The computation of the representation matrices of the generators of the unitary group

A method is presented for the efficient computation of the representation matrices of the unitary group, U(n) in the Gelfand—Tsetlin basis (corresponding to the usual spin-symmetry adapted basis for an N electron CI). The present scheme is conceptually and computationally attractive in that it is formulated directly in terms of Weyl tableaux and also that it permits simultaneous basis vector generation and matrix element evaluation. In addition the basis vectors are ordered so that subsequent restriction to the three dimensional rotation group is facilitated. An illustrative example is also presented.Taken in part from a thesis submitted to the University of London in partial fulfilment of the requirements for the degree of PhD.     

Particles and holes in the unitary group method

Based on the definition for complementary Gel'fand states, we proved the simple relationship between the matrix elements of particle states and those of hole states by unitary calculus.     

A scheme for representation matrices of a permutation group using spin-paired functions

In the present note, a linked form of spin-paired functions for an N-electron system in spin state S is suggested. This is found to lead to a simple scheme for generating the representation matrices of the elements of permutation group without searching for linkages in the superposition diagrams. The program based on this is found to generate the representation matrices more efficiently than do currently available procedures. © 1993 John Wiley & Sons, Inc.     

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.     

Configuration interaction matrix elements. II. Graphical approach to the relationship between unitary group generators and permutations

The explicit expressions for the matrix elements of unitary group generators between geminally antisymmetric spin-adapted N-electron configurations in terms of the orbital occupancies and spin factors, given as spin function matrix elements of appropriate orbital permutations, are derived using the many-body time-independent diagrammatic techniques. It is also shown how this approach can be conveniently combined with graphical methods of spin algebras to obtain explicit expressions for the spin factors, once a definite coupling scheme is chosen. This method yields explicit expressions for the orbital permutations defining the spin factors. However, if desired, the explicit determination of line-up permutations can be avoided in this approach, since they are implicitly contained in the orbital diagrams. It also clearly indicates why the geminally antisymmetric spin functions have to be used when a simple formalism is desired.     

Configuration interaction matrix elements. I. Algebraic approach to the relationship between unitary group generators and permutations

Matrix elements of unitary group generators between spin-adapted antisymmetric states are shown to be proportional to spin matrix elements of so-called “line-up” permutations. The proportionality factor is given explicitly as a simple function of the orbital occupation numbers. If one bases the theory on ordered orbital products, the line-up permutations are given a priori. The final formulas have a very simple structure; this is a direct consequence of the fact that the spin functions have been taken to be geminally antisymmetric.     

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.     

Decomposition of unitary matrices for finding quantum circuits: application to molecular Hamiltonians

Constructing appropriate unitary matrix operators for new quantum algorithms and finding the minimum cost gate sequences for the implementation of these unitary operators is of fundamental importance in the field of quantum information and quantum computation. Evolution of quantum circuits faces two major challenges: complex and huge search space and the high costs of simulating quantum circuits on classical computers. Here, we use the group leaders optimization algorithm to decompose a given unitary matrix into a proper-minimum cost quantum gate sequence. We test the method on the known decompositions of Toffoli gate, the amplification step of the Grover search algorithm, the quantum Fourier transform, and the sender part of the quantum teleportation. Using this procedure, we present the circuit designs for the simulation of the unitary propagators of the Hamiltonians for the hydrogen and the water molecules. The approach is general and can be applied to generate the sequence of quantum gates for larger molecular systems.     

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.     

Application of the unitary group approach to crystal field theory

An analytical treatment for a strong crystal field in an octahedral symmetry by using the unitary group approach is given. It shows that the convenience of the unitary group approach is comparable with that of the Racah method.     

Clifford algebra and unitary group formulations of the many-electron problem

An essential role of Clifford algebras for quantum-chemical finite-dimensional orbital models of many-electron systems is pointed out. The relationship between Clifford algebra matric units, the generators of the unitary group approach (UGA) and the higher order replacement or excitation operators, as well as between their first and second quantized realizations, is elucidated. The usefulness of higher order replacement operators in the spin-adaptation of various many-body theories is briefly outlined and illustrated on the orthogonally spin-adapted coupled-pair approach. A natural connection with the Clifford algebra UGA is explored and new possibilities for its exploitation in large scale configuration interaction calculations are suggested.Dedicated to Professor J. Koutecký on the occasion of his 65th birthdayKillam Research Fellow 1987–8     

Aspects of size extensivity in unitary group adapted multi-reference coupled cluster theories: the role of cumulant decomposition of spin-free reduced density matrices

We present in this paper a comprehensive study of the various aspects of size extensivity of a set of unitary group adapted multi-reference coupled cluster (UGA-MRCC) theories recently developed by us. All these theories utilize a Jeziorski–Monkhorst (JM) inspired spin-free cluster Ansatz of the form \(|\varPsi \rangle = \sum\nolimits_\mu \varOmega _\mu |\phi _\mu \rangle c_\mu\) with \(\varOmega _\mu =\{\exp ({T_\mu })\}\) , where \(T_\mu\) is expressed in terms of spin-free generators of the unitary group \(U(n)\) for n-orbitals with the associated cluster amplitudes. \(\{...\}\) indicates normal ordering with respect to the common closed shell \(core\) part, \(|0\rangle\) , of the model functions, \(\{\phi _\mu \}\) which is taken as the vacuum. We argue and emphasize in the paper that maintaining size extensivity of the associated theories is consequent upon (a) connectivity of the composites, \(G_\mu\) , containing the Hamiltonian \(H\) and the various powers of \(T\) connected to it, (b) proving the connectivity of the MRCC equations which involve not only \(G_\mu\) s but also the associated connected components of the spin-free reduced density matrices (RDMs) obtained via their cumulant decomposition and (c) showing the extensivity of the cluster amplitudes for non-interacting groups of orbitals and eventually of the size-consistency of the theories in the fragmentation limits. While we will discuss the aspect (a) above rather briefly, since this was amply covered in our earlier papers, the aspect (b) and (c), not covered in detail hitherto, will be covered extensively in this paper. The UGA-MRCC theories dealt with in this paper are the spin-free analogs of the state-specific and state-universal MRCC developed and applied by us recently.We will explain the unfolding of the proof of extensivity by analyzing the algebraic structure of the working equations, decomposed into two factors, one containing the composite \(G_\mu\) that is connected with the products of cumulants arising out of the cumulant decomposition of the RDMs and the second term containing some RDMs which is disconnected from the first and can be factored out and removed. This factorization ultimately leads to a set of connected MRCC equations. Establishing the extensivity and size-consistency of the theories requires careful separation of truly extensive cumulants from the ones which are a measure of spin correlation and are thus connected but not extensive. We have discussed in detail, using diagrams, the factorization procedure and have used suitable example diagrams to amplify the meanings of the various algebraic quantities of any diagram. We conclude the paper by summarizing our findings and commenting on further developments in the future.     

Geometry of density matrices. VI. Superoperators and unitary invariance

We examine and compare ways of dividing into subspaces the space whose elements are density matrices or other operators for the class of model problems defined by a finite one-particle basis set. One method of decomposition makes the significance of the subspaces apparent. We show that this decomposition is also complete, in the group-theoretic sense, for the group of unitary transformations of the set of one-electron basis functions. The irreducible subspaces are labeled by particle number and by an additional integer we call the reduction index. For spaces of particle-number-conserving operators, all subspaces with the same reduction index are isomorphic, and an analogous isomorphism exists for non-particle-number-conserving cases. The general linear group also plays a key role, and we introduce the term “canonical superoperators” to characterize those superoperators which commute with this group. When an appropriate basis set is chosen for the matrix spaces, the supermatrices corresponding to these superoperators have a particularly simple form: a block structure with the only nonzero blocks being multiples of unit matrices. The superoperators of interest can be constructed in terms of two operators, , and these two have been expressed simply in terms of creation and annihilation operators. When only real orthogonal transformations of the basis are considered, a further decomposition is possible. We have introduced superoperators associated with this decomposition.     

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

A 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.     

Efficient representation of the linear density-density response function

We present a thorough derivation of the mathematical foundations of the representation of the molecular linear electronic density-density response function in terms of a computationally highly efficient moment expansion. Our new representation avoids the necessities of computing and storing numerous eigenfunctions of the response kernel by means of a considerable dimensionality reduction about from 10³ to 10¹. As the scheme is applicable to any compact, self-adjoint, and positive definite linear operator, we present a general formulation, which can be transferred to other applications with little effort. We also present an explicit application, which illustrates the actual procedure for applying the moment expansion of the linear density-density response function to a water molecule that is subject to a varying external perturbation potential. © 2019 The Authors. Journal of Computational Chemistry published by Wiley Periodicals, Inc.     

Analytical method for the representation of atoms-in-molecules densities

We present analytic refinements and applications of the deformed atomic densities method [Fernández Rico, J.; López, R.; Ramírez, G. J Chem Phys 1999, 110, 4213-4220]. In this method the molecular electron density is partitioned into atomic contributions, using a minimal deformation criterion for every two-center distributions, and the atomic contributions are expanded in spherical harmonics times radial factors. Recurrence relations are introduced for the partition of the two-center distributions, and the final radial factors are expressed in terms of exponential functions multiplied by polynomials. Algorithms for the practical implementation are developed and tested, showing excellent performances. The usefulness of the present approach is illustrated by examining its ability to describe the deformation of atoms in different molecular environments and the relationship between these atomic densities and some chemical properties of molecules.     

Alternative implementation of the unitary group approach to the atomic shell theory

Subduction coefficients adapted to the group chain, which appeared in Racah's treatment of fⁿ configurations, are defined and calculated in the unitary group approach. The coefficients are then utilized to construct successively adapted term functions and evaluate other interesting coefficients. In addition the simplified expressions for the Coulomb and spin-orbit operators are obtained in terms of generators.     

Universal method for computation of electrostatic potentials

A computational approach to determine electrostatic interaction and gravitational potentials by performing direct numerical integration is presented. The potential is expanded using finite-element functions of arbitrary order. The method does not involve any solutions of systems of linear equations. The potential is instead obtained as a sum of differential contributions. Thus, no boundary conditions for the potential are needed. It is computationally efficient and well suited for parallel computers, since the innermost loops constitute matrix multiplications and the outer ones can be used as parallel indices. Without using prescreening or other computational tricks to speed up the calculation, the algorithm scales as N4/3 where N denotes the grid size.     

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.