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Unitary group approach (UGA) to the many‐electron correlation problem is generalized by embedding the unitary group U(n) in a much larger group U(2n) via the rotation groups SO(m) with m=2n or 2n+1 and their covering group Spin (m). Exploiting the spinorial Clifford algebra basis associated with Spin (m), it is shown that an arbitrary N‐electron configuration state can be represented as a linear combination of two‐box Weyl tableaux of U(2n), and the explicit representation for U(n) generators as simple linear combinations of U(2n) generators is given. The problem of U(n) generator matrix element evaluation for two‐column irreducible representations then reduces to an elementary problem of evaluation of generator matrix elements for the totally symmetric two‐box representation of U(2n). Thus a general N‐electron problem is effectively reduced to a number of two‐boson problems. The proposed formalism also enables us to exploit other than Gelfand–Tsetlin coupling schemes and particle nonconserving operators.

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... In the G-T case, this basis is orthonormal and corresponds to the Yamanouchi-Kotani coupling scheme. Although the G-T basis may not be the most appropriate one from the chemical viewpoint (see, e.g., [32,33]), its great advantage is the availability of the explicit expressions for the generator matrix elements. While in the general case the latter are rather unwieldy and the G-T patterns involve n(n + 1)/2 parameters, the entire formalism can be drastically simplified when dealing with the two-column (a, b, c) irreps that are relevant when studying many-electron systems [1]. ...

... All of the above listed developments are based on the G-T type canonical bases representing CSFs. As already mentioned, and as discussed in some detail in [32] (see also [33]), the G-T chain is rather artificial from the viewpoint of the molecular electronic structure applications. This also applies to the so-called generator states [9,[111][112][113] which are, moreover, nonorthogonal and over-complete, thus requiring a special attention (such as Gram-Schmidt or Löwdin orthogonalization so that the resulting states are no longer true generator states). ...

... A step forward in this direction was the development of the Clifford algebra UGA (CAUGA) [32,81,98,114,115] that is based on the work of Sarma and collaborators [116,117]. Here, in lieu of G-T chain one exploits the imbedding of U(n) in a much larger group U(2 n ) via the special orthogonal group SO(m), m = 2n or m = 2n + 1 (i.e., the classical LAs B n and D n ) and their covering group Spin(m), i.e., the group chain ...

The unitary group approach (UGA) to the many-fermion problem is based on the Gel’fand–Tsetlin (G–T) representation theory of the unitary or general linear groups. It exploits the group chain \(\mathrm {U}(n) \supset \mathrm {U}(n-1) \supset \cdots \supset \mathrm {U}(2) \supset \mathrm {U}(1)\) and the associated G–T triangular tableau labeling basis vectors of the relevant irreducible representations (irreps). The general G–T formalism can be drastically simplified in the many-electron case enabling an efficient exploitation in either configuration interaction (CI) or coupled cluster approaches to the molecular electronic structure. However, while the reliance on the G–T chain provides an excellent general formalism from the mathematical point of view, it has no specific physical significance and dictates a fixed Yamanouchi–Kotani coupling scheme, which in turn leads to a rather arbitrary linear combination of distinct components of the same multiplet with a given orbital occupancy. While this is of a minor importance in molecular orbital based CI approaches, it is very inconvenient when relying on the valence bond (VB) scheme, since the G–T states do not correspond to canonical Rumer structures. While this shortcoming can be avoided by relying on the Clifford algebra UGA formalism, which enables an exploitation of a more or less arbitrary coupling scheme, it is worthwhile to point out the suitability of the so-called Verma basis sets for the VB-type approaches.

... It is straightforward to generalize the particular example of the class 2 MF-solvable Hamiltonians given in Eq. (14) to all Hamiltonians where two unitary operations from the Lie group are sufficient for obtaining all eigenstateŝ ...

... are the only modification that are needed to generalize Eq. (14). As before, the important constraint onÛ 2 its commutation withP ⊥ 1 . ...

... The operators {â † p ,â p ,Ê p q ,Ê pq ,Ê pq } are sufficient to construct generators of the compact so(2N + 1) Lie algebra. 13,14 This construction is facilitated by introducing the Majorana operatorŝ ...

Necessary and sufficient conditions for quantum Hamiltonians to be exactly solvable within mean-field theories have not yet been formulated. To resolve this problem, first, we define what mean-field theory is, independently of a Hamiltonian realization in a particular set of operators. Second, using a Lie-algebraic framework we formulate multiple classes of mean-field solvable Hamiltonians both for distinguishable and indistinguishable particles. For the electronic Hamiltonians, a new approach provides mean-field solvable Hamiltonians of higher fermionic operator powers than quadratic. Some of the mean-field solvable Hamiltonians require different sets of quasi-particle rotations for different eigenstates, which reveals a more complicated structure of such Hamiltonians.

... We use the fact that E k − E 0 is non negative, as the true ground state energy will always be lower than an inexact estimate. It is trivial to see that the equality in (18) holds only if|ψ 0 is exactly equal |ψ 0 . Hence, the coefficients ψ k |ψ 0 become the absorbing element in multiplication for all k = 0. ...

... In practice, several software programs already written by the research group of Jiří Pittner and from other open source domains were used in the simulation process. Most notably, it was CAUGA, the software for full configuration interaction (FCI) calculations utilizing, as the name suggests, the Clifford algebra unitary group approach to many-electron correlation problems, resulting from the original paper [18]. Then also GENCCSD, the software used for general unitary coupled cluster simulation, hence the variational quantum eigensolver. ...

In this paper we discuss the utilization of Variational Quantum Solver (VQE) and recently introduced Generalized Unitary Coupled Cluster (GUCC) formalism for the diagonalization of downfolded/effective Hamiltonians in active spaces. In addition to effective Hamiltonians defined by the downfolding of a subset of virtual orbitals we also consider their form defined by freezing core orbitals, which enables us to deal with larger systems. We also consider various solvers to identify solutions of the GUCC equations. We use N$_2$, H$_2$O, and C$_2$H$_4$, and benchmark systems to illustrate the performance of the combined framework.

... In closing this series, let us at least mention some other-more "exotic"-developments related to UGA and to fermionic algebras in general (for a brief overview see Ref. [39]). In particular, we should note the fundamental role played here by Clifford algebras and by the related group U(2 n ) while exploiting the group chain [34,[40][41][42][43][44][45][46] where m = 2n + 1 or m = 2n . These ideas led to a development of the so-called Clifford algebra UGA (CAUGA) [34,[40][41][42][43][44][45][46], which proved to be useful in applications that are based on the coupled cluster (CC) approaches or in those exploiting the valence bond (VB) formalism, using the semi-empirical Pariser-Parr-Pople (PPP) Hamiltonian in the PPP-VB scheme [47][48][49]. ...

... In particular, we should note the fundamental role played here by Clifford algebras and by the related group U(2 n ) while exploiting the group chain [34,[40][41][42][43][44][45][46] where m = 2n + 1 or m = 2n . These ideas led to a development of the so-called Clifford algebra UGA (CAUGA) [34,[40][41][42][43][44][45][46], which proved to be useful in applications that are based on the coupled cluster (CC) approaches or in those exploiting the valence bond (VB) formalism, using the semi-empirical Pariser-Parr-Pople (PPP) Hamiltonian in the PPP-VB scheme [47][48][49]. Recently we also investigated the use of Verma bases for this purpose [50]. ...

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.

... Geometric Algebra has been applied successfully to a wide range of fields such as computer vision, robotics, classical mechanics and theoretical physics. Over the past 3 decades its applications in mathematical/theoretical chemistry has reaped numerous advances: starting in the 1980s, Geometric Algebras were used as part of a unitary group approach to generate quantum chemical, finite-dimension, orbital models of many-electron systems [1,2]. Additionally, Geometric Algebra was used to realise and exploit the Rumer-Weyl basis of valence bonds [3,4]. ...

... N C [1] = N C [2] = N C [5] = N C [6] = N C [9] = N C [10] = 1 10 . ...

This paper presents a novel and generalised mathematical formulation for mathematical and theoretical chemistry. The formulation presented employs network theoretic methods and standard Conformal Geometric Algebra (Conformal Clifford Algebra) with a unique algebraic extension that introduces special non-integer basis vectors, known as ‘shades’. The methodology developed lays preliminary foundations for both generic and more specialised characterisations and modelling of chemical systems by computing a multi-vector function, termed the ‘hyperfield’ (an algebra-geometric characterisation of a many-agent chemical system). In this paper, the formulation has been applied to inorganic compounds, in the context of their emergent properties (such as boiling and melting points) and intra-molecular characteristics. Case studies have been presented for the calculation of melting point, boiling point, maximum solubility in water and equilibrium bond length of 21 diatomic molecules. This work finds that a network theoretic model of many-agent systems modelled in \({\mathbf{R}}^{\mathrm{4,1}}\) space, using an algebraic extension of non-integer basis vectors, presents a promising and unifying method for theoretical mathematical chemistry.

... One disadvantage is that the physical significance of the states is somewhat obscure. For instance, in the context of quantum chemistry, Paldus and Sarma have remarked on "the unphysical nature" of the Gelfand-Zetlin basis and have observed that this is a crucial flaw in the valence bond scheme [33]. (See Figure 1 of [33] for an illustration of this in the case of benzene.) ...

... For instance, in the context of quantum chemistry, Paldus and Sarma have remarked on "the unphysical nature" of the Gelfand-Zetlin basis and have observed that this is a crucial flaw in the valence bond scheme [33]. (See Figure 1 of [33] for an illustration of this in the case of benzene.) Another is that the lowering operators involved are complicated. ...

The representation theory of the unitary groups is of fundamental significance in many areas of physics and chemistry. In order to label states in a physical system with unitary symmetry, it is necessary to have explicit bases for the irreducible representations. One systematic way of obtaining bases is to generalize the ladder operator approach to the representations of SU(2) by using the formalism of lowering operators. Here, one identifies a basis for the algebra of all lowering operators and, for each irreducible representation, gives a prescription for choosing a subcollection of lowering operators that yields a basis upon application to the highest weight vector. Bases obtained through lowering operators are particularly convenient for computing matrix coefficients of observables as the calculations reduce to the commutation relations for the standard matrix units. The best known examples of this approach are the extremal projector construction of the Gelfand-Zetlin basis and the crystal (or canonical) bases of Kashiwara and Lusztig. In this paper, we describe another simple method of obtaining bases for the irreducible representations via lowering operators. These bases do not have the algebraic canonicity of the Gelfand-Zetlin and crystal bases, but the combinatorics involved are much more straightforward, making the bases particularly suited for physical applications.

... We use the fact that E k − E 0 is non negative, as the true ground state energy will always be lower than an inexact estimate. It is trivial to see that the equality in (18) holds only if|ψ 0 is exactly equal |ψ 0 . Hence, the coefficients ψ k |ψ 0 become the absorbing element in multiplication for all k = 0. ...

... In practice, several software programs already written by the research group of Jiří Pittner and from other open source domains were used in the simulation process. Most notably, it was CAUGA, the software for full configuration interaction (FCI) calculations utilizing, as the name suggests, the Clifford algebra unitary group approach to many-electron correlation problems, resulting from the original paper [18]. Then also GENCCSD, the software used for general unitary coupled cluster simulation, hence the variational quantum eigensolver. ...

A new method of quantum molecular system simulation prepared for implementation on a quantum computer is presented and tested using the formalism of hybrid variational quantum eigensolver (VQE) and the new concept of subsystem embedding subalgebras (SES), to obtain full molecular energies through subspace methods of the active orbital space, which causes a reduction in the number of cluster amplitudes and hence the difficulty of the calculated problem.
The method is tested in all component method combinations that are benchmarked and compared, and shows promising accuracy in achieving low energies.
Several novel methods of improving convergence are drafted and tested.
Keywords: variational quantum eigensolver, generalized coupled cluster, singles doubles, diagonalization, downfolded Hamiltonian, excitation subalgebra, subsystem embedding

... An appropriate dynamical group here is the unitary group U(2 n ) leading to the socalled Clifford algebra UGA (CAUGA) (see Refs. [5,[36][37][38] and loc. cit.). ...

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.

... The SESCC formalism hinges upon the notion of excitation sub-algebra of the algebra g (N) generated by E a l i l = a † a l ai l operators (for a more detailed discussion of many-body Lie algebras, the reader is referred to Refs. [84][85][86]. The SESCC formalism utilizes an important class of sub-algebras of g (N) , which contain all possible excitations E a 1 ...a m i 1 ...i m that excite the electrons from a subset of active occupied orbitals (denoted as R) to a subset of active virtual orbitals (denoted as S). ...

In this paper, we discuss extending the sub-system embedding sub-algebra coupled cluster formalism and the double unitary coupled cluster (DUCC) ansatz to the time domain. An important part of the analysis is associated with proving the exactness of the DUCC ansatz based on the general many-body form of anti-Hermitian cluster operators defining external and internal excitations. Using these formalisms, it is possible to calculate the energy of the entire system as an eigenvalue of downfolded/effective Hamiltonian in the active space, which is identifiable with the sub-system of the composite system. It can also be shown that downfolded Hamiltonians integrate out Fermionic degrees of freedom that do not correspond to the physics encapsulated by the active space. In this paper, we extend these results to the time-dependent Schrödinger equation, showing that a similar construct is possible to partition a system into a sub-system that varies slowly in time and a remaining sub-system that corresponds to fast oscillations. This time-dependent formalism allows coupled cluster quantum dynamics to be extended to larger systems and for the formulation of novel quantum algorithms based on the quantum Lanczos approach, which has recently been considered in the literature.

... The SES-CC formalism hinges upon the notion of excitation sub-algebra of algebra g (N ) generated by E a l i l = a † a l a i l operators (for a more detailed discussion of manybody Lie algebras the reader is referred to Refs. [84][85][86]). The SES-CC formalism utilizes an important class of sub-algebras of g (N ) , which contain all possible excitations E a1...am i1...im that excite electrons from a subset of active occupied orbitals (denoted as R) to a subset of active virtual orbitals (denoted as S). ...

In this paper, we discuss extending the sub-system embedding sub-algebra coupled cluster (SES-CC) formalism and the double unitary coupled cluster (DUCC) ansatz to the time domain. As we demonstrated in earlier studies, it is possible, using these formalisms, to calculate the energy of the entire system as an eigenvalue of downfolded/effective Hamiltonian in the active space, that is identifiable with the sub-system of the composite system. In these studies, we demonstrated that downfolded Hamiltonians integrate out Fermionic degrees of freedom that do not correspond to the physics encapsulated by the active space. We extend these results to the time-dependent Schr\"odinger equation, showing that a similar construct is possible to partition a system into a sub-system that varies slowly in time and a remaining subsystem that corresponds to fast oscillations. This time dependent formalism allows coupled cluster quantum dynamics to be extended to larger systems and for the formulation of novel quantum algorithms based on the quantum Lanczos approach, which have recently been considered in the literature.

Relying on a unitary group approach (UGA) to coupled-cluster (CC) theory for open-shell systems we formulate a spin-free analogue of Thouless' theorem and derive stability conditions for spin-restricted, open-shell Hartree-Fock (ROHF) solutions. The formalism is illustrated on the ground states of the oxygen and boron molecules. For the O2 molecule, the broken-space-symmetry solution appears already for a slightly stretched O=O bond. For the B2 molecule we find a broken-symmetry solution even at the equilibrium geometry and, in fact, in almost an entire region of intermolecular separations. Content:text/plain; charset="UTF-8"

A spin-free method is presented for evaluating electronic matrix elements over a spin-independent many-electron Hamiltonian. The spin-adapted basis of configuration state functions is obtained using a nonorthogonal spin basis consisting of projected spin eigenfunctions. The general expressions for the matrix elements are given explicitly, and it is demonstrated how the matrix elements may be obtained simply from the knowledge of the irreducible characters of the permutation group jN. The presented formulas are very general and may be applied in connection with both spin-coupled valence bond studies and in conventional configuration interaction (CI) methods based on an orthonormal orbital basis.

An attempt has been made to understand the structure of the Clifford algebra unitary group adapted many-particle states from the conventional symmetric group point of view. Emphasizing the symmetric group result that the consideration of the spin-independent Hamiltonian matrix over the many-particle configuration functions (CFs) entails a particular subspace of their spatial parts only, attention is confined entirely in this subspace. Question of adapting the functions therein to the unitary group subduction chain is then shown to bring out an interesting lead to the Clifford algebra unitary group approach (CAUGA) states, thus underlining the motive and the essential gains of the CAUGA formulation. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 77: 607–614, 2000

We overview our valence bond (VB) approach to the π-electron Pariser-Parr-Pople (PPP) model Hamiltonians referred to as the PPP-VB method. It is based on the concept of overlap enhanced atomic orbitals (OEAOs) that characterizes modern ab initio VB methods and employs the techniques afforded by the Clifford algebra unitary group approach (CAUGA) to carry out actual computations. We present a sample of previous results, as well as some new ones, to illustrate the ability of the PPP-VB method to provide a highly correlated description of the π-electron PPP model systems, while relying on conceptually very simple wave functions that involve only a few covalent structures.

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.

Methods based on the coupled-cluster (CC) Ansatz and most widely applied for the computation of molecular properties and electronic structure are reviewed. The applications of each method are presented, and its performance is evaluated. Following introductory remarks and a brief historical overview of CC methodology and its applications, we first outline the scope of our review, establish the required notation, and recall the foundations and origins of the CC Ansatz. We begin with single-reference CC approaches, which have greatly matured since the first ab initio study in early seventies, and are currently used to solve many diverse problems. The handling of nondegenerate closed-shell states being nowadays routine, we focus on open-shell systems and the important role of spin adaptation, as well as on methods extending or improving the ubiquitous CC method with singles and doubles (CCSD). Multireference CC approaches still defy routine usage and are therefore given only a cursory treatment. Their main purpose is to provide a basis for the so-called state-selective, or state-specific, CC approaches that employ a single, yet possibly multideterminantal, reference. Perturbative and the so-called quadratic configuration interaction methods are briefly addressed from the viewpoint of CC theory. We also make an attempt to assess and interrelate numerous CC-based approaches to molecular properties, ranging from those related to the shape of the potential-energy surfaces (geometry, harmonic force fields, etc.) to properties characterizing the interaction with electromagnetic fields (static moments, polarizabilities, etc.). We then conclude the methodological part of our review with a few comments concerning the computational aspects. The subsequent part, devoted to applications, presents various examples that illustrate the scope, efficiency, and reliability of various CC methods, particularly those of recent provenance. We focus on the general account of correlation effects, potential-energy surfaces and related properties, ionization potentials and electron affinities, vertical electronic excitation energies, and various static and dynamical properties. The final section of this review attempts to summarize the status quo of the CC methodology and its applications, as well as to ponder its future prospects.

The problem of bond length alternation in linear extended ϕ-electron systems with conjugated double bonds is examined using the valence bond theory applied to a simple model of cyclic polyenes CNHN with N = 4v and N = 4v + 2 sites as described by the Pariser-Parr-Pople Hamiltonian. Overlap enhanced atomic orbitals are employed in order to achieve the optimal treatment with only two Kekulé structures. The predicted bond length alternation and its magnitude are in good agreement with earlier molecular orbital based calculations and with experiment. Special attention is given to the discussion of the origin of bond length alternation in long polyenic chains and to the role of the resonance energy leading to stabilization of undistorted, symmetric structures for small aromatic (N = 4v + 2) cycles. © 1996 John Wiley & Sons, Inc.

The use of group symmetric localized molecular orbitals as configuration generating orbitals of many-electron valence bond functions, namely bonded tableaux (BTs ), greatly simplifies the calculation of the Hamiltonian matrix elements. A practical algorithm and program for such calculations was developed, which includes the calculation of the overlap integral between the BTs and the action of the generators and of the generator products of the unitary group on the BT states, and the Hamiltonian matrix elements are expressed in terms of a linear combination of nonequivalent two-centre and four-centre integrals and geometric factors based on the Wigner-Eikart law.

An exploitation of the Rumer-Weyl basis of the valence bond (VB) formalism within the Clifford algebra unitary group approach (CAUGA) scheme is formulated. Simple rules are given for the construction of the relevant non-orthogonal bi-spinor Clifford-Weyl basis, whose vectors can be conveniently labeled with UGA step numbers, as well as simple rules for the action of U(n) generators on these states, which enable an efficient evaluation of the Hamiltonian matrix elements in this basis. Possible applications within both the VB- and MO-type formalisms are discussed and the results of a test program implementing the suggested algorithm are presented. A possible extension to non-orthogonal bases is also briefly outlined.

A new implementation of the orthogonally spin-adapted open-shell (OS) coupled-cluster (CC) formalism that is based on the unitary group approach to many-electron correlation problem is described. Although the emphasis is on the so-called state specific single-reference but multiconfigurational OS CC approach, the developed algorithms as well as the actual codes are also amenable to multireference CC applications of the state-universal type. A special attention is given to simple OS doublets and OS singlet and triplet cases, the former being applicable to the ground states of radicals and the latter to the excited states of closed shell systems. The encoding of the underlying formalism is fully automated and is based on a convenient decomposition of the Hamiltonian into the effective zero-, one-, and two-orbital contributions as well as on the general strategy that focuses on the excitation operator driven evaluation of individual absolute, linear, quadratic, etc., coupled cluster coefficients, rather than on the standard molecular (spin) orbital driven algorithms. In this way unnecessary duplications are avoided and efficient codes are developed both for the general formula generation and final executable modules. A thorough testing of this procedure on a number of model cases is described and several illustrative applications at the abinitio level are provided.

The recently developed and implemented state selective, fully spin-adapted coupled cluster (CC) method that employs a single, yet effectively multiconfigurational, spin-free reference and the formalism of the unitary group approach (UGA) to the many-electron correlation problem, has been employed to calculate static electric properties of various open-shell (OS) systems using the finite field (FF) technique. Starting with the lithium atom, the method was applied at the first-order interacting space single and double excitation level (CCSD(is)) to several first- and second-row hydrides having OS ground state, namely to the CH, NH, OH, SiH, PH, and SH radicals. In the case of NH we also considered three OS excited states. In all cases the dipole moment and polarizability were determined using a high quality basis set and compared with the experiment, whenever available, as well as with various configuration interaction results and other theoretical results that are available from the literature. The agreement of our CCSD(is) values with experiment is very satisfactory except for the 3Σ− ground state of the NH radical, where the experimentally determined dipole moment is too small. No experimental data are available for the corresponding polarizabilities. It is also shown that the FF technique is not suitable for calculations of higher order static properties, such as the hyperpolarizability β and γ tensors. For this reason we formulate the linear response version of our UGA-based CCSD approach and discuss the aspects of its future implementation. Key words: static molecular properties, dipole moments, polarizabilities, free radicals, unitary group based coupled cluster method, linear response theory, finite field technique.

An alternative formalism leading to a vectorizable algorithm for a large scale full or limited configuration interaction (CI) calculations, employing partially spin-adapted bi-spinor or Hartree-Waller-type determinantal basis, is outlined and its relationship to existing approaches and algorithms is briefly discussed.

A new spin‐dependent unitary group approach to the many‐electron correlation problem is investigated. It is demonstrated that the matrix elements of the U(2n) generators, in the U(n)×U(2) adapted electronic Gel’fand basis, are determined by the matrix elements of a single U(n) adjoint tensor operator, herein denoted by Δij(1≤i, j≤n), where Δ is a polynomial of degree two in the U(n) matrix E=[Eij]. The method is then applied, in the second paper of the series, to derive a simple segment level formula for the matrix elements of all U(2n) generators. The advantages of this new procedure for computer implementation are discussed and the central role played by the matrix Δ for the determination of molecular spin densities is highlighted.

The performance of recently introduced coupled cluster (CC) method exploiting the unitary group approach (UGA) to many‐electron systems, truncated at the first order interacting space level [UGA‐CCSD(is)] and using the 6‐31G∗ basis set, in computations of equilibrium bond lengths and harmonic vibrational frequencies, is examined for a series of open‐shell (OS) states of the first row diatomics and hydrides. Altogether, 48 distinct electronic states are considered for 9 diatomic hydrides (BeH, BH, CH, CH+, NH, NH+, OH, OH+ and FH) and 18 diatomics (BeF, BN, BO, C2, C2+, C2−, CN, CO, CO+, CF, N2+, NO, NO−, NF, O2, O2+, OF and F2+), involving both high and low spin cases. Very good agreement with the available experimental data is found in all cases, except when the experimental values are marked as ‘‘uncertain’’ or where only the ΔG(1/2) values of harmonic frequencies are available. For the so‐called ‘‘difficult’’ systems, namely NO(X 2Π), O2(X 3Σg−), O2+(X 2Πg), OF (X 2Π) and F2+ (X 2Πg), the geometries and vibrational frequencies are also calculated using the TZ2P [5s4p2d] basis sets, and the results are compared with both the experiment and existing perturbation theory and CC results. All results indicate that UGA CCSD(is) represents a versatile, reliable and computationally affordable method that can handle a great variety of OS states, including OS singlets. © 1996 American Institute of Physics.

It is shown that the Clifford algebra unitary group approach, which is based on the subgroup chain U(2n)⊇SO(2n+1)⊇SO(2n)⊇U(n), may be described in terms of the para‐Fermi algebra. Applications to the development of efficient algorithms for the evaluation of matrix elements of U(n) generators and of their products are also briefly discussed.

Recently introduced state‐specific coupled‐cluster method, which exploits the formalism of the unitary group approach to the many‐electron correlation problem and enables a properly spin‐adapted treatment of open‐shell states, is applied to several low lying singlet and triplet electronic states of ozone at the double‐zeta plus polarization level of approximation. The method employs a nonstandard cluster Ansatz, based on a single spin‐free reference built from either the ground state restricted Hartree–Fock orbitals or from the restricted open‐shell Hartree–Fock orbitals, specific for each excited state. The results are compared with available experimental data and with other ab initio calculations, particularly with those employing spin‐orbital based, spin nonadapted multireference coupled‐cluster approaches of both state universal and valence universal types, as well as equation‐of‐motion coupled‐cluster method. The general agreement is satisfactory except for the B2 states, where the difference between the multireference spin nonadapted and our state specific, but spin adapted, approaches amounts to as much as 0.64 eV. It is shown that this difference arises due to the spin contamination that is present in the spin‐orbital based multireference approaches. © 1995 American Institute of Physics.

A new direct graphical formula in terms of Young tableaus is derived inductively, which provides a convenient and rapid method of evaluating arbitrary matrix elements of generators and of the products of generators of U(n).

Summary A novel approach of space symmetry adaptation is developed for multiconfigurational (MC) functions in fully optimized reaction space and complete active space SCF calculations. The bonded tableau and two box symmetric tableau are basic representations (rep) of configuration functions; the group symmetric localized orbitals are used as one-electron orbitals. The method is proposed for generating a complete and orthonormal set of MC single excited functions. The redundant variable in MCSCF can be eliminated by symmetry adaptation.

Explicitly connected many-body perturbation expansion for the energy of the first-order exchange interaction between closed-shell atoms or molecules is derived. The influence of the intramonomer electron correlation is accounted for by a perturbation expansion in terms of the Mo&slash;ller–Plesset fluctuation potentials WA and WB of the monomers or by a nonperturbative coupled-cluster type procedure. Detailed orbital expressions for the intramonomer correlation corrections of the first and second order in WA+WB are given. Our method leads to novel expressions for the exchange energies in which the exchange and hybrid integrals do not appear. These expressions, involving only the Coulomb and overlap integrals, are structurally similar to the standard many-body perturbation theory expressions for the polarization energies. Thus, the exchange corrections can be easily coded by suitably modifying the existing induction and dispersion energy codes. As a test of our method we have performed calculations of the first-order exchange energy for the He2, (H2)2, and He–H2 complexes. The results of the perturbative calculations are compared with the full configuration interaction data computed using the same basis sets. It is shown that the Mo&slash;ller–Plesset expansion of the first-order exchange energy converges moderately fast, whereas the nonperturbative coupled-cluster type approximations reproduce the full configuration interaction results very accurately.

The determinant full-CI algorithm of Zarrabian, Sarma and Paldus (Chem.
Phys. Letters 158 (1989) 183) has been implemented for efficient
operation on parallel vector computers. For few electrons ( n) in many
orbitals ( m) and nCI determinants, the floating point
operation count is O ( nCIm2n2),
dominated by matrix multiplication. Timings reported include
5.6×10 7 and 7.7×10 7 determinant
calculations on oxygen and its anion in 5s4p3d2f1g and 4s3p2d1f+spd
basis sets respectively. Consideration is given to exact manipulation of
CI expansions much larger than those used here.

With the advancement of our ability to account for many-electron correlation effects in atomic and molecular electronic structure calculations, particularly when exploring systems undergoing chemical reactions or other dissociative or associative processes, more and more emphasis is being placed on a proper size-extensive (or size-consistent) behavior of the theories employed. The requirement of size-extensivity (i.e., an exact additivity of the energy when applied to non-interacting systems) is, of course, absolutely crucial when we deal with extended systems. This is why this characteristic was automatically required in earlier developments of the general many-body perturbation theory (MBPT) by Brueckner (1955), Goldstone (1957), Hugen-holtz (1957), and others, since its primary domains of application at that time were an infinite nuclear matter (e.g., de Shalit and Feshbach, 1974; Eisenberg and Greiner, 1972) and various models of solid state physics (Hubbard, 1957,1958), notably the electron gas model (Gell-Mann and Brueckner, 1957; Quinn and Ferrell, 1958). The importance of size-extensivity in finite atomic and molecular systems was first recognized by Primas (1965), even though the term itself was coined and employed only later (Pople et al. 1976,1977,1978; Bartlett and Purvis, 1978,1980). Although both terms are often used interchangeably, we shall understand by the size-extensivity the additivity of energy for noninteracting systems or, equivalently, proportionality of the energy to the number of noninteracting systems (electrons, atoms, diatomics, etc.) involved.

One of the main challenges in the Variational Quantum Eigensolver (VQE) framework is construction of the unitary transformation. The dimensionality of the space for unitary rotations of $N$ qubits is $4^N-1$, which makes the choice of a polynomial subset of generators exponentially difficult process. Moreover, due to non-commutativity of generators, the order in which they are used strongly affects results. Choosing the optimal order in a particular subset of generators requires testing the factorial number of combinations. We propose an approach based on the Lie algebra - Lie group connection and corresponding closure relations that systematically eliminates the order problem.

An extension of the single reference coupled cluster method truncated to 1- and 2-body cluster components (CCSD) to quasidegenerate systems, where 3-and 4-body connected cluster components play an important role, is proposed. The basic idea is to extract the information concerning the 3- and 4-body clusters from some independent source, similarly as was implicitly done in the so-called ACPQ or ACC(S)D methods, and correct accordingly the absolute term in the CCSD equations. As a source of these approximate 3- and 4-body clusters, simple valence bond (VB) type wave functions are employed, since they are capable of describing electronic structure of various molecular systems for a wide range of nuclear conformations including their dissociation. The cluster analysis of these VB wave functions, that provides the desired information concerning the connected 3- and 4-body cluster components, is outlined and the explicit form of required correction terms to the CCSD equations is given.

The recently proposed valence bond (VB) corrected single reference (SR) coupled cluster method with singly and doubly excited cluster components (CCSD) [Paldus and Planelles, Theor Chim Acta 89, 13-31 (1994)] is tested using a number of simple yet typical Pariser-Parr-Pople (PPP) π-electron model systems, including both cyclic and linear polyenes. The cluster analysis of various approximate VB wave functions, obtained with the PPP-VB approach [Li and Paldus, J Mol Struct (Theochem) 229, 249 (1991)], is carried out and the resulting three- and four-body connected cluster components are employed in the VB corrected CCSD method. The cluster structure and the correlation energies obtained are compared to full configuration interaction (FCI) or full VB (FVB) results, representing the exact solutions for these models, and the performance and potential of the CCSD-VB approach are discussed.

In this paper, we discuss properties of single-reference coupled cluster (CC) equations associated with the existence of sub-algebras of excitations that allow one to represent CC equations in a hybrid fashion where the cluster amplitudes associated with these sub-algebras can be obtained by solving the corresponding eigenvalue problem. For closed-shell formulations analyzed in this paper, the hybrid representation of CC equations provides a natural way for extending active-space and seniority number concepts to provide an accurate description of electron correlation effects. Moreover, a new representation can be utilized to re-define iterative algorithms used to solve CC equations, especially for tough cases defined by the presence of strong static and dynamical correlation effects. We will also explore invariance properties associated with excitation sub-algebras to define a new class of CC approximations referred to in this paper as the sub-algebra-flow-based CC methods. We illustrate the performance of these methods on the example of ground- and excited-state calculations for commonly used small benchmark systems.

Spinor representations of generators of the Lie algebra of SO(N)(N=2n, 2n+1; n integer, have played a key role in a number of areas of Physics [1–5]. A general approach to these representations in a form suitable for practical applications has been of recent origin [6–8]. Starting with the unitary algebra of U(2n), the generators of SO(N) and U(n) were realised in the chain U(2n) ⊃ SO(2n+1) ⊃ SO(2n) ⊃ U(n). It was found that the symmetric bispinor basis spanning the representation [2 Ȯ] of U(2n) could be used to subduce the spin-free configurations spanning the representations [2N/2-s, 12s, Ȯ] of U(n). Some preliminary studies of generating the configuration space in this manner have recently been carried out for basis adapted for the chains U(n)⊃...⊃ U(1) [8] and U(n) ⊃ SO(n) ...⊃ SO(2) [9]. From a practical point of view this approach has a basic drawback. This is the fact that the simple one electron orbital description of spin-free configurations is masked in using the spinor basis. In the present study we examine the possibility of inducing a basis spanning the representations of U(2n) and SO(N) starting with the antisymmetric representations, [1N, Ȯ] (0≤N≤n; N integer) of U(n). The aim is to provide a simple interpretation of the spinor basis in terms of the tensor (integer) representations of U(n).

A tribute to the life and work of Josef Paldus is presented in the form of a brief biographical sketch and an extensive bibliography.

The unitary group approach (UGA) to many-fermion correlation problem may be regarded as a direct outgrowth of the original ideas as laid down by Hermann Weyll in his “Gruppentheorie and Quantenmechanik”. Although these advances have in the past been overshadowed by the simplicity of Slater determinant based formalisms, and have even been referred tol at one time as a “group pest”, today we find Weyl’s original ideas very much alive and well, particularly in the many-electron correlation problem. For instance, the Gelfand-Tsetlin (GT) representation theory’ of the unitary group is a natural extension of Weyl’s branching rule presented in the very last Section of his book.1

A new statistical approach to the configuration interaction problem is proposed. It will potentially reduce the size of CI vectors significantly and thus alleviate one of the main hindrances to doing very large CI calculations. This method will also provide us with statistical confidence limits on lower and upper bounds. These confidence bounds are qualitative different from mathematically more rigorous lower bounds discussed elsewhere. The approach is discussed and possible algorithms presented.

The Molecular Mechanics Valence Bond (MM-VB) method is a hybrid MM/QM method that uses a Heisenberg Hamiltonian (VB) for the QM part combined with force field methods. The method works with hybrid atoms that have both VB linkages to other atoms as well as classical MM interactions. The MM and QM parts are fully parametrized. The use of the VB method allows modelling of ground and excited state processes. In dynamics computations the nuclei are propagated using the gradient while the wavefunction propagation is treated via semi-classical equations which include the non-adiabatic coupling. Applications to molecular structure and dynamics for photochemistry and photophysics will be discussed.

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.

It is a great honor to contribute to this volume, dedicated to Professor Larry C. Biedenharn at the occasion of his 70th anniversary. His seminal papers on representation theory of semi-simple Lie groups [1], boson and pattern calculus [2], canonical tensor operators in the unitary group [3], as well as his and J.D. Louck’s authoritative monographs on Racah-Wigner algebras [4,5], represent a fundamental point of departure for any development based on the unitary group symmetry of various physical or chemical systems. Although the standard SU(2) spin angular momentum formalism is in principle adequate when exploring molecular electronic structure, the exploitation of higher unitary group symmetries, that naturally arise when using the molecular orbital (MO) formalism, proved to be very beneficial and useful not only for various methodological advances at various levels of approximation [6–11], but also for the design and development of efficient algorithms and actual computer codes (for references, see [11,12]). In fact, the boson calculus based formalism of unitary group representation theory proved to be a useful source of various new concepts in quant urn-chemical methodology [6–16] that continue to play an important role in diverse approaches to molecular electronic structure, be they of variational or perturbative nature. Moreover, it also provided a unified viewpoint and better insight into various existing schemes, that are based either on the permutation group S
N
invariance properties of systems involving N indistinguishable particles [17] or the classical spin angular momentum formalism [18].

Relying on a semiempirical Pariser‐Parr‐Pople (PPP) model, it is shown that classical valence bond (VB) theory, using only a few covalent structures, can provide highly correlated energies and wave functions and thus a useful description of a molecular electronic structure, assuming that it is based on appropriate overlap‐enhanced atomic orbitals. Approximating the σ‐energy contribution by a quasi‐harmonic empirical potential, this model provides a faithful yet simple and transparent explanation of the interplay between electron derealization or resonance effects and the π‐electron localization tendency leading to bond length alternation in various π‐electron systems possessing aromatic, nonaromatic, or antiaromatic characters.

A detailed exposition of explicit formulas used in the evaluation of
raising-raising forms of two-body-operator matrix elements is presented.
The methods are based on the use of the unitary-group distinct-row
tabular-graphical representation of the many-particle basis. All matrix
elements are expressible in the form of scalar and 2×2 matrix
factors. In order to facilitate the derivations of simple,
computationally efficient forms for the matrix factors we develop a
calculus based on elementary graphs. The methods are applicable to
systems of particles involving spins greater than (1/2 and reduce to
previously known results for the case of spin (1/2.

The well known symmetry (invariance, degeneracy)
groups or algebras of quantum mechanical Hamiltonians provide quantum numbers (conservation laws, integrals of motion) for state labeling and the associated selection rules. In addition, it is often advantageous to employ much larger groups, referred to as the dynamical groups (noninvariance groups, dynamical algebras, spectrum generating algebras), which may or may not be the invariance groups of the studied system [4.1,2,3,4,5,6,7]. In all known cases, they are Lie groups (LGs), or rather corresponding Lie algebras (LAs), and one usually requires that all states of interest of a system be contained in a single irreducible representation (irrep). Likewise, one may require that the Hamiltonian be expressible in terms of the Casimir operators of the corresponding universal enveloping algebra [4.8,9]. In a weaker sense, one regards any group (or corresponding algebra) as a dynamical group if the Hamiltonian can be expressed in terms of its generators [4.10,11,12]. In nuclear physics, one sometimes distinguishes exact (baryon number preserving), almost exact (e.g., total isospin), approximate (e.g., SU(3) of the “eightfold way”) and model (e.g., nuclear shell model) dynamical symmetries [4.13]. The dynamical groups of interest in atomic and molecular physics can be conveniently classified by their topological characteristic of compactness. Noncompact LGs (LAs) generally arise in simple problems involving an infinite number of bound states, while those involving a finite number of bound states (e.g., molecular vibrations or ab initio models of electronic structure) exploit compact LG's.
We follow the convention of designating Lie groups by capital letters and Lie algebras by lower case letters, e.g., the Lie algebra of the rotation group SO(3) is designated as so(3).

This is a chapter on the history of theoretical organic chemistry that is principally concerned with the structure of molecules. The covalent bond in organic molecules can be understood using quantum mechanics. The development phase has been divided into 3 periods—1850–1875, 1910–1935, 1955–1980 with interludes in-between. The first of these periods (1850–1875) witnessed the birth of the structural formula and its development from formal representation to a reflection of physical reality. The second period (1910–1935) saw the advent of quantum mechanics and the concepts of the electron pair, resonance and mesomerism, and hybridisation. The third period (1955–1980), is the period of successful application of molecular orbital theory to chemical reactions. There is mention of the ways in which in the first period the line between two atoms only meant the mutual saturation of valencies and this sequence of discoveries culminated with the realization that molecules exist in a three-dimensional space. There is the mention of Cannizzaro's introduction of the true atomic (and molecular) weights. In the second period, the electronic structure of benzene was finally understood. The story of resonance starts in the middle of this period. The relationship of the two main quantum–chemical methods was established. It became known that at large distances the relief of kinetic pressure lowers the energy. The third period saw the important progress in reaction dynamics. The wave packet method was developed to treat reactions that start on the excited state potential energy surface. Whereas in the early dynamics calculations the potential energy surface was no more than a model surface or the result of a semi-empirical calculation, nowadays ab initio and fully geometry optimized surfaces are available. The problem was dealt with the concept of molecular structure.

An indexing scheme for determinantel states of spin-orbitals that enables a dense contiguous labeling of configurations even for truncated limited configuration-interaction expansions of the many-electron wavefunction is presented.

A method called algebraic resonance quantization (ARQ) is presented for highly excited multidimensional systems. This approach, based on the Heisenberg form of the correspondence principle, is a fully quantum mechanical matrix method. At the same time, it uses modern nonlinear classical mechanics to greatly simplify the Hamiltonian matrix. For a model system of coupled Morse oscillators, a nonlinear resonance analysis shows that the Hamiltonian matrix is dominated by a few leading terms. This leads to an effective truncated sparse matrix whose diagonalization yields eigenvalues in excellent agreement with the exact values, even high in the chaotic regime. A new finding is that quantum couplings corresponding to rapidly oscillating, nonresonant terms can be important, and not just the higher‐order resonant terms. The generalization of ARQ via a numerical semiclassical technique to many‐dimensional systems with arbitrary couplings is outlined. The applicability of contemporary vector methods from quantum chemistry to ARQ of high vibrational levels is considered. The feasibility of sparse matrix ARQ methods for fitting spectra in the highly chaotic, ‘‘unassignable’’ regime is discussed.

A time dependent theory is developed for the SO(2N+1) wave function which yields both paired and unpaired modes of Fermion. The time dependent SO(2N+1) equation is derived and shown to have two kinds of solution, the Hartree-Bogoliubov solution which describes even Fermion
systems and the improper Hartree-Bogoliubov solution which describes odd Fermion systems. The time dependent improper Hargree-Bogoliubov
equation leads to a new random phase approximation which gives a self-consistent unified description of Bose and Fermi type
collective excitations in an odd Fermion system.

An approach to the configuration interaction method based on symmetric groups (SGA) is developed. The formalism is an alternative of the unitary group approach (UGA). In many aspects the present formulation seems to be superior to UGA. In particular, in SGA the orbital and the spin parts of the configuration state functions may be processed separately. In consequence its graphical formulation is much simpler and the coupling constant expressions are more compact than the UGA analogs. A special emphasis is put on direct CI implementations. In addition to formulas for coupling constants, explicit expressions allowing for separation of external and internal space contributions are also presented.

A thorough analysis of the direct CI method as applied to the case of a general set of reference configurations coupled to all single and double substitutions is presented. It is pointed out that there is no single strategy which proves optimal under all circumstances. A variety of procedures are therefore presented together with rules to enable the selection of the most favourable under a given circumstance. Much emphasis has been placed on organizing the calculations via a series of matrix multiplications, which enables a vector or array processing computer to be used to best effect. Some consideration is given to using an atomic integral (rather than molecular integral) driven scheme for some interactions, thus removing the necessity for a complete transformation of the molecular integrals to a molecular orbital basis, and the advantages and disadvantages of so doing are discussed. Improved procedures for carrying out both full and partial transformations of the molecular integrals are described. A number of test case calculations involving configuration lists of the order of 10 4 to 10 5 have been analysed in detail, to give a clear picture of the cost of the various interaction types which arise, and indicating that integrals carrying two external molecular orbital indices account for approximately 60 per cent of the cost in typical cases. Typically, a calculation involving 10 5 configurations requires approximately two minutes of CRAY-1 computer time, allowing for ten iterations of the diagonalization procedure. The cost of the calculations are found, somewhat surprisingly, to be approximately linear in the dimension of the configuration space, indicating that calculations involving 10 6 configurations are now quite feasible.

The chapter discusses the time-independent diagrammatic approach to perturbation theory of fermion systems. The chapter explores the perturbation theory for a non-degenerate level. The formulas derived serves as a starting point for the subsequent consideration of the excitation and ionization energies. The advantages of the direct calculation of excitation energies, compared with the approach in which the total energies of the pertinent electronic states are calculated separately for each state and then the excitation energies are obtained by subtracting the appropriate state energies, are quite obvious. The Rayleigh-Schrödinger (RS) perturbation theory (PT) for the case of a non-degenerate level of some Hamiltonian operator is discussed. The chapter discusses that even the Rayleigh-Schrodinger perturbation expressions for the direct calculation of the excitation energies may be obtained in a rather simple way without the involvement of the Green function formalism. On the contrary, our simple approach using the ordinary perturbation theory for separate levels presents certain desirable features of the Green function formalism. The chapter explains the diagrammatic representation of Wick's theorem and resulting diagrams. General explicit formulas for the second- and third-order excitation energy contributions are given in the chapter.

A formalism for an efficient generation of spin‐symmetry adapted configuration interaction (CI) matrices of the N‐electron atomic or molecular systems, described by nonrelativistic spin‐independent Hamiltonians, is presented. The Gelfand and Tsetlin canonical basis for the finite dimensional irreducible representations of the unitary groups is used as an N‐electron CI basis. A simplified Gelfand‐type pattern pertaining to the N‐electron problem is introduced, which considerably simplifies the canonical basis generation and, more importantly, the calculation of representation matrices of the (infinitesimal) generators of the pertinent unitary group in this basis. The calculation of the CI matrices for the above mentioned systems is then straightforward, since any particle number conserving operator may be written as a sum of n‐degree forms in the unitary group generators. The computation of CI matrices for various Hamiltonians as well as the problems of the space‐symmetry adaptation of the Gelfand‐Tsetlin basis and of limited CI calculations are briefly discussed.

Total orbital momentum and total spin are good quantum numbers for a complete basis which we construct as explicit linear combinations of orthonormal vectors (m1m2m3), where integers | mi | ≤ 4. Matrix elements of generators of the orbital momentum group are found explicitly in the basis (m1m2m3) and a one-to-one correspondence between (m1m2m3) and Gel'fand—Tseitlin patterns of SU (9) is shown. The multiplicity problem is solved systematically by simplicity conventions similar to those introduced by Racah for the cases where his classification scheme for higher fn configurations failed. Every angular momentum multiplet we construct belongs to one irreducible representation of O (9) only.

The generators of the rotation groups SO(N) (N=2n, 2n+1) have been realized using a restriction of the unitary group U(2n) defined on the 2n ‐dimensional fundamental representation space of spinors. These generators have been used to subduce multispinor representations of SO(N) from those of U(2n). The procedure has been illustrated for the two‐spinor vector representations 〈10〉 and 〈1000〉 of SO(5) and SO(8), respectively.

All Clebsch-Gordan coefficients required for calculations in the meson sector of a proposed SO(8) model of elementary particles are obtained (as SU(4) singlet factors) by an extension of the general formalism of Gel'fand.

A picture of reality drawn with a few sharp lines cannot be expected to be adequate to the variety of all its shades. Yet even so the drafsman must have courage lo draw the lines firm. H. Weyl in Philosophy of Mathematics and Natural Science

It is shown that if α denotes an n × n antisymmetric matrix of operators αpq,p,q = 1, 2, …, n, which satisfy the commutation relations characteristic of the Lie algebra of SO(n), then α satisfies an nth degree polynomial identity. A method is presented for determining the form of this polynomial for any value of n. An indication is given of the simple significance of this identity with regard to the problem of resolving an arbitrary n‐vector operator into n components, each of which is a vector shift operator for the invariants of the SO(n) Lie algebra.

Zusammenfassung Die Möglichkeit, dem quantenmechanischen Mehrkörperproblem gerecht zu werden durch zwei äußerlich ganz verschiedene Methoden (Koordinatenraum-methode und zweite Quantelung), hängt zusammen mit einer mathematischen Wechselbeziehung zwischen den Darstellungen der symmetrischen Permutations-gruppen und der linearen Gruppen.

A computational approach to the direct configuration interaction method is described. The method is formulated using the calculus of the generators of the unitary group. The simple structure of the generator matrices within the harmonic excitation level scheme is exploited to give a computational method that is competitive with traditional approaches. A new scheme for basis set truncation in the case of partial configuration interaction is devised employing orbital populations. It is also shown that the block structure of the generator matrices leads to the definition of a new order parameter for perturbation methods which is both effective and convenient.

We study in this series the group theoretical structure of Fermion many-body systems arising from the canonical anticommutation
relation of the annihilation-creation operators. Owing to the canonical anticommutation relation, a Fermion system with N single particle states has at least six Lie algebras of Fermion operators, a U(N), an SO(2N), an SO(2N+1), an SO(2N+2) and two U(N+1) Lie algebras. There are also two Clifford algebras of 2N and 2N+1 dimensions. The Fermion space is shown to belong to the spinor representations of the SO(2N), SO(2N+1) and SO(2N+2) groups. The canonical transformations generated by the U(N), SO(2N) and SO(2N+1) Lie algebras are characterized as the transformations to induce the linear U(N), SO(2N) and SO(2N+1) transformations for the Clifford algebras. The independent (quasi-) particle type wave functions of three kinds including
the Hartree-Fock and Hartree-Bogoliubov wave functions are constructed by means of the canonical transformations and their
relationship is studied. We derive three exact generator coordinate representations for state vectors in which the generator
coordinates are the U(N), SO(2N) or SO(2N+1) group and the generating functions are the independent (quasi-) particle type wave functions. We characterize the structures
of state vectors in the generator coordinate representations and study the relationship of the three representations.

A straightforward derivation of the matrix elements of the U(n) generators is presented using algebraic infinitesimal techniques. An expression for the general fundamental Wigner coefficients of the group is obtained as a polynomial in the group generators. This enables generalized matrix elements to be defined without explicit reference to basis states. Such considerations are important for treating groups such as Sp(2n) whose basis states are not known.

A realization of the spinor algebra of the rotation group SO(N), N=2n or 2n+1, in the covering algebra of U(2n) is exploited to obtain explicit representation matrices for the SO(N) generators in the basis adapted to the subgroup chain SO(N)⊃U(n)&supuline;U(n−1)⊃⋅⋅⋅⊃U(1). As a special case the computation of matrices of U(n) representations characterized by a k-column Young tableau is reduced to the evaluation of at most k-box totally symmetric representations of U(2n).

A purely algebraic approach to the evaluation of the fundamental Wigner coefficients and reduced matrix elements of O(n) and U(n) is given. The method employs the explicit use of projection operators which may be constructed using the polynomial identities satisfied by the infinitesimal generators of the group. As an application of this technique, a certain set of raising and lowering operators for O(n) and U(n) are constructed. They are simpler in appearance than those previously constructed since they may be written in a compact product form. They are, moreover, Hermitian conjugates of one another, and therefore are easily normalized.

In a previous paper, many nucleon states in the shell model are characterized through the group C2 used for an exact treatment of charge independent pairing interaction. This scheme was proved to be equivalent to Flowers' seniority-reduced-isospin scheme. The matrix elements of pair creation operator (a generator of the group C2) are connected with the coefficients of fractional parentage of the type
which play an essential role in seniority scheme. Then a simple method of calculation of the matrix elements of pair operator is developed, and they are tabulated up to λ1+λ2≤4 (complete for up to f7/2 shell). A recursion formula of c.f.p. in seniority scheme is derived, which is more general than that of de-Shalit and Talmi

A new direct CI method is presented, which is particularly suited for large CI expansions in a small orbital space. These are the type of expansions which are common in the CAS SCF method. Only one-electron coupling coefficients are stored, which leads to reduced elapsed times and storage requirements compared to earlier approaches. The two-electron coupling coefficients are implicitly created in the diagonalization step. The algorithm for updating the CI vector is formulated as the trace of a product of three matrices, Tr(A · D · I). By ordering the one-electron coupling coefficients (A) in a certain way the matrix D is easilly created as a sparse scalar product between these coefficients and the trial CI vector. The main computational step is then a simple matrix multiplication between the matrix D and the symmetry blocked integral matrix (1). This operation vectorizes very well on most vector processors. Another sparse scalar product between the resultant matrix and the coupling coefficients leads to the update of the CI coefficients. In a calculation on CRAY-1 with 30700 configurations, the two-electron part in a CI iteration required 10 s of which half went into the handling of the one-electron formula tape.

A new method is given for deriving the angular factors of energy
matrices in l/sup n/ atomic configurations, and the need for coefficients of
fractional parentage, Racah coefficients, algebraic formulas, and chain
calculations is avoided. Instead, matrix elements are obtained directly by a
comparatively simple digital counting procedure. (auth)

A hole-particle version of the unitary group approach to the many-electron correlation problem is given. To achieve a unique state labeling, the subgroup chain of Flores and Moshinsky for the nuclear many-body problem is modified, and the rules for the construction of relevant irreducible representations and of the orthonormal spinadapted states are given. The structure of these hole-particle formalism spin-adapted bases is illustrated on several examples for all the three possible types of the hole-particle defect pertaining to the excitation, electron attachment, or detachment processes relative to the reference state used. The tensor character of the particle formalism electron-number-preserving generators is pointed out, and their basic classification in the hole-particle formalism is given.

Graphical methods of spin algebras are used to derive the expressions for the matrix elements of the total particle-number-conserving operators in the basis of hole-particle states, adapted to the chain U(n′+n′′)⊃U(n′)⊗U(n′′), where n′ and n′′ designate dimensions of particle and hole subspaces, respectively. The matrix elements are expressed as a product of segment values, each associated with one orbital level as in the particle formalism, and of an additional segment value, representing a linkage of the hole and particle subspaces. It is shown that the particle formalism segment values can be used throughout except for the link segment, whose possible values are derived. An example of hole-particle bases and of their graphical representations is given and the advantages of the hole-particle formalism in shell-model calculations are outlined. An extension of this formalism to particle-number-nonconserving operators, needed in applications involving the mp-nh propagators with m≠n, m, n=0, 1, and 2, is discussed.

The power of the Young tableau scheme for labeling a complete spin-adapted basis set in the theory of complex spectra lies in one's ability to evaluate matrix elements of irreducible tensor operators directly in terms of the tableau labels and shapes. We show that the matrix-element rules stated by Harter for one-body operators can be easily derived from simple vector-coupling considerations. The graphical method of angular momentum analysis is used to derive closed-form expressions for the matrix elements of two-body operators. This study yields several interesting new relationships between spin-dependent operators and purely orbital operators.

Recent theoretical research by Paldus and by Shavitt has strongly suggested that the unitary group approach to the many body problem may be useful in molecular electronic problems. The graphical unitary group approach (GUGA) has now been developed into an extraordinarily powerful theoretical method. The theoretical/methodological contributions made here include a solution of the upper walk problem, the restriction of configuration space employed to the multireference interacting space, and the restructuring of the Hamiltonian in terms of loop types. Several test calculations are examined in detail to illustrate the unique features of the method. For large general multireference configuration interaction (CI) problems, computation times are typically only 15% of those reported using state‐of‐the‐art conventional techniques. Finally, these methods are applied to the vertical electronic spectrum of ketene, and excellent agreement with experiment is found.

The graphical techniques of spin algebras are combined with a diagrammatic approach based on the time independent Wick theorem to yield the spin‐adapted form of the coupled cluster theory. The general rules for the implementation of this formalism are formulated and illustrated on the basic coupled‐pair many‐electron theory pertaining to closed shell ground states, for which case the explicit spin‐adapted equations are derived. The advantages of the spin‐adapted form of the theory are discussed along with the new insights it affords.

A hierarchy of tensor identities, satisfied by the generators of the general linear group GL(n), is obtained in terms of two different sets of invariants. An application to the identification of irreducible representations and the decomposition of reducible representations is described. Similar results are obtained for the generators of orthogonal, pseudo‐orthogonal, and symplectic groups.

The direct CI method is generalized to the case of all single and double replacements from an arbitrary set of reference configurations. This is a continuation of the work and ideas presented in an earlier paper on first order wave functions. The analysis is done using the unitary group formulation of the correlation problem, and the resulting method is a combination of the direct CI method and the unitary group approach as formulated particularly by Paldus and Shavitt. The main idea in the present work is the factorization of the coupling coefficients appearing in the direct CI formalism, into a complicated internal part and a simple external part. The general philosophy is like in all direct CI methods to allow long CI expansions by avoiding the storage and retrieval of a large formula tape. The longest CI expansion treated in this paper is an application on the system CH2(3B1)+H2→CH3+H with five reference states, resulting in 16 096 configurations. The barrier height for the reaction is calculated to be 11.3 kcal/mol and predicted to be slightly below 10 kcal/mol.

After recalling the duality between the general linear group GL(m), represented by its N‐fold inner product, and the permutation group SN, we have given a survey of its quantum chemical consequences. It causes the one‐to‐one correspondence between the total spin quantum number and the permutation symmetry of N‐electron spin functions, and, via the Pauli principle which imposes permutation symmetry on the spatial part also, it leads to specific properties of antisymmetric spin eigenfunctions under orbital transformations. Such functions can be classified according to the irreducible representations of GL(m). For special orbital transformations, often occurring in quantum chemistry, which mix only orbitals in different subsets among each other, we have derived how the transformation of the N‐electron wavefunctions simplifies, by a reduction of the representations of GL(m). The theory is illustrated by an example and some applications are discussed.

A new method for the approximate solution of Schrödinger’s equation for many electron molecular systems is outlined. The new method is based on the unitary group approach (UGA) and exploits in particular the shape of loops appearing in Shavitt’s graphical representation for the UGA. The method is cast in the form of a direct CI, makes use of Siegbahn’s external space simplifications, and is suitable for very large configuration interaction (CI) wave functions. The ethylene molecule was chosen, as a prototype of unsaturated organic molecules, for the variational study of genuine many (i.e.,≳2) body correlation effects. With a double zeta plus polarization basis set, the largest CI included all valence electron single and double excitations with respect to a 703 configuration natural orbital reference function. This variational calculation, involving 1 046 758 spin‐ and space‐adapted 1Ag configurations, was carried out on a minicomputer. Triple excitations are found to contribute 2.3% of the correlation energy and quadruple excitations 6.4%.

The recent electronic orbital tableau ("jawbone") formulas of Harter and
Patterson for matrix representatives of elementary generators of U(n) in
the canonical basis are shown to be simply related to our expressions
for the latter, which are based on a simplified Gelfand tableau
formalism pertinent to N-electron problems, a brief survey of which is
given in a form particularly suitable for computer implementation.

A self-contained review of the symmetric group approach to configuration interaction methods is given. Benefits resulting from an explicit separation of the N-particle configuration space to the orbital and the spin subspaces are discussed in detail. In particular, the internal structure of both the subspaces has been explored using their graphical representations. In effect an optimum configuration interaction algorithm has been formulated. Complete sets of formulas, necessary in both conventional and direct modes of implementations, are given in a compact, tabular form.

The first two members of the cyclic polyene homologous series are studied over a wide range of the coupling constant using the Hubbard and Pariser–Parr–Pople model Hamiltonians. The full and various limited configuration interaction (CI) correlation energies and wave functions are calculated exploiting the unitary group approach. The formalism for the cluster analysis of the exact wave function expressed through the unitary group formalism electronic Gelfand states is developed and applied to the full CI wave functions of the cyclic polyene models studied. It is shown that the connected tetraexcited clusters become essential in the fully correlated limit and that their contribution also significantly increases with electron number even for the coupling constant corresponding to the spectroscopic parametrization of the model Hamiltonians used.

In this article we derive a segment-level formula for the matrix elements of the U(2n) generators in a basis symmetry adapted to the subgroup U(n) × U(2) (i.e., spin-orbit basis), for the representations appropriate to many-electron systems. This enables the direct evaluation of the matrix elements of spin-dependent Hamiltonians.

This is the final paper in a series of three directed toward the evaluation of spin-dependent Hamiltonians. In this paper we derive the reduced matrix elements of the U(2n) generators in a basis symmetry adapted to the subgroup U(n) × U(2) (i.e., spin-orbit basis), for the representations appropriate to many-electron systems. This enables a direct evaluation of the matrix elements of spin-dependent Hamiltonians in the spin-orbit basis. An alternative (indirect) method, which employs the use of U(2n) ↓ U(n) × U(2) subduction coefficients, is also discussed.

In this note a method is presented for quick implementation of configuration interaction (CI) calculations in molecules. A spin-free Hamiltonian for anN electron system in a spin stateS, expressed in terms of the generators for the unitary group algebra, is diagonalized over orbital configurations forming a basis for the irreducible representation [21/2N-S
12S
] of the permutation group S
N
. It has been found that the basic algebraic expressions necessary for the CI calculation involve a limited category of permutations. These have been displayed explicitly.

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.

When dealing with the second quantized picture with nonorthogonal basis, the creation and annihilation operators, which are hermitian conjugate, no longer satisfy the usual anticommutation relations. If, on the other hand, we introduce an appropriate dual basis we have for the ordimary creation and the dual-annihilation operators the usual anticommutation relations, but these operators are no longer hermitian conjugate. Nevertheless, the dual basis approach proves very convenient as we can still use all the familiar group theoretical procedures of the second quantized picture, at the expense of dealing with irreducible representations of linear rather than unitary groups. In problems in which nonorthogonal single particle states are used (e.g., in molecules), O(3) is in general not a symmetry group and thus we are free to characterize our n-particle states by the irreducible representations of a mathematically convenient chain of subgroups. This turns out to be the canonical chain of subgroups of the linear group giving rise to Gel'fand states. We discuss the evaluation of matrix elements of one and two body operators with respect to these states and illustrate our analysis with the simplest nontrivial example in which the number of both particles and orbital single particle states is 3.

The five-dimensional quasi-spin formalism is used to factor out the n, T dependent parts of shell-model matrix elements in the seniority scheme and derive reduction formulae which make it possible to express matrix elements for states of definite isospin T in the configuration jn in terms of the corresponding matrix elements for the configuration jv. The n, T dependent factors for one- and two-nucleon c.f.p. and for the matrix elements of one-body operators and the two-body interaction are expressed in terms of generalized R(5) Wigner coefficients. The needed R(5) Wigner coefficients are calculated in the form of general algebraic expressions for the seniorities v and reduced isospins t corresponding to the simpler R(5) irreducible representations. In this first contribution, the R(5) representations are restricted to ([omega]10), (), (tt), and the states of ([omega]11) with n-v = 4k-2T, (k is an integer). Explicit expressions are given for the diagonal matrix elements of the general, charge-independent, two-body interaction and the isovector and isotensor parts of the Coulomb interaction for seniorities v = 0 and 1, and the v = 2 states with n = 4k +2-2T. Peer Reviewed http://deepblue.lib.umich.edu/bitstream/2027.42/33284/1/0000676.pdf