Symmetrical overlap transformations of function basis sets: the LCAO MO and quantum similarity practical cases

Quantum chemical computational procedures, like LCAO MO theory and quantum similarity, use non orthogonal function basis sets, which define finite dimensional subspaces of a Hilbert space. Based on the original overlap metric matrices, generated by the chosen finite non orthogonal basis sets, there are several symmetrical overlap and basis set transformations possible. This study tries to find out the general point of view, from where all these procedures can be studied in a clear generalized perspective.     

Some comments on the properties of unitary transformations on linear spaces having an indefinite metric and the connection with the theory of spin

After a brief review of the history of the discovery of the spin, some fundamental properties of linear spaces having an indefinite metric are being discussed. The study starts with an elementary survey of the theory of matrices and their stability problem. It is emphasized that—by a similarity transformation—all matrices may be brought to classical canonical form characterized by the diagonal elements called eigenvalues, their multiplicities, their Jordan blocks, and their Segré characteristics. In connection with the reduced Cayley-Hamilton equation, the existence of the product projection operators and their main properties is briefly discussed. Particular attention is paid to the concept of a basis for the linear space and the associated metric matrix, which is self-adjoint and may be brought to diagonal form with the eigenvalues ± 1 by a unitary transformation, which reveals the indices of inertia, p and q. The Minkowski space having p = 3 and q = 1 is used as an example. After this introduction, some properties of linear operators defined on an indefinite space are discussed, and it is pointed out that self-adjoint operators and unitary operators may now have a rather peculiar and unexpected behavior, and the special Lorentz transformations are used as an example. It is then shown that these features are of essential importance in studying rotations as special cases of unitary transformations defined on an indefinite space. The rotations are here defined by means of their reduced Cayley-Hamilton equation, and their properties are studied by means of the associated product projection operators, which are idempotent, mutually exclusive, and form a resolution of the identity. In a previous article, it was shown that, in a positive definite space, there is a close connection between the requirement that all rotations around an external axis form a group and the existence of an anticommutator algebra leading to the concept of spinors. The rotations are expressed in the exponential form U=exp(iO), where O is a self-adjoint operator which is independent of any choice basis, coordinate system, etc., and which is, hence, a true invariant. It is shown that this approach may now be extended also to indefinite spaces and may lead to operators O which are both rotationally and relativistically invariant. In this connection, the full Lorentz transformations are given a particularly simple form. The article may be considered as a simple exercise in linear algebra, in which the mathematical connection between the concept of rotations and the existence of spinors is strongly emphasized. © John Wiley & Sons, Inc.     

A comparison of geometric methods with other methods for the characterization of density matrices

Some recently developed geometric methods for characterizing the subset of density matrices within the space of Hermitian matrices are compared with methods commonly used for the approximate characterization of reduced density matrices. The decomposition of a density matrix into components in terms of the reducing basis set is compared with decomposition in terms of representations of U(r).     

Symmetry aspects of diatomics-in-molecules (DIM) calculations. II. Construction of symmetry-adapted diatomic fragment subspaces

A new general procedure is presented for the construction of symmetry adapted diatomic fragment sub-spaces of spin-adapted polyatomic DIM bases obtained by direct diagonalization of the polyatomic spin-square operator. The spin part of the construction is accomplished by the diagonalization in the DIM basis of the diatomic spin-square operators. The spatial part of relevant transformations is given in terms of quantum numbers of the constituent atomic functions. Simple expressions for the spatial symmetry and spin designation of the resulting symmetry-adapted diatomic fragment states are obtained. Special attention is paid to consistent phases of diatomic fragment functions corresponding to equivalent fragments. The approach eliminates such mental operations from the construction of symmetry-adapted diatomic fragment subspaces which are difficult to formulate in a way suitable for straightforward computer implementation.     

The symmetrized random matrix approach, an efficient method to obtain relativistic molecular symmetry adapted functions

In relativistic quantum chemical calculation of molecules, where the spin-orbit interaction is included, the electron orbitals possess both the double point group symmetry and the time-reversal symmetry. If symmetry adapted functions are employed as the basis functions of electron orbitals, it would allow a significant reduction of the computational expense. The point group symmetry adapted functions can be obtained by the group projection operators via its actions on the atomic orbital functions. We have proposed an efficient and simple method to obtain all irreducible representation matrices, which are the basis of the group projection operators, of any finite double point group. Both double point group symmetry and time-reversal symmetry are automatically imposed on the representation matrices. This is achieved by the symmetrized random matrix (SRM) approach, where the SRM is constructed in the regular representation space of a finite group and the eigenfunctions of SRM provide all irreducible representation matrices of the given point group.     

Symmetry adapted functions belonging to the dirac groups

An earlier analysis of the canonical form of a pair of invertible operators obeying the exchange rule is extended to cover a set of operators, between each pair of which a relation of this type exists; and for which a power of each operator is the unit matrix. Such relations define a system which may be regarded as a generalization of the Dirac matrices of relativistic quantum mechanics. We concentrate upon the group theoretic aspects of such a system and its matrix representations. Applications arise from the fact that all projective representations of finite abelian groups take the form of a Dirac Group. In particular, the representations of the magnetic space groups, which are projective representations of the lattice groups, arise in this manner.     

On formulas in closed form for Van Vleck expansions

Canonical transformations have been widely used to simplify Hamiltonians and other operators. In molecular and in solid state theory, the so-called Van Vieck expansion is usually employed for this purpose while in theories of particles interacting with fields a combination of canonical transformations in closed form with Van Vleck type expansions has been found effective. For some of the transformations used in applications formulas in closed form are well known. It will be shown here that such formulas can be derived whenever the transformation function is bilinear in the canonical variables, and further that the use of matrix operators makes it possible to simplify these derivations substantially. The Cayley-Hamilton theorem is then used to express the expansions for the matrix operators in closed form. The number of separate operator terms appearing in the formulas thus obtained is the same as the rank of the matrices used. To calculate the coefficients of these operator terms a new type of special functions is introduced. The resulting linear canonical transformations include generalized rotations in both ordinary and phase-space. Explicit results have been obtained for several two- to four-dimensional problems.     

Dyad algebra and multiplication of graphs. I. Directed graphs

The ket-bra algebra for quantum mechanics and for the quantum chemistry.in valence shells was made by this author fully covariant recently. The resulting principle of linear covariance allowed diverse approaches such as molecular orbital, valence bond, localized orbital theories to come out as special cases in particular basis frames not necessarily orthonormal. The principal also led to the pictorial VIF (valency interaction formula) methods for deducing qualitative quantum chemistry directly from the structural formulas of molecules. The present set of two papers (II on undirected graphs) develops graphs and graph rules for abstract linear vector spaces, bras, kets, and abstract operators as ket-bra dyads. Multiplications of such operators are carried out with graphs of two kinds of lines and two kinds of vertices. The theorems are demonstrated on some examples and are useful, e.g., with the recent method of moments and in deriving Lie algebras pertinent to quantum chemistry.     

On the decomposition of the unitary group and the multiple coupling of angular momenta

The decomposition of the direct product group, S_n ⊗ U_N, for a system of n identical particles having access to N one-particle states is considered from the point of view of its reduction to invariant subspaces. A double-factor matrix approach is developed in terms of the frequency of occurrence of the invariant subspaces and the frequency of configurations arising from the set of one-particle states. Simple formulas are obtained for both these frequencies. It is shown that this approach is especially useful when treating the problem of multiple coupling of the quantum numbers of the states. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 62 : 331–341, 1997     

Studies in perturbation theory XIII. Treatment of constants of motion in resolvent method,partitioning technique,and perturbation theory

After a brief survey of some basic concepts in the theory of linear spaces, the eigenvalue problem is formulated in the resolvent technique based on the introduction of a reference function φ and a complex variable ?. This leads to a series of fundamental concepts including the trial wave function, the inhomogeneous equation, and finally the transition and expectation values of the Hamiltonian, of which the former renders a “bracketing function” for the energy. In order to avoid the explicit limiting procedures in this approach, the eigenvalue problem is then reformulated in terms of the partitioning technique which, in turn, leads to a closed form of infinite-order perturbation theory. The eigenvalue problem is greatly simplified if the Hamiltonian H has a constant of motion Λ or has symmetry properties characterized by the group G = {g}, and the question is now how these simplifications can be incorporated into the partitioning technique and into perturbation theory. In both cases, there exists a set of projection operators {Q_k} which lead to a splitting of the Hilbert space into subspaces which have virtually nothing to do with each other. It is shown that, in the partitioning technique, it is sufficient to consider one of these subspaces at a time, and the results are then generalized to perturbation theory. It turns out that the finite-order expansions are no longer unique, and the commutation rules connecting the various forms are derived. The infinite-order results are finally presented in such a form that they are later suitable for the evaluation of upper and lower bounds to the energy eigenvalues.     

Usage de l'Algèbre de Lie su (n) dans l'Etude des Systèmes Quantiques à n Etats. II. Transformation de l'Espace des Observables,Problèmes d'Evolution

In the previous paper we examined, for a quantum system, the relation between its n-dimensional state space and the su (n) Lie algebra. The present paper is devoted to relations between unitary transformations in the state space and orthogonal transformations in Lie's algebra. Two cases can happen. First, the transformations are independently chosen in the two spaces; this amounts to changing the former relation. On the other hand, the relation is maintained and the unitary operators are then related to some of the orthogonal operators. This second case is used to study the evolution operators.     

Energy partitioning schemes

The paper gives an overview, generalization and systematization of the different energy decomposition schemes we have devised in the last few years by using both the 3-D analysis (the atoms are represented by different parts of the physical space) and the Hilbert space analysis in terms of the basis orbitals assigned to the individual atoms. The so called "atomic decomposition of identity" provides us the most general formalism for analyzing different physical quantities in terms of individual atoms or pairs of atoms. (The "atomic decomposition of identity" means that we present the identity operator as a sum of operators assigned to the individual atoms.) By proper definitions of the atomic operators, both Hilbert-space and the different 3-D decomposition schemes can be treated on an equal footing. Several different but closely related energy decomposition schemes have been proposed for the Hilbert space analysis. They differ by exact or approximate treatment of the three- and four-center integrals and by considering the kinetic energy as a part of the atomic Hamiltonian or as having genuine two-center components, too. (Also, some finite basis correction terms may be treated in different manners.) The exact schemes are obtained by using the "atomic decomposition of identity". In the approximate schemes a projective integral approximation is also introduced, thus the energy components contain only one- and two-center integrals. The diatomic energy contributions have also been decomposed into terms of different physical nature (electrostatic, exchange etc.) The 3-D analysis may be performed either in terms of disjunct atomic domains, as in the case of the AIM formalism, or by using the so called "fuzzy atoms" which do not have sharp boundaries but exhibit a continuous transition from one to another. The different schemes give different numbers, but each is capable of reflecting the most important intramolecular interactions as well as the secondary ones--e.g. intramolecular interactions of type C-H[...]O.     

Reduced hamiltonian orbitals. III. Unitarily invariant decomposition of hermitian operators

A decomposition of an N-particle operator as a sum of N + 1 components is defined such that, in the case of a model system employing a finite one-particle basis set, the decomposition is invariant under unitary transformations of the basis set. Applied to a two-particle Hamiltonian, this decomposition gives rise to the distinction between the independent-particle energy and the coupling energy defined in previous papers. Applied to the reduced density operator for a quantum state, the decomposition corresponds to partitioning the density into irreducible components. This partitioning is illustrated by graphs of electron density for the water molecule.     

Self-consistent,nonorthogonal group function approximation: An ab initio approach for modelling interacting fragments and environmental effects

The reformulation of the single determinantal, closed shell wavefunction into an antisymmetrized product of nonorthogonal group functions (NOGF) is reviewed. It is shown that by introducing the idea of a reciprocal group function, i.e. a group function defined as a product of reciprocal orbitals, the resulting expressions for one- and two-electron operators are formally identical with the equations obtained using strong-orthogonal group functions. Orbital equations are given for the NOGF wavefunction which are derived by formulating a variation principle in terms of group energy functionals, where the presence of the other groups is expressed in terms of Coulomb and exchange operators in the group's Hamiltonian, To ensure that the group's orbitals do not violate the Pauli exclusion principle, a coupling or screening operator is introduced into the variational equations. The effectiveness of the coupling operator is discussed and it is demonstrated that it fully screens the group's orbitals from collapsing or distorting into forbidden regions of function space. To provide techniques for modelling and analyzing intermolecular interactions, the procedure for calculating the NGOF wavefunction can be reformulated into a series of steps which allows the components of the interaction energy, i.e. Coulomb, exchange, polarization and charge transfer, to be evaluated. This approach leads to considerable simplification and reduces the computational effort required to determine the wavefunction. The decomposition is used to analyze many-body effects in linear water chains and a model of a helical hydrogen bonding. The basis set superposition error (BSSE) in the NOGF approximation is discussed and methods for its evaluation are given, and it is shown that the BSSE is inherently less in the NOGF wavefunction than in the corresponding HF-SCF wavefunction. In the final parts of the paper, additional methods are given which further reduce computation time when both interacting fragments and their immediate environment must be considered at the quantum chemical level. These techniques are then applied to a study of the effect of environment on ion pair formation and proton transfer. The results of these studies demonstrate the remarkably strong modulating effect of molecules hydrogen bonded to the interacting pair.This paper is dedicated to Dr. Bess-Gene Holt, whose untimely death was a sad reminder to the author that our knowledge is still far from complete. Dr. Holt was a close friend whose insights and philosophy served as a strong guide to the development of the author's moral and philosophical views during his graduate student career at Iowa State University, Ames, Iowa.     

Characteristic operators and unitarily invariant decomposition of hermitian operators

A set of characteristic operators {F} is proposed for performing the decomposition of p-particle Hermitian operators {D^p} to constitute irreducible components {D} of the unitary group D = FD^p, q = 0,1,2,…,p. For a deeper expolration of the properties of the characteristic operators, a few theorems are presented. As an illustration, the expected values for symmetric p-particle Hermitian operators are obtained as a number of terms having invariant group-theoretical meaning.     

Non-Hermitian Hydrogen atom

We have constructed a set of non-Hermitian operators that satisfy the commutation relations of the $SO(3)$ SO ( 3 ) -Lie algebra. Using these set of operators we have constructed a non-Hermitian Hamiltonian corresponding to the Hydrogen atom that includes a complex term but with the same spectra as in the Hermitian case. It is also found a non-Hermitian Runge–Lenz vector that represents a conserved quantity. In this way, we obtain a set of non-Hermitian operators that satisfy the commutation relations of the $SO(4)$ SO ( 4 ) -Lie algebra.     

Calculating multidimensional discrete variable representations from cubature formulas

Finding multidimensional nondirect product discrete variable representations (DVRs) of Hamiltonian operators is one of the long standing challenges in computational quantum mechanics. The concept of a "DVR set" was introduced as a general framework for treating this problem by R. G. Littlejohn, M. Cargo, T. Carrington, Jr., K. A. Mitchell, and B. Poirier (J. Chem. Phys. 2002, 116, 8691). We present a general solution of the problem of calculating multidimensional DVR sets whose points are those of a known cubature formula. As an illustration, we calculate several new nondirect product cubature DVRs on the plane and on the sphere with up to 110 points. We also discuss simple and potentially very useful finite basis representations (FBRs), based on general (nonproduct) cubatures. Connections are drawn to a novel view on cubature presented by I. Degani, J. Schiff, and D. J. Tannor (Num. Math. 2005, 101, 479), in which commuting extensions of coordinate matrices play a central role. Our construction of DVR sets answers a problem left unresolved in the latter paper, namely, the problem of interpreting as function spaces the vector spaces on which commuting extensions act.     

A simple representation of energy matrix elements in terms of symmetry‐invariant bases

When a system under consideration has some symmetry, usually its Hamiltonian space can be parallel partitioned into a set of subspaces, which is invariant under symmetry operations. The bases that span these invariant subspaces are also invariant under the symmetry operations, and they are the symmetry‐invariant bases. A standard methodology is available to construct a series of generator functions (GFs) and corresponding symmetry‐adapted basis (SAB) functions from these symmetry‐invariant bases. Elements of the factorized Hamiltonian and overlap matrix can be expressed in terms of these SAB functions, and their simple representations can be deduced in terms of GFs. The application of this method to the Heisenberg spin Hamiltonian is demonstrated. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2010     

Poly-α-ester degradation studies. II. Thermal degradation of poly(isopropylidene carboxylate)

The thermal degradation of poly(isopropylidene carboxylate) has been studied over the temperature range 200–800°C by using the kinetic and analytical techniques described in Part I of this series. Over a wide range of temperature, tetramethyl glycollide, acetone, carbon monoxide, and, to a lesser extent, methacrylic acid are observed when the products are rapidly swept away from the reaction zone. As decomposition temperature is increased, tetramethyl glycollide takes on the role of an intermediate, the more stable products carbon monoxide and acetone being formed from it. At the highest temperature examined, carbon monoxide begins to predominate, and its formation is accompanied by formation of small amounts of a carbonaceous residue. When the reaction products are allowed to remain in the reaction zone, which is possible at the lower end of the temperature range, methacrylic acid becomes the major product. This is interpreted as a dual decomposition route, involving the tetramethyl glycollide intermediate. Kinetic studies have shown that the decomposition is a first-order process and that the first-order rate constant k is related to temperature (T) by the expression: The results are interpreted in terms of an intramolecular ester interchange process involving tetramethyl glycollide as the primary decomposition product.     

A second quantized formulation of valence bond theory

Valence bond theory is formulated in terms of second quantized operators and is related to the theory of the unitary group of spin-free orbital transformations. The construction of Weyl basis states, the evaluation of matrix elements, and the application to a linked-diagram valence bond perturbation theory are all discussed.