transformations on a reference matrix. The reference matrix elements may be appropriate integrals or parameters. The methods of normalized irreducible tensorial matrices (NITM) are employed to express elements of the generated matrix in terms of those of the reference matrix without performing the actual transformations. Only NTTM components of the reference matrix with the same transformation properties as the group-generated matrix will contribute to its elements. The elements of invariant symmetry-generated matrices are proportional to simple averages of certain elements of the reference matrix. This relation is substantially more efficient than previous techniques for evaluating matrix elements of octahedral and tetragonal d-type ligand-field Hamiltonians.     

Expression of theSU(2) rotation matrices D( j ) in terms of normalized irreducible tensorial matrices

TheSU(2) rotation matricesD ^(j), specified in terms of axis and angle of rotation, are expressed as linear combination of normalized irreducible tensorial matricces (NITM) of rankl = 0 to 2j rotated to the polar angles of the axis. The rotated NITM are constructed from spherical harmonics of the same rank. Since this formulation requires no matrix products, it may be computationally more efficient than Euler angle formulas, particularly for largej. Rotated NITM and formulas for theD ^(j) withj = 1/2 andj = 1 are written out explicitly. A formula for the structure constants of the products of conformable NITM is also given in terms of 3-j and 6-j symbols.     

Relevant space within the spin-adapted reduced Hamiltonian theory. II. Study of the π cloud in benzene and naphthalene

The properties of the spin-adapted reduced Hamiltonian (SRH) matrices and of their eigenvectors permit in many cases a projection of the two-electron matrices, which amounts to an effective truncation of the basis at the stage of the calculations which are time and as memory consuming. Besides this effective truncation of the basis, another simplification can be introduced by segregating an n-electron cloud from the N electrons of the system. Thus, the energy and the electron distribution of a smaller electronic cloud, for instance, the π or the σ cloud in aromatic systems, can be calculated; their separability being subsequently analyzed. Different relevant spaces have been examined in the study of the π-electron cloud in benzene and naphthalene. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65 : 107–119, 1997     

Discrete variable representation method applied to the determination of rotation-vibration bound states of NO2

The discrete variable representation method is applied to the determination of the rotation-vibration energy levels of the fundamental electronic state of NO₂. The Hamiltonian is expressed in Johnson hyperspherical coordinates and developed on a DVR basis for each internal coordinate, while parity-adapted linear combinations of Wigner functions are used to describe the rotational motion. The diagonalization of the Hamiltonian matrix is performed using the Lanczos algorithm for large symmetric and Hermitian matrices. Results for rovibrational states up to J = 11 for the first five vibrational energy levels are presented. © 1997 John Wiley & Sons, Inc.     

Hamiltonian matrix and reduced density matrix construction with nonlinear wave functions

An efficient procedure to compute Hamiltonian matrix elements and reduced one- and two-particle density matrices for electronic wave functions using a new graphical-based nonlinear expansion form is presented. This method is based on spin eigenfunctions using the graphical unitary group approach (GUGA), and the wave function is expanded in a basis of product functions (each of which is equivalent to some linear combination of all of the configuration state functions), allowing application to closed- and open-shell systems and to ground and excited electronic states. In general, the effort required to construct an individual Hamiltonian matrix element between two product basis functions H(MN) = M|H|N scales as theta (beta n4) for a wave function expanded in n molecular orbitals. The prefactor beta itself scales between N0 and N2, for N electrons, depending on the complexity of the underlying Shavitt graph. Timings with our initial implementation of this method are very promising. Wave function expansions that are orders of magnitude larger than can be treated with traditional CI methods require only modest effort with our new method.     

Analytical block-diagonalization of matrices using unit spherical tensors

A new method for block-diagonalizing large Hamiltonian matrices, in closed form, is described. The method is based on (i) a general unitary transformation due to Slichter, and (ii) Fano's unit spherical operatorsÛ _Q ^K (I _i,I _i). The method is illustrated with a simple three spin 1/2 dipolar coupled spin system, characterized by off-block-diagonal unit spherical tensorsÛ ₀ ² (3/2,1/2,) andÛ ₀ ² (3/2,1 /2,). In addition, it is pointed out that any Hamiltonian matrix can be re-labelled in terms of fictitious spin labels, enabling a wide variety of unit spherical tensors to be used in block-diagonalization. For example, a single spin 5/2 matrix can be re-labelled using three spin labels 1/2, 1/2, and 1/2, respectively. Thus the tensor algebra required to block-diagonalize a 6 x 6 matrix is determined solely by the properties of the Pauli spin matrices. Finally, it is shown that re-labelling within the unit spherical tensor framework provides a unifying framework for standard basis operators, fictitious spin 1/2 and 1 operators, and others. The fictitious spin 1 / 2 unit spherical operators discussed in this paper differ from those of Vega and Pines.     

A Note on Nonlinear Parameters in Variational Methods

The dependence of the eigenvalues of matrices representing a quantum-chemical Hamiltonian in a model space on the nonlinear parameters of the trial functions is analyzed. Several theorems useful in determining the dependence of the matrix eigenvalues on the parameters are presented and their implications on the choice of the orbital basis sets are briefly discussed. A simple method of optimization of the parameter values is formulated.     

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.     

Interpretation of energy bands of polyethylene in terms of effective interactions between first- and second-neighboring CH2 groups

The alternative way of solving secular problems for the Hamiltonian matrices of regular quasi-one-dimensional systems developed previously [V. Gineityte, Int. J. Quant. Chem. 60 (3), 717 (1996)] has been applied to polyethylene. An implicit form of the dispersion relation has been obtained in terms of three local-structure-determined energy-dependent functions δ(ε), τ(ε), and η(ε), describing the effective interactions inside a separate CH₂ group and those between first- and second-neighboring CH₂ groups, respectively. The actual shapes of dispersion curves proved to be determined by relative mean values of the functions τ(ε) and η(ε) within the ε region under interest. The unusual minimum within the low-energy branch of dispersion curves situated at a low-symmetry point of the first Brillouin zone (k≈0.6π/a) has been established to appear owing to considerable values of effective interactions between the second-neighboring CH₂ groups within the respective energy interval. Just the latter type of interactions has been concluded to be responsible also for non-one-dimensionality of the polyethylene chain. The Hamiltonian matrix eigenfunctions of this chain have been expressed as the Bloch sums of eigenvalue-dependent local-structure-determined basis orbitals. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 64 : 481–494, 1997     

Approximate solutions of Heisenberg Hamiltonians

Even when each atom of a 2n-center cluster or molecule only brings one active electron in one atomic orbital, the size of the Heisenberg Hamiltonian matrix increases as C. Simple truncations of this matrix would result in size consistence defects, as evident from the isomorphism between Heisenberg and configuration interaction (CI ) matrices. Geometry-dependent Heisenberg Hamiltonians derived from accurate ab initio calculations on the two-center systems have proved to be very efficient for conjugated hydrocarbons and for alkali metals; in order to apply this approach to intermediate size systems (10–20 centers), a rational procedure is proposed consisting of the selection of a truncated set of determinants (of low energy) and a dressing of the truncated Hamiltonian matrix under the perturbation of the other determinants. The second order dressing is analogous to a so-called “shift B_k procedure” or Generalized Degenerate Perturbation theory and is weakly dependent on an E₀ parameter. Tests performed on various 8- and 10-atom systems show the accuracy of the procedure. An iterative selection of the truncated basis set and proper choices of the E₀ values allow one to obtain the whole lower part of the spectrum. The calculated geometries are satisfactory. Some preliminary applications are reported concerning the C₁₂H₁₄ dodecahexene linear chain, perfectly fitting with previous extrapolations.     

Hermiticity of Hamiltonian Matrix using the Fourier Basis Sets in Bond-Bond-Angle and Radau Coordinates

In quantum calculations a transformed Hamiltonian is often used to avoid singularities in a certain basis set or to reduce computation time. We demonstrate for the Fourier basis set that the Hamiltonian can not be arbitrarily transformed. Otherwise, the Hamiltonian matrix becomes non-hermitian, which may lead to numerical problems. Methods for correctly constructing the Hamiltonian operators are discussed. Specific examples involving the Fourier basis functions for a triatomic molecular Hamiltonian (J=0) in bond-bond angle and Radau coordinates are presented. For illustration, absorption spectra are calculated for the OClO molecule using the time-dependent wavepacket method. Numerical results indicate that the non-hermiticity of the Hamiltonian matrix may also result from integration errors. The conclusion drawn here is generally useful for quantum calculation using basis expansion method using quadrature scheme.     

A new method for analysis of disulfide-containing proteins by matrix-assisted laser desorption ionization (MALDI) mass spectrometry

A simple and high-throughput method for the identification of disulfide-containing peptides utilizing peptide-matrix adducts is described. Some commonly used matrices in MALDI mass spectrometry were found to specifically react with sulfhydryl groups within peptide, thus allowing the observation of the peptide-matrix adduct ion [M+n+n′ matrix+H]⁺ or [M+n+n′ matrix+Na]⁺ (n = the number of cysteine residues, n′=1, 2,…, n) in MALDI mass spectra after chemical reduction of disulfide-linked peptides. Among several matrices tested, α-cyano-4-hydroxycinnamic acid (CHCA, molecular mass 189 Da) and α-cyano-3-hydroxycinnamic acid (3-HCCA) were found to be more effective for MALDI analysis of disulfide-containing peptides/proteins. Two reduced cysteines involved in a disulfide bridge resulted in a mass shift of 189 Da per cysteine, so the number of disulfide bonds could then be determined, while for the other matrices (sinapinic acid, ferulic acid, and caffeic acid), a similar addition reaction could not occur unless the reaction was carried out under alkaline conditions. The underlying mechanism of the reaction of the matrix addition at sulfhydryl groups is proposed, and several factors that might affect the formation of the peptide-matrix adducts were investigated. In general, this method is fast, effective, and robust to identify disulfide bonds in proteins/peptides.     

Configuration interaction matrix elements for atoms using permutation group algebra

A procedure is described for the efficient evaluation of the energy matrix elements necessary for atomic configuration-interaction calculations. With the orbital configurations of an N electron system in spin state S written as the irreducible representations [2^1/2N?S, 1^2S] of the permutation group S( N ), it is possible to evaluate readily the energy matrix elements of a spin-free Hamiltonian expressed in terms of the generators of the unitary group. We show how the use of angular momentum ladder operators permits the effective generation of a basis of eigenstates of ??², ??_z as well as ??² and ??_z, for which the energy matrix elements may be evaluated with ease.     

Energy levels of paramagnetic ions: Algebra. III. The case of dN ions in cubical symmetry

The formalism developed in the first two papers of this series is applied to the investigation of a new weak-field model. This crystal-field model lies on the use of a symmetry-adapted weak-field basis and an effective Hamiltonian involving in a symmetrical way both spin- and orbit-dependent contributions. Some general properties of this Hamiltonian are studied and complete calculation of its matrix elements is conducted in a symmetry-adapted weak-field basis in the case of an arbitrary configuration nl^N in any symmetry. The case of a configuration nd^N in octahedral symmetry is fully explored. In this case, the proposed weak-field model is restricted to a 12-parameter model which accounts for isotropic and anisotropic Coulomb interactions, isotropic and anisotropic spin-orbit interactions, and crystal-field interactions. A comparison between this 12-parameter weak-field model and the 14-parameter strong-field model is established. Equivalence between the latter two models requires two constraint relations to be satisfied for some strong-field parameters. These two relations are examined with various viewpoints.     

Density matrix structure for saturated molecules

Conclusion The existence of a common Hamiltonian matrix structure for saturated systems results in common structural properties of the density matrices for the whole class of molecules, such as the zero occupation of AO in the first approximation, the density matrix perturbations due to a heteroatom, etc. This fact can be taken as a quantum-mechanical foundation for viewing saturated molecules as a separate class of compounds. The endowment of this system with the transferability of electronic structure properties, relative to atoms and bonds, to high accuracy, within the framework of the effective Hamiltonian method follows from an analysis of the general expressions for the density matrix elements. The transferability of the saturated system Hamiltonian matrix elements requisite for this is supported by a comparison among the self-consistent Fock matrix elements of various hydrocarbons in a localized orbital basis [9]. Independently of the detailed structure of the actual molecules, the influence of a heteroatom on the electron density distribution in saturated systems dies off quickly with distance from the heteroatom. From an analysis of expressions for the nondiagonal elements of the density matrix corresponding to nonneighboring AO we establish a connection between the degree of electron localization in saturated systems and the size or certain Hamiltonian matrix elements.There is a consequent analogy between saturated and alternatively conjugated hydrocarbons, which, starting from the common structure of the Hamiltonians, also leads to common properties of the density matrices [14]. However, the study of the influence of heteroatoms on the density matrices in these systems by means of perturbation theory is complicated by the dependence of the matrix N(4) on the molecular structure, which makes it necessary to introduce highly simplified approaches for the solution of Eq. (2) [8]. Therefore, for alternating hydrocarbons we have succeeded in establishing only the sign of the orbital — orbital polarizability (alternating polarity theorem [14]), while, as for saturated systems, the equality N=1 permits an analytic expression for the polarizability.V. Kapsukas Vilnius State University. Translated from Zhurnal Strukturnoi Khimii, Vol. 29, No. 5, pp. 3–8, September–October, 1988.     

Spin-independent three-body effective valence-shell operators: Application to molecular oxygen

A mathematical construction is presented that uniquely defines a set of spin-independent effective valence-shell Hamiltonian (H^v) three-body matrix elements. These spin-independent H^v matrix elements separate direct and exchange portions of the three-body H^v matrix elements and therefore provide the most natural form for comparisons with parameterization schemes of semiempirical electronic structure methods in which the three-body matrix elements are incorporated into semiempirical one- and two-body Hamiltonian matrix elements in an averaged manner. Ab initio H^v three-body matrix elements of O₂ are computed through third order of quasidegenerate perturbation theory and are analyzed as a function of internuclear distance and atomic orbital overlap to aid in understanding how these three-body matrix elements may be averaged into semiempirical one- and two-body matrix elements. © 1992 John Wiley & Sons, Inc.     

A note on the importance of d orbitals in the calculation of effective interactions and the excitation spectra of atoms and molecules

The focal point of our discussion is the examination of truncated basis sets used in obtaining an accurate first principles clculation of the effective valence shell Hamiltonian by the canonical transformation-cluster expansion approasch. Subsequent diagonalization of this effecitve valence shell hamiltonian yields the valence shell transition energies. A detailed analysis of numerical results obtained using a number of different basis sets of hydrogen-like orbitals together with rigorous symmetry arguments celarly demonstrates the special role played by d orbitals in computing the ³P → ¹D transition energy in carbon. The failure of early attempts to calculate the effective Hamiltonian for ethylene from first principles is examined in the light of recent ab initio calculations on ethylene involving d orbitals and the computations reported in this paper. We conclude that accurate calculations of the effective valence shell Hamiltonian for molecules must consider d orbitals in the excited orbital basis set.     

Efficient evaluation of the algebrants of VB wave functions using the successive expansion method. I. Spin S = 0, ½

An efficient expansion method for the evaluation of VB matrix elements is introduced. The overlaps of VB wave functions of N electrons can be treated as algebrants, i.e., generalized determinants, of N × N matrices. An algebrant can be expanded with subalgebrants of lower orders in a successive way. By choosing Rumer spin bases and appropriately arranging the expansion, it is found that the number of unique subalgebrants involved in the expansion increases in a quite moderate way with N. In contrast to the traditional symmetric group approach, which explicitly utilizes all N! representation matrices, the new strategy incorporates the group theoretical factors in a simple way in the successive expansion. As only the unique subalgebrants are further expanded, the computational effort required by the new strategy scales in a very acceptable manner with the increasing number of electrons. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 62: 245–259, 1997