Direct MRCI method for the calculation of relativistic many-electron wavefunctions. I. General formalism

The formalism of a quasi- or full-relativistic multireference CI method has been developed and implemented. The scheme is appropriate for the calculation of molecular systems in which the relativistic effects are of the same order of magnitude as the correlation contributions. In this contribution some important symmetry aspects of a relativistic many-electron wave function are discussed and the consequences for the CI matrix structure are shown. An efficient CI strategy in the form of a direct CI is presented, which avoids the construction of the whole CI matrix. Based on a determinantal expansion of molecular spinor products, the individual one- and two-electron molecular integrals are processed, and the molecular symmetry is easily accounted for by a proper linear combination of Slater determinants in the CI starting vector. For an efficient CI organization some powerful techniques of the graphical unitary group approach have been transferred to the relativistic case.     

On the SCF calculation of excited states: Singlet states in the two-electron problem

The problem of determining SCF wave functions for excited electronic states is examined for singlet states of two-electron systems using a Lowdin natural orbital transformation of the full CI wave function. This analysis facilitates the comparison of various SCF methods with one another. The distribution of the full CI states among the natural orbital MCSCF states is obtained for the S states of helium using a modest Gaussian basis set. For SCF methods that are not equivalent to the full CI wave functions, it is shown that the Hartree-Fock plus all single excitation wave functions are equivalent to that of Hartree-Fock plus one single excitation. It is further shown that these wave functions are equivalent to the perfect pair or TCSCF wave functions in which the CI expansion coefficients are restricted to have opposite signs. The case of the natural orbital MCSCF wave function for two orbitals is examined in greater detail. It is shown that the first excited state must always be found on the lower natural orbital MCSCF CI root, thus precluding the use of the Hylleras-Undeim-MacDonald (HUM) theorem in locating this state. It is finally demonstrated that the solution obtained by applying the HUM theorem (minimizing the upper MCSCF CI root with respect to orbital mixing parameters) is an artifact of the MCSCF method and does not correspond to any of the full CI states.     

Analytic expression of the second derivatives of electronic energy for full configuration interaction wave functions

Summary A new analytic second derivative expression of the electronic energy is derived for full configuration interaction (CI) wave functions. This formula is shown to be free from the derivative terms of both CI and MO coefficients. The second-order relationships between CI and MO coefficients for full CI wave functions are also presented.     

Exploiting regularity in systematic sequences of wavefunctions which approach the full CI limit

Summary The ground state total energy and related 1-electron properties are computed for three small molecules (N₂, H₂O, and H₂CN) using several systematic sequences of wavefunctions which approach the full CI. These sequences include multireference CI, averaged coupled pair functional and quasidegenerate variational perturbation theory wavefunctions. It is demonstrated that sufficient regularity exists in the sequence of variationally computed energies to permit extrapolation to the full CI limit using simple analytic expressions. It is furthermore demonstrated that a subset of the original list of configurations employed in the normal singles and doubles CI procedure can be selected using second order perturbation theory without adversely affecting the extrapolation to the full CI limit. This significantly broadens the range of applicability of the method. Along these lines, a scheme is proposed for the extrapolation of the selected CI results to the zero threshold (i.e. unselected) values in cases where the numbers of configurations associated with the latter would render the calculations intractable. Due to the vast reduction in the number of configurations which are handled variationally, the proposed scheme makes it possible to derive estimates of the full CI limit in cases where explicit full CI is either very difficult or currently impossible.Dedicated to Prof. Klaus RuedenbergThe Pacific Northwest Laboratory is operated for the U.S. Department of Energy by Battelle Memorial Institute under contract DE-AC06-76RLO 1830     

A new direct CI method for large CI expansions in a small orbital space

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.     

Role of the configuration interaction in the CNDO/S calculation of optical rotatory strengths

The rotatory strengths calculated directly by the CNDO/S method exhibit a pronounced dependence on the size of configuration interaction (CI). In order to elucidate the role of highly excited configurations in such calculations the perturbation theory is employed. It is shown that the restriction of the CI size to 20–40 may be quite inadequate in some cases. The calculations of rotatory strengths of several optically active molecules containing carbonyl and amide chromophores has shown that the best results can be obtained for half of full CI but sometimes it is possible to restrict the CI size to 100 configurations. The agreement with experiment for all molecules considered is satisfactory.     

Cluster chemical ionization for improved confidence level in sample identification by gas chromatography/mass spectrometry

Upon the supersonic expansion of helium mixed with vapor from an organic solvent (e.g. methanol), various clusters of the solvent with the sample molecules can be formed. As a result of 70 eV electron ionization of these clusters, cluster chemical ionization (cluster CI) mass spectra are obtained. These spectra are characterized by the combination of EI mass spectra of vibrationally cold molecules in the supersonic molecular beam (cold EI) with CI-like appearance of abundant protonated molecules, together with satellite peaks corresponding to protonated or non-protonated clusters of sample compounds with 1-3 solvent molecules. Like CI, cluster CI preferably occurs for polar compounds with high proton affinity. However, in contrast to conventional CI, for non-polar compounds or those with reduced proton affinity the cluster CI mass spectrum converges to that of cold EI. The appearance of a protonated molecule and its solvent cluster peaks, plus the lack of protonation and cluster satellites for prominent EI fragments, enable the unambiguous identification of the molecular ion. In turn, the insertion of the proper molecular ion into the NIST library search of the cold EI mass spectra eliminates those candidates with incorrect molecular mass and thus significantly increases the confidence level in sample identification. Furthermore, molecular mass identification is of prime importance for the analysis of unknown compounds that are absent in the library. Examples are given with emphasis on the cluster CI analysis of carbamate pesticides, high explosives and unknown samples, to demonstrate the usefulness of Supersonic GC/MS (GC/MS with supersonic molecular beam) in the analysis of these thermally labile compounds. Cluster CI is shown to be a practical ionization method, due to its ease-of-use and fast instrumental conversion between EI and cluster CI, which involves the opening of only one valve located at the make-up gas path. The ease-of-use of cluster CI is analogous to that of liquid CI in ion traps with internal ionization, and is in marked contrast to that of CI with most other standard GC/MS systems that require a change of the ion source.     

Asymptotic multipolar expansion of collision-induced properties

The collision-induced (CI) dipole moment, CI polarizability, and CI hyperpolarizability are considered for three H(2)-rare gas (Rg) pairs (Rg = He, Ne, Ar). In this study, the symmetry-adapted (SA) components, the projection of the CI dipole, polarizability, and hyperpolarizability on an appropriately tailored set of spherical harmonics are calculated. A set of equations for the respective SA components is derived. The Cartesian components of the CI properties calculated by quantum chemistry methods for three intermolecular geometries are used in our calculations as input data. The analytical, multipolar long-range behavior of the CI properties studied is considered within a multipole-induced multipole model. Taking the SA components at large distances, the ab initio SA numerical results and the model semianalytical data were compared. In general, a good agreement has been found. The results of our study are expected to be of value in spectral line shape analysis and in modeling of processes in the Earth's and planetary atmospheres.     

Ab initio CI calculations on benzene with an extended basis set

An extended basis set of triple zeta plus polarization quality is employed to carry out configuration interaction (CI ) calculations of the three lowest singlet and triplet excited states of benzene. The CI calculation is carried out by taking into account single and double excitations of π and σ electrons. In the CI , composite natural orbitals (CNO s), which are constructed from the natural orbitals of the ground state of ethylene, are used as virtual orbitals. The aim of using CNOs is to reduce the number of virtual orbitals to be used in constructing configuration-state functions, thus cutting down CI dimensions without losing reasonable accuracy. The excitation energies resulting from the CI are in fairly good agreement with experiment. The root mean square of the deviation is 0.22 eV for the six calculated energies and the largest disagreement is 0.37 eV for the third singlet excited state. To obtain better excitation energies by an ab initio calculation, it seems likely that we need to take into account more electron correlation than in the present calculation. © 1994 John Wiley & Sons, Inc.     

Computational intelligence techniques in bioinformatics

Computational intelligence (CI) is a well-established paradigm with current systems having many of the characteristics of biological computers and capable of performing a variety of tasks that are difficult to do using conventional techniques. It is a methodology involving adaptive mechanisms and/or an ability to learn that facilitate intelligent behavior in complex and changing environments, such that the system is perceived to possess one or more attributes of reason, such as generalization, discovery, association and abstraction. The objective of this article is to present to the CI and bioinformatics research communities some of the state-of-the-art in CI applications to bioinformatics and motivate research in new trend-setting directions. In this article, we present an overview of the CI techniques in bioinformatics. We will show how CI techniques including neural networks, restricted Boltzmann machine, deep belief network, fuzzy logic, rough sets, evolutionary algorithms (EA), genetic algorithms (GA), swarm intelligence, artificial immune systems and support vector machines, could be successfully employed to tackle various problems such as gene expression clustering and classification, protein sequence classification, gene selection, DNA fragment assembly, multiple sequence alignment, and protein function prediction and its structure. We discuss some representative methods to provide inspiring examples to illustrate how CI can be utilized to address these problems and how bioinformatics data can be characterized by CI. Challenges to be addressed and future directions of research are also presented and an extensive bibliography is included.     

SMAI和CI法制备的Ni—Ag／SiO2催化剂的结构与催化性质研究

应用溶剂化金属原子浸渍（ＳＭＡＩ）法和普通浸渍（ＣＩ）法制备了金属含量相同的ＳｉＯ２负载Ｎｉ－Ａｇ双金属催化剂。ＸＲＤ和磁测定结果表明ＳＭＡＩ催化剂中Ｎｉ和Ａｇ的粒度均小于金属含量相同的ＣＩ催化剂，ＳＭＡＩ催化剂中Ｎｉ和Ａｇ未形成合金，而ＣＩ催化剂中Ｎｉ和Ａｇ形成了合金。ＳＭＡＩ和ＣＩ催化剂都具有超顺磁性。研究了这些催化剂在甲苯加氢反应的催化性质，结果表明与组成相同的普通浸渍法催化相比，ＳＭＡＩ催     

Elastic scattering of low-energy electrons by N2 including the effect of target electronic correlation

A procedure to use configuration-interaction (CI) target wave-functions in the electron–molecule collision theory is applied to study the elastic e⁻–N₂ scattering in the (5–20) eV incident energy range. Correlated static and exchange contributions to the interaction potential are presented. Two different atomic basis sets are used. Differential cross sections (DCS) obtained by using Hartree–Fock or CI wave-functions are presented and compared. In the CI case, single and double, and single, double and triple excitations are considered. The effect of electron correlation is analyzed in all the cases. The continuum wave-functions were obtained via the Schwinger variational iterative method. The influence on the DCS of both the size of the atomic basis set and the inclusion of higher-order excitations in the CI calculation is discussed.     

The effect of inner-shell polarization on diagonal stretching force constants. Some large-basis SCF and CI results

Hartree-Fock force constant calculations employing the 6–311G^** basis augmented with (optimized) inner-shell (short-range) polarization functions are presented for OH⁻, HF, and HCN. A significant lowering of the quadratic stretching force constant is observed at the SCF level. Also presented are determinations of the quadratic stretching force constant of HF and the CN stretching force constant in HCN at the TZ+2P CI level of theory in which one of the sets of polarization functions has been “optimized” to account for inner-shell or short-range polarization. The hydrogen fluoride CI quadratic stretching force constant agrees with the experimentally determined value to within one third of one percent. The CI hydrogen cyanide CN stretching force constant is much improved over the TZ+P CI value, but the improvement over the TZ+2P CI SD value of Pulay et al. is only modest.     

Analysis of coupled cluster methods

Summary A procedure is developed that leads from CI to size-extensive CI (ECI) by stepwise cancellation of disconnected terms in the CI equations. The ECI methods thus obtained are identical with the corresponding coupled cluster (CC) methods with the exception of CISD and CISDT, which convert to size-extensive quadratic CI (QCISD) and ECISDT. The latter method has similar properties as CCSDT, but does not offer any significant time-savings as compared to CCSDT. Therefore, the idea of extending CI methods to size-extensive CI methods does not lead to a hierarchy of independent CC methods. However, the procedure of obtaining ECI methods lays the basis for deriving QCI methods that are truly size-extensive. This is accomplished by (a) deleting the first linear term of thep-fold CI excitation equations (p 3) since this term always represents a disconnected term and (b) including just the connected part of appropriate quadratic correction terms in all but the energy equation. In this way, size-extensive QCISDT and QCISDTQ are obtained and their properties are discussed in comparison with QCISD(T) and CCSDT.This paper is dedicated to Prof. Werner Kutzelnigg on the occasion of his sixtieth birthday     

Numerical Convergence of the Sinc Discrete Variable Representation for Solving Molecular Vibrational States with a Conical Intersection in Adiabatic Representation

Within the Born-Oppenheimer (BO) approximation, nuclear motions of a molecule are often envisioned to occur on an adiabatic potential energy surface (PES). However, this single PES picture should be reconsidered if a conical intersection (CI) is present, although the energy is well below the CI. The presence of the CI results in two additional terms in the nuclear Hamiltonian in the adiabatic presentation, i.e., the diagonal BO correction (DBOC) and the geometric phase (GP), which are divergent at the CI. At the same time, there are cusps in the adiabatic PESs. Thus usually it is regarded that there is numerical difficulty in a quantum dynamics calculation for treating CI in the adiabatic representation. A popular numerical method in nuclear quantum dynamics calculations is the Sinc discrete variable representation (DVR) method. We examine the numerical accuracy of the Sinc DVR method for solving the Schr?dinger equation of a two dimensional model of two electronic states with a CI in both the adiabatic and diabatic representation. The results suggest that the Sinc DVR method is capable of giving reliable results in the adiabatic representation with usual density of the grid points, without special treatment of the divergence of the DBOC and the GP. The numerical uncertainty is not worse than that after the introduction of an arbitrary vector potential for accounting the GP, whose accurate form usually is not easy to obtain.     

Generic implementation of semi-analytical CI gradients for NDDO-type methods

Generic semi-analytical energy gradients are derived and implemented for NDDO-type methods, by using numerical integral and Fock matrix derivatives in the context of an otherwise analytical approach for configuration interaction (CI) and other non-variational treatments. The correctness, numerical precision, and performance of this hybrid approach are established through comparisons with fully numerical and fully analytical calculations. The semi-analytical evaluation of the CI gradient is generally much faster than the fully numerical computation, but somewhat slower than a fully analytical calculation, which however shows the same scaling behavior. It is the method of choice whenever a fully analytical CI gradient is not available due to the lack of analytical integral derivatives. The implementation is generic in the sense that it can easily be extended to any new NDDO-type Hamiltonian. The present development of a semi-analytical CI gradient will facilitate studies of electronically excited states with recently proposed NDDO methods that include orthogonalization corrections. Dedicated to Professor Karl Jug on the occasion of his 65th birthday     

Matrix‐covariant representation of high‐order configuration interaction and coupled cluster theories

We present the closed form of the reduced density matrices (RDMs) of arbitrary order for configuration interaction (CI) wave functions at any excitation level, up to the full CI. A special operator technique due to Bogoliubov is applied and extended. It focuses on constructions of matrix‐covariant expressions independent of the basis set used. The corresponding variational CI equations are given in an explicit form containing the matrices related to conventional excitation operators. A subsequent transformation of the latter to an irreducible form makes it possible to generate the matrix‐covariant representation for coupled cluster (CC) models. Here this transformation is performed for a simplified high‐order CC scheme somewhat reminiscent of the quadratic CI model. A generalized spin‐flip approximation closely related to high‐order CI and CC models is presented, stressing on a possible inclusion of nondynamical and dynamical correlation effects for multiple bond breaking. A derivation of the full CI and simple CC models for systems involving effective three‐electron interactions is also given, thereby demonstrating the capability of the proposed method to deal with complicated many‐body problem. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Probing the Effects of Reagent Gas Pressure and Ion Source Temperature for Dimethyl Ether Chemical Ionization of Tricyclic Antidepressants in an External Source Ion Trap Mass Spectrometer

This study examines the reagent gas pressure and ion source temperature dependence for dimethyl ether chemical ionization (DME CI) mass spectra recorded with an external source ion trap mass spectrometer (ITMS). Information for better controls of the reagent gas pressure in order to obtain fair CI spectra is provided. The origin of M^+? ions observed in DME CIMS is discussed in detail. Furthermore, the ion source temperature effect on the DME CI is also investigated.     

SCF MO CI perturbation calculations of inner shell and valence shell vertical ionization potentials

A CI method for calculating inner and valence shell vertical ionization potentials is presented. It is based on ab initio SCF MO calculations for the neutral closedshell ground state followed by CI perturbation calculations for the ground and ion states including all spin and symmetry adapted singly and doubly excited configurations with respect to the main configurations of the state of interest. The state energy is computed by performing a CI calculation for a set of selected configurations, and then adding the contributions of the remaining configurations as estimated by second order Brillouin-Wigner perturbation theory. The use of the same set of MO's for all states together with the CI perturbation method makes the method rather rapid. The numerical results are, in spite of the limited Gaussian basis sets used, in good agreement with experiment.