On the choice of active space orbitals in MCSCF calculations

We describe a procedure which may be used to aid selection of the active space in multiconfigurational self-consistent field (MCSCF) calculations for general chemical systems. Starting from a restricted Hartree-Fock calculation, we define a hierarchy of interacting virtual orbitals for every occupied orbital. The most strongly interacting orbitals are then taken to constitute the active space in a configuration interaction (CI) calculation. The natural orbital occupation numbers obtained from the CI calculation are then used to choose the active space to be used in a subsequent MCSCF calculation. We illustrate our method on a number of systems (Li₂, B₂, C₂, carbonyl oxide and the transition state for oxidation of H₂S by dioxirane). In all these cases, ‘intuitive’ active spaces are inadequate, as are active spaces derived from the natural orbitals of unrestricted Hartree-Fock calculations.     

Geometrical energy derivative evaluation with MRCI wave functions

The theory of MCSCF and CI energy derivatives with respect to geometrical variations is briefly reviewed with special attention given to the MCSCF and MRCI energy gradients. A computational procedure is proposed for MRCI energy gradients that does not require the solution to any “coupled-perturbed MCSCF ” equations, it does not require any expensive direct-CI matrix-vector products involving derivative integrals, and it does not require any derivative integrals to be transformed from the AO basis to the MO basis. An additional feature is that it does not require any changes to existing MCSCF gradient evaluation programs in order to compute MRCI gradients. The only difference in the two cases is the exact nature of the data passed to the gradient evaluation program from the previous steps in the computational procedure. The additional effort required to compute the entire MRCI energy gradient vector is approximately that required for one additional iteration of the MRCI diagonalization procedure and for one additional MCSCF iteration. For large scale MRCI wave functions, the MRCI energy gradient evaluation should only require about 10% of the effort of computing the wave function itself. This computational procedure removes a major computational botleneck of potential energy surface evaluation.     

Virtual orbital transformation prior to configuration interaction with localized orbitals

A virtual orbital transformation is proposed involving pairing of localized occupied orbitals with virtual orbitals. The virtual orbitals are transformed so that each virtual orbital is as “close” as possible to its occupied counterpart, where closeness is the inverse of the particular definition of localization. The appropriate transformation is derived for the special case of Foster–Boys localization, and an illustrative CNDO /2 calculation on HNO is presented. INDO CI results on the series N₂, CO, BF indicate that use of this transformation may reduce the number of energetically significant configurations.     

The optimization of MCSCF functions by application of the generalized brillouin theorem: The LiH2 potential energy surface

The generalized Brillouin theorem is used to construct an optimization procedure for MCSCF functions by iterative contracted CI calculations. Special attention is paid to the MO transformation step in each iteration. In this method the MCSCF calculation may easily be augmented by a restricted CI calculation involving a configuration set which is uniquely determined by the trial function. An application to the calculation of the potential energy surface for linear LiH₂ in the reaction LiH + HLi + H₂ leads to the conclusion that this restricted CI is necessary, in order to obtain satisfactory results for the potential energy barrier in this reaction.     

Ab initio MRD-CI calculations for breaking a chemical bond in a molecule in a crystal or other solid environment. III. Me2N?NO2 decomposition of dimethylnitramine in a large crystalline environment

Recently we extended our strategy for MRD-CI (multireference double excitation-configuration interaction) calculations, based on localized/local orbitals and an “effective” CI Hamiltonian, for molecular decompositions of large molecules to breaking a chemical bond in a molecule in a crystalline or other solid environment. Our technique begins with an explicit quantum chemical SCF calculation for a reference molecule surrounded by a number of other molecules in the multipole environment of more distant neighbors. The resulting canonical molecular orbitals are then localized, and the localized occupied and virtual orbitals in the region of interest are included explicitly in the MRD-CI with the remainder of the occupied localized orbitals being folded into an “effective” CI Hamiltonian. The MRD-CI calculations are then carried out for breaking a bond in the reference molecule. This method is completely general in that the space treated explicitly, as well as the surrounding space, may contain voids, defects, deformations, dislocations, impurities, dopants, edges and surfaces, boundaries, etc. Dimethylnitramine is the smallest prototype of the energetic R₂N—NO₂ nitramines, such as the 6-member ring RDX or the 8-member ring HMX. Decomposition of energetic compounds is initiated in the solid by a breaking of the target bond. Thus, it is crucial to know the difference in energy between breaking a bond in an isolated energetic molecule versus in the molecule in a solid. In the present study, we have carried out MRD-CI calculations for the Me₂N—NO₂ dissociation of dimethylnitramine in a dimethylnitramine crystal. The cases we investigated were one dimethylnitramine molecule (surrounded by 53 and 685 neighboring dimethylnitramine molecules represented by multipoles), three dimethylnitramine molecules, and three dimethylnitramine molecules (surrounded by 683 neighbors). All multipoles were cumulative atomic multipoles up through quadrupoles. The MRD-CI calculations on dimethylnitramine required large numbers of reference configurations from which were allowed all single and double excitations.     

Application of many-body perturbation theory in the localized representation for the all-trans conjugated polyenes

Ab initio self-consistent field (SCF ) calculations are performed with the standard 6-31G* basis set for all-trans conjugated oligomers C_2n+2H_2n+4. The canonical occupied and virtual molecular orbitals (MO s) are separately localized by unitary transformations. Due to the localization, the perturbation operator is partitioned into two-particle and into single-particle terms; the MBPT is, therefore, a double-perturbation expansion in this case. By using the localized representation of the MBPT , the correlation energy contributions can be partitioned into local and nonlocal effects. It can be shown that the local effects are very important and well transferable, which makes possible the calculation of the correlation energy of larger molecules if the localized molecular orbitals (occupied and virtual) of smaller related molecules are known. © 1994 John Wiley & Sons, Inc.     

Extended Koopmans' theorem ionization potentials for beryllium atom shake-up transitions

The ionization potentials were calculated for Be using the extended Koopmans' theorem (EKT ) using several full configuration interaction (CI ) and multiconfigurational-self-consistent-field (MCSCF ) wave functions as reference wave functions. The wave functions used account for 89.7–96.7% of the correlation energy. Comparisons are made with experimental values and with δCI values calculated as the difference in energy obtained from CI wave functions for Be and Be⁺. The best EKT IP differed from the δCI value by 0.0003 eV for the lowest IP and by 0.0006 eV for ionization into the lowest ²P state of Be⁺. A calculation of ionization into the second ²P state of Be⁺ requires diffuse orbitals that are unimportant in the wave function for the ground state of Be. This results in small natural orbital occupation numbers for natural orbitals needed in the EKT calculation. © 1994 John Wiley & Sons, Inc.     

大体系多电子相关研究中应用群对称定域轨道的构想

大体系多电子相关研究中应用群对称定域轨道的构想周泰锦，刘爱民（厦门大学化学系，厦门３６１００５）关键词：组态相关，多构型自治叠代，多中心积分，群对称定域轨道，对称约化有关原子簇化合物及化学吸附、过渡态、激发态、催化反应等大体系的量子化学研究，对于探讨...     

SCF theory of intermolecular interactions without basis set superposition error

Special SCF LCAO MO type equations are derived, permitting “supermolecule” calculations for intermolecular interactions, excluding basis set superposition error (BSSE) from the beginning on the basis of the “chemical Hamiltonian approach”. (No additional “monomer” calculations are necessary to correct for BSSE.) The formalism excluding the BSSE results in a non-Hermitean Fock matrix; an algorithm is proposed to obtain the required molecular orbitals, in which no integral transformation is needed.     

One electron orbitals intrinsic to the reduced hamiltonian

One electron orbitals are determined from the reduced hamiltonian by a simple one-step diagonalization. These reduced hamiltonian orbitals (RHO's) are uniquely determined and virtual orbitals obtained in this procedure are on a par with filled orbitals. These RHO's appear well suited for CI calculations. Minimum basis set calculations are presented for H₂O and compared with similar SCF studies.     

Approximate MCSCF optimization for multiconfigurational spin‐tensor electron propagator method (MCSTEP): The vertical ionization potentials of CO,HCN, HNC,H2CO,and O3

The multiconfigurational spin tensor electron propagator method (MCSTEP) was developed as an implementation of electron propagator/single particle Green's function methods. MCSTEP was specifically designed for open shell and highly correlated (nondynamically correlated) initial states. The initial state used in MCSTEP is typically a small complete active space (CAS) with multiconfigurational self‐consistent field (MCSCF) state. In some cases, because of our use of a small CAS in MCSTEP, the Lagrangian eigenvalues of the MCSCF reference state are in an undesired order (u). The desired order (d) can usually be obtained by excluding one or more orbital rotations in MCSCF optimization between the doubly occupied and partially occupied orbitals. We systematically examine several cases where the undesired order occurs for the low‐lying vertical MCSTEP ionization potentials (IPs) of the molecules CO, HCN, HNC, H₂CO, and O₃ with our recently established CAS choices for MCSCF/MCSTEP. By excluding one or more orbital rotations between the partially and doubly occupied orbitals, an approximate MCSCF reference state with the same CAS choice is obtained for use in standard MCSTEP calculations that, in general, gives more reliable vertical MCSTEP IPs. © 2007 Wiley Periodicals, Inc. J Quantum Chem, 2008     

A theoretical introduction to “Combinatory SYBR®Green qPCR Screening”, a matrix-based approach for the detection of materials derived from genetically modified plants

The detection of genetically modified (GM) materials in food and feed products is a complex multi-step analytical process invoking screening, identification, and often quantification of the genetically modified organisms (GMO) present in a sample. “Combinatory qPCR SYBR®Green screening” (CoSYPS) is a matrix-based approach for determining the presence of GM plant materials in products. The CoSYPS decision-support system (DSS) interprets the analytical results of SYBR®GREEN qPCR analysis based on four values: the C _t- and T _m values and the LOD and LOQ for each method. A theoretical explanation of the different concepts applied in CoSYPS analysis is given (GMO Universe, “Prime number tracing”, matrix/combinatory approach) and documented using the RoundUp Ready soy GTS40-3-2 as an example. By applying a limited set of SYBR®GREEN qPCR methods and through application of a newly developed “prime number”-based algorithm, the nature of subsets of corresponding GMO in a sample can be determined. Together, these analyses provide guidance for semi-quantitative estimation of GMO presence in a food and feed product.     

Ab-Initio MRD-CI calculations for breaking a chemical bond in a molecule in a crystal or other solid environment. II.H3C—NO2 decomposition of nitromethane in a nitromethane crystal with voids

Recently we extended our strategy for MRD-CI (multireference double excitation-configuration interaction) calculations based on localized/local orbitals and an “effective” CI Hamiltonian for molecular decompositions of large molecules to breaking a chemical bond in a molecule in a crystal or other solid environment. Our technique involves solving a quantum chemical ab-initio SCF explicitly for a system of a reference molecule surrounded by a number of other molecules in the multipole environment of more distant neighbors. The resulting canonical molecular orbitals are then localized and the localized occupied and virtual orbitals in the region of interest are included explicitly in the MRD-CI with the remainder of the occupied localized orbitals being folded into an “effective” CI Hamiltonian. The MRD-CI calculations are carried out for breaking a bond in the reference molecule. This method is completely general. The space treated explicitly quantum chemically and the surrounding space can have voids, defects, deformations, dislocations, impurities, dopants, edges and surfaces, boundaries, etc. We previously applied this procedure successfully to the H₃C? NO₂ bond dissociation of nitromethane in a nitromethane crystal with extensive testing of the number of molecules that have to be included explicitly in the SCF and how many molecules have to be represented by more distant multipoles. The results indicated that it took more energy to dissociate the H₃C? NO₂ bond when the nitromethane molecule was in the crystal than it did to dissociate that bond in the free nitromethane molecule. In this present study we have investigated the effect of voids (both in the nitromethane molecules treated explicitly in the SCF and those in the environment represented by multipoles) on the calculated H₃C? NO₂ bond dissociation energies.     

A discussion of the bonding in ferrocene from an ab initio LCAO MO SCF calculation

The nature of the bonding in ferrocene is discussed on the basis of an ab initio LCAO MO SCF calculation, with the emphasis put on a number of controversial points such as: i) the relative ordering of the orbitals e_2g(3d) and a_1g(3d); ii) the relative ordering of the 3_1u and e_1g ligand π orbitals; and iii) the degree of participation of the metal 4s and 4p orbitals to the bonding.     

Calculation of energies of excited states using MNDO

The energies of the lowest singlet (S₁) and triplet (T₁) states of 28 molecules have been calculated by the “half-electron” (MNDO -HE ) and spin-unrestricted (UMNDO ) versions of MNDO . While most of the calculated values are too negative, because of overestimation of the correlation energy in MNDO -HE and UMNDO , the errors are systematic and depend in an understandable way on the nature of the molecular orbitals (MO S) involved. When appropriate corrections are applied, the calculated energies agree with experiment almost as well as they do for ground states. This justifies the use of MNDO -HE or UMNDO for studies of excited state processes.     

Molecular symmetry. IV. The coupled perturbed Hartree–Fock method

Symmetry methods employed in the ab initio polyatomic program HONDO are extended to the coupled perturbed Hartree–Fock (CPHF) formalism, a key step in the analytical computation of energy first derivatives for configuration interaction (CI) wavefunctions, and energy second derivatives for Hartree–Fock (HF) wavefunctions. One possible computational strategy is to construct Fock-like matrices for each nuclear coordinate in which the one- and two-electron integrals of the usual Fock matrix are replaced by the integral first derivatives. “Skeleton” matrices are constructed from the unique blocks of electron-repulsion integral derivatives. The correct matrices are generated by applying a symmetrization operator. The analysis is valid for many wavefunctions, including closed- or open-shell spin-restricted and spin-unrestricted HF wavefunctions. To illustrate the method, we compare the computer time required for setting up the coupled perturbed HF equations for eclipsed ethane using D_3h symmetry point group and various subgroups of D_3h. Computational times are roughly inversely proportional to the order of the point group.     

Explaining the “large numbers” via a unified (classical) theory of strong and gravitational interactions

An advantage of modified virtual orbitals of Hartree–Fock method is discussed in the calculation of the second-order perturbation energy. All the modified virtual orbitals can be fitted for the intermediate virtual states in the perturbation expansion, only if the molecular orbitals are expanded in terms of infinite basis functions and the set of molecular orbitals is infinite and complete. If the molecular orbitals are expanded in terms of finite basis functions, only the modified virtual orbitals with lower energies are appropriate to describe the intermediate virtual states, but the modified virtual orbitals with higher energies become inadequate. To explain the usefulness of the modified virtual orbitals, the calculation by the modified Hartree–Fock method without CI are compared with the calculation by the traditional Hartree–Fock method with complete CI .     

Why is there a molecular relativistic effect?

The almost exclusive association of the molecular geometry dependence of the relativistic correction with the valence orbital contribution to the mass-velocity and Darwin terms is investigated using SCF and MCSCF wavefunctions. The requirement of orthogonality of the valence orbitals to the core orbitals is confirmed to be the mechanism responsible for the increase in (the absolute value of) relativistic energy upon decrease of the internuclear distance. Certain “fingerprint”-type features of the valence relativistic correction, revealing the identity of the particular core orbital giving rise to it, are identified.