An investigation of the reliability of the Galerkin-Petrov method

The Galerkin-Petrov method is applied to the determination of the ground state energy of the beryllium atom. The basis set of the coordinate subspace consists of correlated wavefunctions of the combined configuration-interaction-Hylleraas type. The basis set of the projective subspace is of the configuration-interaction type. The usefulness of a previously proposed way of characterization of pairs of subspaces has been further confirmed. Experience in constructing close pairs of subspaces gathered in the case of two-electron systems is used for the larger system. Two methods for constructing pairs of subspaces are used in the calculations.     

A characterization of pairs of subspaces for quantum chemical applications of the Galerkin–Petrov method

A method is proposed for characterizing pairs of finite dimensional subspaces of the Hilbert space. The method might be helpful for the proper selection of coordinate and projective spaces leading to reliable realizations of the Galerkin–Petrov method for use in quantum chemistry. In order to illustrate the method some numerical results are presented.     

On the Presumptive Similarity of Kohn–Sham and Brueckner Orbitals

For states of many-electron systems disclosing various degrees of quasidegeneracy, we have carried out comparative studies of Kohn–Sham orbitals (KSO) generated for several xc-potentials, Brueckner orbitals (BO) represented by the Brueckner-coupled cluster orbitals, and Hartree–Fock (HF) orbitals by means of criteria directly related to the orbital structure which are based on relative distance indices for various pairs of equidimensional subspaces defined by the KSO and BO basis sets. We have found that both for weak and strong quasidegeneracy there are systems for which the KSO–BO distances are larger than the BO–HF ones. For strongly quasidegenerate states it is found that the distance indices are the largest for hybrid potentials, and that the subspaces spanned by KSOs are closer to those spanned by HF orbitals than by BOs. Hence, our results do not support the recently formulated expectations concerning the similarity of Brueckner orbitals and Kohn–Sham orbitals, including those corresponding to purely local exchange-correlation potentials.     

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     

Complete TDA and RPA Calculations on the Electronic Transitions of Fullerene‐C60 in the CNDO/S and INDO/S Approximations*

Complete single‐excitation mixing calculations on the electronic transitions of the icosahedral C₆₀ molecule have been carried out with the Tamm–Dancoff approximation (TDA) and random‐phase approximation (RPA) schemes in the CNDO/S and INDO/S approximations. The complete space of 14,400 (1p–1h) pairs is partitioned into subspaces classified according to the irreducible representations of the Ih group. For this purpose, matrix representations of the group generators are obtained on a fixed set of basis functions and are used to construct the projection operators. Degenerate molecular orbitals in each energy level are symmetry‐adapted to these projection operators. Degenerate (1p–1h) pairs or singly excited configuration wave functions are similarly symmetrized. In addition, the Clebsch–Gordan coefficients are obtained and listed in an Appendix. The TDA and RPA equations are then solved for each irreducible representation separately. Both schemes with the projection operators and with the Clebsch–Gordan coefficients gave the same results as expected, indicating that the calculations were correctly done. The transition energies from the ground state 1¹A_g to low‐lying singlet and triplet excited states and the oscillator strengths for the allowed transitions (n¹T_1u–1¹A_g) are given in tables. A proper way to normalize is discussed for the eigenvectors of the RPA‐type matrix equation. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001     

Optimization of molecular orbitals: Easy parametrizations of little rotations suitable for functional iteration

The method of functional iteration applied to orbital optimization is proposed. The main problem, i.e., the most economical parametrization of the rotations of the molecular orbitals, is resolved. First and second derivatives of rotations with respect to the parameters are given. The fact that the orbital space can be decomposed in subspaces such that energy is invariant against transformations of these subspaces into themselves is explicitly taken into consideration.     

Geometry of density matrices. VI. Superoperators and unitary invariance

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

Estimating the chemical rank of three-way data arrays by a simple linear transform incorporating Monte Carlo simulation

Estimating an appropriate chemical rank of a three-way data array is very important to second-order calibration. In this paper, a simple linear transform incorporating Monte Carlo simulation approach (LTMC) to estimate the chemical rank of a three-way data array was suggested. The new method determines the chemical rank through performing a simple linear transform procedure on the original cube matrix to produce two subspaces by singular value decomposition. One of two subspaces is derived from the original three-way data array itself and the other is derived from a new three-way data array produced by the linear transformation of the original one. Projection technique incorporating the Monte Carlo approach acts as distinguishing criterion to choose the appropriate component number of the system. Simulated three-way trilinear data arrays with different noise types (homoscedastic and heteroscedastic), various noise level as well as high collinearity are used to illustrate the feasibility of the new method. The results have shown that the new method could yield accurate results with different conditions appended. The feasibility of the new method is also confirmed by two real arrays, HPLC-DAD data and excitation-emission fluorescent data. All the results are compared with the other three factor-determining methods: factor indicator function (IND), core consistency diagnostic (CORCONDIA) and two-mode subspace comparison (TMSC) approach. It shows that the newly proposed algorithm can objectively and quickly determine the chemical rank to fit the trilinear model.     

Model of elementary particles with substructures moving in positive and negative time directions

A new theory of elementary particles is developed with the assumption that all real and virtual particles are not point-like. The structure, the diameter, the real mass, the bare charge and the spin for the simplest leptons are determined. The volume of the elementary particle is composed of several three-dimensional curved volumes lying in a higher dimensional flat space, and the mass and the different charges are located in these different subspaces. The properties of the elementary particles can be explained by the zero point energy and by the curvatures of the subspaces. The fundamental difference between the gravitational and the electric field strength could be presented.     

基于化学子空间对线性变换稳定的秩估计方法

提出了一个新的化学计量学方法,即线性变换下稳定的化学子空间法,用于二维数据的化学秩估计.该方法的基本思想是具有化学意义的子空间对在一个方向上的二维数据的线性变换是最稳定的.据此,两个新的定量指标投影残差法和子空间夹角法被提出用于衡量两个子空间之间的差异性.对两个近红外数据的分析结果表明,这一方法为二维数据的化学秩估计提供了一个很有价值的工具.     

A classical versus quantum mechanics study of the \hbox{OH}\,+\,\hbox{CO} \rightarrow\,\hbox{H}\,+\,\hbox{CO}_2 (J?=?0) reaction

When appropriately used, the multiconfigurational self-consistent field (MCSCF) approximation is useful in discerning correct electronic structure results. However, with the increasing size of chemical systems of interest, MCSCF rapidly becomes unfeasible due to the requirement of larger active spaces, which lead to computationally unmanageable numbers of configurations. This situation is especially true for complete active space self-consistent field (CASSCF). In particular, reducing this computational expense by using restricted active spaces in solving for gradients and nonadiabatic couplings (NACs) during dynamics runs would save computer time. However, the validity of such restricted spaces is not well-known even for recovering the majority of the nondynamical correlation and inevitably varies between chemical systems across a range of nuclear geometries. As such, we use the recently implemented coupled perturbed–occupation restricted multiple active space (CP-ORMAS) equations (West et al., unpublished) to verify the accuracy of this approximation for gradients and NACs vectors around two specific conical intersection geometries for the silaethylene and butadiene systems. Overall, no excitations between appropriate subspaces show qualitatively reasonable results while single excitations significantly improve ORMAS results relative to the CASSCF level in these particular systems. However, single excitation schemes do not always lead to the correct orbital subspaces, and as a result, seemingly smooth potential energy surfaces (PES) do not always result in smooth analytical gradients and NACs. In addition, while some of the single excitation ORMAS and CASSCF schemes have improper orbitals rotate into the active space, the schemes without excitations (even with more subspaces) do not exhibit this behavior.     

Adiabatic perturbation theory for the degenerate case. I. Perturbation of isolated degenerate or quasidegenerate levels

Adiabatic formulae for secular operators and contracted Hamiltonians in an arbirary combination of degenerate or quasidegenerate subspaces are derived. A detailed consideration of the adiabatic limit in the power series is given, and “stability” of proper linear combinations with respect to a transformation produced by the S_α-matrix is proved.     

Quantitation of chromatographic peaks in the 0.1 to 1.0% range

Summary This report was conceived as a source for the relations between five pneumatic parameters of carrier gas-inlet and outlet pressure, inlet and outlet velocity, and average velocity. Any pair of these parameters can be independent while the remaining three can be expressed as functions of the independent pair. As the total number of the independent pairs is ten, thirty different functions describing relations between arbitrary independent pairs and dependent parameters can be identified. All thirty were derived below together with the complete set of bounds for variations of independent parameters. To derive some relations, a third order rational equation had to be solved. Some properties of that solution are discussed. A simple case of vacuum operations with zero outlet pressure has also been considered.     

Gaussian expansions from STOs by the distance between subspaces

Expansions of STO orbitals with GTO s for the first-row atoms have been obtained by the method of the distance between subspaces. The expansion coefficients and exponential parameters were simultaneously varied when the distance between subspaces, which are generated from STO and GTO functions, is minimized. The ζ; exponents (or scale factors) for the atomic orbitals that are optimized for these atoms are also shown to be almost independent of the number of Gaussian functions. Comparisons carried out with Stewart's least-squares method produce equivalent results when exponents for 2s and 2p functions are different. Some examples and applications for several atomic properties of the first-row atoms are included: energies and expectation values of rⁱ and pⁱ for the several expansions. These new minimal basis sets were tested for diatomic and polyatomic molecules containing these atoms. © 1993 John Wiley & Sons, Inc.     

A strategy to derive new internal coordinates by partitioning the internal configuration space according to invariance properties

A method to derive optimally orthogonal curvilinear coordinates for N-body systems is proposed. The invariance of certain subspaces under groups of linear transformations is employed to partition the configuration subspace into internal and external components. The construction is initially carried out locally by orthogonalizing typical group invariant vector fields. Integration is performed subsequently by means of integrating factors. Simple examples of orthogonal invariants illustrate the discussion.     

Simple constructive proof of Karadakov's extended pairing theorem

A simple proof is given for Karadakov's recent extension of the pairing theorem to the virtual subspaces. The proof consists of an explicit construction of the virtual orbitals with the required pairing properties, utilizing the pairing of the occupied orbitals.     

Picosecond kinetic study of the photoinduced homolysis of benzhydryl acetates: the nature of the conversion of radical pairs into ion pairs

The conversion of benzhydryl acetate geminate radical pairs to contact ion pairs following photoinduced homolysis in solution is studied using picosecond pump-probe spectroscopy. The dynamics for the decay of the geminate radical pairs into contact ion pairs is modeled within a Marcus-like theory for nonadiabatic electron transfer. A second decay channel for the geminate radical pairs is diffusional separation to free radicals. The kinetics of this latter process reveals an energy of interaction between the two radicals in the geminate pair.     

Diabatic representation for the excited states of the Na2 molecule: application to the associative ionization reaction between two excited sodium atoms

A two-electron model potential method is proposed to compute diabatic electronic excited states for Na₂. The configuration space is first divided into two subspaces corresponding to singly and doubly excited configurations respectively. Next this partition is modified to ensure a correct dissociation limit for the ground state. The matrix element of the electronic Hamiltonian between the two subspaces can be extrapolated along a Rydberg series up to the ionization continuum. The first order M.Q.D.T. treatment of Giusti (1980) is then used to estimate the cross-sections for the reaction Na(3p)+Na(3p)→Na ₂ ⁺ +e ^?, considering various symmetries of the intermediate Na₂ molecule. A marked selectivity in favour of the³Σ _u ⁺ symmetry is found and the estimated cross-section σ ～ 5 Ų for a collision energy of 0.05 eV is in satisfactory agreement with the experimental results.     

块定域波函数方法及其应用

提出了块定域波函数方法以定量分析分子内的电子定域现象或分子间的电荷传递效应。对于一个假想的严格定域的分子,我们通过将全部的电子和基轨道配分成几个子空间来构造其相应的波函数。其中每一个分子轨道只对某一个子空间展开,各子空间内的分子轨道相互正交,但不同子空间内的分子轨道间是非正交的。Hartree-Fock波函数和块定域波函数之间的能量之差即为分子内的电子定域能或分子间的电荷传递能。我们应用块定域波函数方法讨论了丁二烯分子中的旋转势垒。