Generators Of Unitary Group Research Articles

Matrix elements of unitary group generators in many-fermion correlation problem. III. Green-Gould approach

The third part of our survey series concerning the evaluation of matrix elements (MEs) of the unitary group generators and of their products in the electronic Gel’fand–Tsetlin basis of the two-column irreps of U(n)—which are essential in the unitary group approach (UGA) to the many-electron correlation problem as handled by the configuration interaction (CI) and the coupled cluster (CC) approaches—relies on what we refer to as the Green-Gould (G-G) approach. In addition to the CI and CC methods, the G-G formalism proved to be very helpful in a number of other tasks, particularly in handling of the spin-dependent operators, the density matrices, or partitioned basis sets adapted to a chosen group chain.

Matrix elements of unitary group generators in many-fermion correlation problem. I. tensorial approaches

The objective of this series of papers is to survey important techniques for the evaluation of matrix elements (MEs) of unitary group generators and their products in the electronic Gel’fand–Tsetlin basis of two-column irreps of U(n) that are essential to the unitary group approach (UGA) to the many-electron correlation problem as handled by configuration interaction and coupled cluster approaches. Attention is also paid to the MEs of one-body spin-dependent operators and of their relationship to a standard spin-independent UGA formalism. The principal goal is to outline basic principles, concepts, and ideas without getting buried in technical details and thus to help an interested reader to follow the detailed developments in the original literature. In this first instalment we focus on tensorial techniques, particularly those designed specifically for UGA purposes, which exploit the spin-adapted tensorial analogues of the standard creation and annihilation operators of the ubiquitous second-quantization formalism. Subsequent instalments will address techniques based on the graphical methods of spin-algebras and on the Green–Gould polynomial formalism. In the “Appendix A” we then provide a succinct historical outline of the origins of the Lie group and Lie algebra concepts.

Matrix elements of unitary group generators in many-fermion correlation problem. II. Graphical methods of spin algebras

This second instalment continues our survey inter-relating various approaches to the evaluation of matrix elements (MEs) of the unitary group generators, their products, and spin-orbital U(2n) generators, in the basis of the electronic Gel’fand–Tsetlin (G–T) [or Gel’fand–Paldus (G–P) or P-] states spanning the carrier spaces of the two-column irreducible representations (irreps) of the unitary group approach (UGA) to the many-electron correlation problem. Exploiting an intimate relationship between the U(n), $${\mathcal {S}}_N$$ , and SU(2) groups we rely here on the SU(2) formalism within the confines of UGA rather than on an earlier approach relying on the permutation group $${\mathcal {S}}_N$$ . The key is a close relationship between the P-states of UGA and the Yamanouchi–Kotani (Y–K) states of SU(2) that differ only in a phase. This enables one to exploit the powerful graphical techniques of spin (or angular momentum) algebras as first developed by Jucys. We show the usefulness and power of these techniques in the problem of UGA ME evaluation which, moreover, leads automatically to their segmentation, first derived for single generators by Shavitt and for generator products by Boyle and Paldus. We also show how the same technique can be successfully employed for MEs of spin-dependent U(2n) generators in the UGA P-basis.

Strong Asymptotic Freeness for Free Orthogonal Quantum Groups

AbstractIt is known that the normalized standard generators of the free orthogonal quantum groupO+Nconverge in distribution to a free semicircular system as N → ∞. In this note, we substantially improve this convergence result by proving that, in addition to distributional convergence, the operator normof any non-commutative polynomial in the normalized standard generators ofO+Nconverges asN→ ∞ to the operator norm of the corresponding non-commutative polynomial in a standard free semicircular system. Analogous strong convergence results are obtained for the generators of free unitary quantum groups. As applications of these results, we obtain a matrix-coefficient version of our strong convergence theorem, and we recover a well-knownL2-L∞norm equivalence for noncommutative polynomials in free semicircular systems.

Unitary group approach to the many-electron correlation problem: spin-dependent operators

Following a brief overview of the unitary group approach (UGA) to the many-electron correlation problem, focusing in particular on Shavitt’s contribution via his graphical unitary group approach, we present a short review of our earlier results for the evaluation of matrix elements (MEs) of unitary group generators or products of generators in the electronic Gel’fand–Tsetlin basis with the help of spin-adapted second-quantization-like creation and annihilation vector operators at the unitary group level. This formalism is then extended to a spin-dependent case that is required when accounting for relativistic effects by developing explicit expressions for MEs of spin-orbital creation and annihilation operators in terms of the standard spin-adapted UGA basis. This leads naturally to a segmentation of these MEs and enables the evaluation of spin-dependent one-body operators while relying largely on the segment values of the standard spin-independent UGA.

A Non-antisymmetric Tensor Contraction Engine for the Automated Implementation of Spin-Adapted Coupled Cluster Approaches.

We present a symbolic manipulation algorithm for the efficient automated implementation of rigorously spin-free coupled cluster (CC) theories based on a unitary group parametrization. Due to the lack of antisymmetry of the unitary group generators under index permutations, all quantities involved in the equations are expressed in terms of non-antisymmetric tensors. Given two tensors, all possible contractions are first generated by applying Wick's theorem. Each term is then put down in the form of a non-antisymmetric Goldstone diagram by assigning its contraction topology. The subsequent simplification of the equations by summing up equivalent terms and their factorization by identifying common intermediates is performed via comparison of these contraction topologies. The definition of the contraction topology is completely general for non-antisymmetric Goldstone diagrams, which enables our algorithm to deal with noncommuting excitations in the cluster operator that arises in the unitary group based CC formulation for open-shell systems. The resulting equations are implemented in a new code, in which tensor contractions are performed by successive application of matrix-matrix multiplications. Implementation of the unitary group adapted CC equations for closed-shell systems and for the simplest open-shell case, i.e., doublets, is discussed, and representative calculations are presented in order to assess the efficiency of the generated codes.

Feynman Formulas Representation of Semigroups Generated by Parabolic Difference-Differential Equations

We establish that the Laplas operator with perturbation by symmetrised linear hall of displacement argument operators is the generator of unitary group in the Hilbert space of square integrable functions. The representation of semigroup of Cauchy problem solutions for considered functional differential equation is given by the Feynman formulas.

Matrix element evaluation in the unitary group approach to the electron correlation problem

Computationally effective formulations are presented for the evaluation of matrix elements of unitary group generators and products of generators between Gelfand states. These matrix elements are the coefficients of the orbital integrals in the expressions for the Hamiltonian matrix elements in the Gelfand basis, and as such are the key elements in any application of the unitary group approach to wave-function calculations. The present formulations, which, like previous analyses, result in a simple factorization of the generator matrix elements, are based on a graphical representation of the Gelfand basis, and do not require orbital permutations or an interpretation of the Gelfand states in terms of Young tableaus. It is shown that the resulting formalism can lead to very efficient procedures for “direct” configuration-interaction calculations, and probably also for perturbation-theory treatments.

SGA derivation of matrix elements between spin-adapted perturbative wave functions

The symmetric group approach (SGA) is used to derive formulas for the Hamiltonian matrix elements between perturbatively generated, highly contracted configuration state functions (CSF). The first-order Moller–Plesset, Epstein–Nesbet, and Krylov corrections are examples of such CSFs. They have been recently employed within the superdirect configuration interaction (Sup-CI) method, which combines the efficiency and simplicity of the many-body perturbation theory (MBPT) with the robustness of variational methods. The linear Sup-CI expansion in terms of such perturbative correction functions leads to a problem of efficient evaluation of matrix elements of the third power of the Hamiltonian in the usual CSFs. Using SGA and previously developed algebra of unitary group generators and circular operators one may express such matrix elements, similarly to MBPT, in terms of sums of product of two-electron integrals. Maple code enhancing greatly the algebraic manipulations has been developed and is available upon request. All the main steps in the derivation are discussed and the formulas are given in a compact form. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 74: 123–133, 1999

Quantum Mechanics from Symmetry and Statistical Modeling

Quantum theory is derived from a set ofplausible assumptions related to the following generalsetting: For a given system there is a set ofexperiments that can be performed, and for each suchexperiment an ordinary statistical model is defined. Theparameters of the single experiments are functions of ahyperparameter which defines the state of the system.There is a symmetry group acting on the hyperparameters, and for the induced action on the parameters ofthe single experiment a simple consistency property isassumed, called permissibility of the parametricfunction. The other assumptions needed are rather weak. The derivation relies partly on quantumlogic, partly on a group representation of thehyperparameter group, where the invariant spaces areshown to be in 1-1 correspondence with the equivalenceclasses of permissible parametric functions. Planck'sconstant only plays a role connected to generators ofunitary group representations.

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

AbstractWe 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 nc doubly occupied and nv unoccupied (virtual) orbitals, the irreducible tensor operators of the group U(nc) ⊗ U(nV) generate all Gelfand‐Tsetlin (GT) states corresponding to appropriate irreducible representation of U(nc + nv). 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 SM. This provides an alternative to the Nagel‐Moshinsky construction of the GT basis. Since the corresponding cluster amplitudes, which are also U(nc) ⊗ U(ns) tensors, can be shown to be connected, the irreducible tensor operators of U(nc) ⊗ U(nv) 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=nc + nv + ns is not simply reducible under the group U(nc) ⊗ U(ns) ⊗ U(nv). The multiplicity problem is resolved using the group chain U(n) ⊃ U(nc + nv) ⊗ U(ns) ⊃ U(nc) ⊗U(ns)⊗ U(nv) ⊗ U(nv). The labeling of the resulting configuration‐state functions (which, in general, are not GT states when nc > 1) by the irreducible representations of the intermediate group U(nc + nv) ⊗U(ns) 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.

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

AbstractA 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.

The automated solution of second quantization equations with applications to the coupled cluster approach

Theoretical methods in chemistry frequently involve the tedious solution of complex algebraic equations. Then the solutions, sometimes still quite complex, are usually hand-coded by a programmer into an efficient computer language. During this procedure it is all too easy to make an error which will go undetected. A better approach would be to introduce the computer at an even earlier stage in the development of the theory by programming it to first solve the set of equations and then compile the solution into an efficient computer language. In this research a program has been written in the C programming language which can efficiently compute the quasivacuum expectation value of a product of creation and annihilation operators and scalar arrays. The terms in the resulting expressions are then transformed into a canonical form so that all equivalent terms can be combined. Finally, the equations are compiled into a simple representation which can be rapidly interpreted by a Fortran program. This symbol manipulator has been applied to open-shell coupled cluster theory. Two coupled cluster methods using high-spin open-shell references are presented. In one of these methods, the cluster operator contains the unitary group generators, and products thereof, which generate all single and double excitations with respect to the reference. The other uses a simplified cluster operator which generates equations that must be spin-projected. These methods are compared to other descriptions of electron correlation for the CH2 singlet-triplet splitting and the NH2 potential energy surface.

BONDED TABLEAU AND UNITARY GROUP APPROACH FOR MANY-ELECTRON PROBLEMS

A unitary group approach based on the so-called bonded tableaux (VB) states is described. Several different realizations of the bonded tableaux are discussed and their relations are pointed out. From a viewpoint of the symmetric group we reveal a simple structure of the matrix elements of unitary group generators and generator products. This structure makes an efficient approach to the matrix element evaluation.

Bonded tableau unitary group approach to the many‐electron correlation problem

AbstractA spin‐free symmetry‐adapted valence bond (VB) state, named bonded tableau (BT), is deduced from the classical bonded function and labeled by an at most two‐column Weyl tableau. The complete set, which is composed of the BT basis or canonical bonded tableau (CBT), can be constructed from an overcomplete set of BT states. CI CBT and VB CBT are two kinds of complete sets that are constructed in this paper. They can be used, respectively, in the CI and VB theory. It is shown that there is a one‐to‐one correspondence between the labeling scheme for CI CBT and the Gelfand–Tsetlin (GT) basis. This relationship enables an efficient generation and compact representation of the BT basis if one desires to use the known global representation scheme for the GT basis. Effective algorithms for the matrix element evaluation of unitary group generators and products of generators between BT states are presented. In the formulation, the action of a generator on a BT state yields another BT state times a coefficient, so that the matrix elements of an arbitrary multiple product of generators are reduced to a calculation of the overlaps between BT states. The evaluation of the overlaps leads to a simple factorization into cycle contributions, whose values are given explicitly and only depend on the length parameters of the cycles. It is hoped that the presented formalism can facilitate the procedures for handling of the many‐electron correlation problem.

Operator algebra for the many-body problem in the spin eigenfunction basis

In the many-body perturbation theory and related methods equations involve sums over orbital indices. Reduction of the many-body equations to such a form is not trivial when spin eigenfunctions are used. A method to achieve such reduction, based on the algebraical properties of the unitary group generators, is described. As an example of application a formula for the second order MBPT with an arbitrary open shell reference state of the RHF type is derived.

The freeon unitary group procedure and the structure of matter

There is presented a novel, coherent procedure, based on the unitary group, for the computation of the spectra of atoms, molecules, solids, nuclei, and elementary particles. These spectra cover a truly remarkable energy range from a fraction to billions of electron volts. The procedure employs freeon orbitals that are free of energetically inert orbitals; e.g., ordinary spin orbitals for electrons, isospin orbitals for nuclei, and the product of ordinary spin orbitals and color orbitals for baryons. The Pauli principle requires the freeon spaces to be conjugate to the inert spaces. The spaces are spanned by Gel'fand states that have a simple, compact, pictorial representation in the form of Gel'fand structures (tableaux). The Hamiltonian for all systems is a second degree polynomial in the generators of the unitary group with parameters that determine the spectral scale and distribution and whose matrix elements are computed Lie algebraically. The unitary group supplies the unitary group quantum number and is the head group of chains of groups that supply additional quantum numbers.

Alternancy symmetry: A unified viewpoint

A general formulation of the alternancy symmetry adaptation for the semiempirical Pariser–Parr–Pople (PPP) type Hamiltonians is presented at both the spin-orbital and spin-adapted many-electron levels. The derivation of the general form of the alternancy symmetry conjugation operators is based solely on the tight-binding approximation for the short range one-particle part of the Hamiltonian considered. It starts by a simple formulation of the desired invariance properties of the PPP type Hamiltonian. Using algebraic properties of the unitary group generators and of their particle number nonconserving extensions, it leads to a completely explicit and general form for the alternancy symmetry conjugation operators. In this way the prior descriptions, which become special cases of this general formulation, are interrelated and unified. The spin and quasispin character of certain components of these operators are also pointed out and explicitly derived. The spin-adapted version is based on the unitary group formulation of the valence bond-type approach. A completely general formulation is given which applies to many-electron states of an arbitrary multiplicity of neutral systems, either of the normal kind with an identical number of starred and nonstarred sites, or of the radicaloid character with different numbers of starred and nonstarred sites. An explicit form of the dependence of the relative phase factor of the alternancy symmetry conjugation operator on the total spin, total electron number and on the number of starred and nonstarred sites is also given. General rules for the construction of spin and alternancy symmetry-adapted states are illustrated on a few simple examples. Finally, a brief discussion of the implications of the alternancy symmetry is presented.

Small systems of generators of isotropic unitary groups over finite fields of characteristic not two

Some sets of generators of unitary groups on a nonsingular Hermitian space with hyperbolic rank at least one over a finite field of characteristic not 2 with an involution ∗ are studied. All unitary groups are shown to be generated by 3 elements and all symplectic groups are generated by 2 elements.

Efficient method for computation of representation matrices of the unitary group generators

AbstractThe representation matrices of the unitary group generators Ek1 are equivalent to the representation matrices of cyclic permutations (k, k ± l, …, l ∓ 1, l). A method is presented for simultaneous computation of matrices corresponding to all different generators at a cost of less than one multiplication per nonzero element. The number of operations necessary for calculation of individual matrices for single generators or for products of two generators is at most proportional to the number of matrix elements of the final matrix. This approach eliminates the need to store the representation matrices in CI calculations.