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     

Modification of optimal complete active space choices for the multiconfigurational spin‐tensor electron propagator method for ionization potentials

The multiconfigurational spin tensor electron propagator (MCSTEP) method 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. Ionization or electron attachment is always from a state of pure spin symmetry to a state of pure spin symmetry even if the initial state is open shell. MCSTEP can be used as well for molecules with initial states that can be accurately described by a single determinant‐based theory. The initial state that is used in MCSTEP is typically a small complete active space (CAS) multiconfigurational self‐consistent field (MCSCF) state. We previously examined different small CAS choices for MCSTEP initial states and have developed a generally workable scheme. This article further examines some different ways to choose the CAS for MCSTEP. With several logical CAS choices, we have calculated the low‐lying vertical MCSTEP ionization potentials (IPs) of C₂, N₂, linear H₂O, O₂, CH₂, and NH₂, comparing them with large multireference configuration interaction (MRCI) calculations. We conclude that generally a small modification and extension of our previous schemes for choosing the MCSTEP CAS gives IPs that most effectively mimics the results of large scale MRCI IPs in general. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

The multiconfigurational spin tensor electron propagator method (MCSTEP): Comparison with extended Koopmans' theorem results

Summary We applied the multiconfigurational spin tensor electron propagator method (MCSTEP) for determining the lowest few (in energy) vertical ionization potentials (IPs) of HF, H₂O, NH₃, CH₄, N₂, CO, HNC, HCN, C₂H₂, H₂CO, and B₂H₆. We chose these molecules so that we could compare MCSTEP IPs with recently reported extended Koopmans' theorem (EKT) IPs on the same molecules. Using standard Dunning core-valence basis sets with relatively small complete active spaces, MCSTEP results are in very good to excellent agreement with experiment. These MCSTEP IPs are obtained using matrices no larger than 400 × 400. EKT matrices are even smaller; however, to obtain similar but generally slightly worse agreement with experiment, fairly large active spaces are required with EKT.     

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.     

Obtaining positions and widths of scattering resonances from a complex multiconfigurational self‐consistent field state using the M1 method

We present a complex multiconfigurational self‐consistent field (CMCSCF)‐based approach to investigate electron‐atom scattering resonances. It is made possible by the use of second quantization algebra adapted for biorthogonal spin orbitals, which has been applied to develop a quadratically convergent CMCSCF method. To control the convergence to the correct CMCSCF stationary point, a modified step‐length control algorithm is introduced. Convergence to a tolerance of 1.0 × 10^?10 a.u. for the energy gradient is found to be typically within 10 iterations or less. A method involving the first block of the M matrix defined in the multiconfigurational spin tensor electron propagator method (MCSTEP) based on the CMCSCF reference state has been implemented to investigate ²P Be^? shape resonances. The position and width of these resonances have been calculated for different complete active space choices. The wide distribution of the position and width of the resonance reported in the literature is explained by the existence of two distinct resonances which are close in energy. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

Performance of dynamically weighted multiconfiguration self-consistent field and spin-orbit coupling calculations of diatomic molecules of Group 14 elements

The efficacy of several multiconfiguration self-consistent field (MCSCF) methods in the subsequent spin-orbit coupling calculations was studied. Three MCSCF schemes to generate molecular orbitals were analyzed: state-specific, state-averaged, and dynamically weighted MCSCF. With Sn(2)(+) as the representative case, we show that the state-specific MCSCF orbitals lead to discontinuities in potential energy curves when avoided crossings of electronic states occur; this problem can be solved using the state-averaged or dynamically weighted MCSCF orbitals. The latter two schemes are found to give similar results when dynamic electron correlation is considered, which we calculated at the level of multiconfigurational quasidegenerate perturbation theory (MCQDPT). We employed the recently developed Douglas-Kroll spin-orbit adapted model core potential, ZFK3-DK3, and the dynamically weighted MCSCF scheme to calculate the spectroscopic constants of the mono-hydrides and compared them to the results obtained using the older set of potentials, MCP-TZP. We also showed that the MCQDPT tends to underestimate the dissociation energies of the hydrides and discussed to what extent coupled-cluster theory can be used to improve results.     

Accurate calculation of core-electron binding energies: multireference perturbation treatment

Multireference perturbation theory (MRPT) with multiconfigurational self-consistent field (MCSCF) reference functions is applied to the calculations of core-electron binding energies (CEBEs) of atoms and molecules. Orbital relaxations in a core-ionized state and electron correlation are both taken into account in a conventional MCSCF-MRPT procedure. In the MCSCF calculation, the target core ionized state is directly optimized as an excited state and this treatment can completely prevent a variational collapse. Multireference Moller-Plesset perturbation theory and multiconfigurational self-consistent field reference quasidegenerated perturbation theory were used to treat electron correlation. The present method quite accurately reproduced the 1s CEBEs of CH4, NH3, H2O, and FH; the average deviation from the experimental data is 0.11 eV using Ahlrichs' VTZ basis set. The C 1s and O 1s CEBEs of formic acid and acetic acid were calculated and the results are consistent with the bonding characters of the atoms in these molecules. The present procedure can also be applied to CEBEs of higher angular momentum orbitals by including spin-orbit coupling. The calculated CEBEs of Ar 2p, HCl 2p, Kr 3d, and HBr 3d are in reasonable agreement with the available experimental values. In the calculation of the 3d CEBEs, a relativistic correction significantly improves the agreements. The effect of polarization functions is also discussed.     

Symmetry adaptation of configuration basis in MCSCF method

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.     

Algebraic modifications to second quantization for non‐Hermitian complex scaled hamiltonians with application to a quadratically convergent multiconfigurational self‐consistent field method

The algebraic structure for creation and annihilation operators defined on orthogonal orbitals is generalized to permit easy development of bound‐state techniques involving the use of non‐Hermitian Hamiltonians arising from the use of complex‐scaling or complex‐absorbing potentials in the treatment of electron scattering resonances. These extensions are made possible by an orthogonal transformation of complex biorthogonal orbitals and states as opposed to the customary unitary transformation of real orthogonal orbitals and states and preserve all other formal and numerical simplicities of existing bound‐state methods. The ease of application is demonstrated by deriving the modified equations for implementation of a quadratically convergent multiconfigurational self‐consistent field (MCSCF) method for complex‐scaled Hamiltonians but the generalizations are equally applicable for the extension of other techniques such as single and multireference coupled cluster (CC) and many‐body perturbation theory (MBPT) methods for their use in the treatment of resonances. This extends the domain of applicability of MCSCF, CC, MBPT, and methods based on MCSCF states to an accurate treatment of resonances while still using L² real basis sets. Modification of all other bound‐state methods and codes should be similarly straightforward. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

A MCSCF method for ground and excited states based on full optimizations of successive Jacobi rotations

A new multiconfigurational self-consistent field (MCSCF) method based on successive optimizations of Jacobi rotation angles is presented. For given one- and two-particle density matrices and an initial set of corresponding integrals, a technique is developed for the determination of a Jacobi angle for the mixing of two orbitals, such that the exact energy, written as a function of the angle, is fully minimized. Determination of the energy-minimizing orbitals for given density matrices is accomplished by successive optimization and updating of Jacobi angles and integrals. The total MCSCF energy is minimized by alternating between CI and orbital optimization steps. Efficiency is realized by optimizing CI and orbital vectors quasi-simultaneously by not fully optimizing each in each improvement step. On the basis of the Jacobi-rotation based approach, a novel MCSCF procedure is formulated for excited states, which avoids certain shortcomings of traditional excited-state MCSCF methods. Applications to specific systems show the practicability of the developed methods.     

Study of localized molecular orbitals using group theory methods and its approach to the many-electron correlation problem. IV. The symmetry-adaptation of many-center integrals and hamiltonian matrix elements in MCSCF calculations

Group theoretic methods are presented for the transformations of integrals and the evaluation of matrix elements encountered in multiconfigurational self-consistent field (MCSCF) and configuration interaction (CI) calculations. The method has the advantages of needing only to deal with a symmetry unique set of atomic orbitals (AO) integrals and transformation from unique atomic integrals to unique molecular integrals rather than with all of them. Hamiltonian matrix element is expressed by a linear combination of product terms of many-center unique integrals and geometric factors. The group symmetry localized orbitals as atomic and molecular orbitals are a key feature of this algorithm. The method provides an alternative to traditional method that requires a table of coupling coefficients for products of the irreducible representations of the molecular point group. Geometric factors effectively eliminate these coupling coefficients. The saving of time and space in integral computations and transformations is analyzed. © 1994 by John Wiley & Sons, Inc.     

Multiconfiguration self-consistent-field theory based upon the fragment molecular orbital method

The fragment molecular orbital (FMO) method was combined with the multiconfiguration self-consistent-field (MCSCF) theory. One- and two-layer approaches were developed, the former involving all dimer MCSCF calculations and the latter limiting MCSCF calculations to a small part of the system. The accuracy of the two methods was tested using the six electrons in six orbitals complete active space type of MCSCF and singlet spin state for phenol+(H(2)O)(n), n=16,32,64 (6-31G( *) and 6-311G( *) basis sets); alpha helices and beta strands of phenylalanine-(alanine)(n), n=4,8,16 (6-31G( *)). Both double-zeta and triple-zeta quality basis sets with polarization were found to have very similar accuracy. The error in the correlation energy was at most 0.000 88 a.u., the error in the gradient of the correlation energy was at most 6.x10(-5) a.u./bohr and the error in the correlation correction to the dipole moment was at most 0.018 D. In addition, vertical singlet-triplet electron excitation energies were computed for phenol+(H(2)O)(n), (n=16,32,64), 6-31G( *), and the errors were found to be at most 0.02 eV. Approximately linear scaling was observed for the FMO-based MCSCF methods. As an example, an FMO-based MCSCF calculation with 1262 basis functions took 98 min on one 3.0 GHz Pentium4 node with 1 Gbyte RAM.     

Comparison of perturbative and multiconfigurational electron propagator methods

Ionization energies below 20 eV of 10 molecules calculated with electron propagator techniques employing Hartree-Fock orbitals and multiconfigurational self-consistent field orbitals are compared. Diagonal and nondiagonal self-energy approximations are used in the perturbative formalism. Three diagonal methods based on second- and third-order self-energy terms, all known as the outer valence Green's function, are discussed. A procedure for selecting the most reliable of these three versions for a given calculation is tested. Results with a polarized, triple ζ basis produce root mean square errors with respect to experiment of approximately 0.3 eV. Use of the selection procedure has a slight influence on the quality of the results. A related, nondiagonal method, known as ADC (3), performs infinite-order summations on several types of self-energy contributions, is complete through third-order, and produces similar accuracy. These results are compared to ionization energies calculated with the multiconfigurational spin-tensor electron propagator method. Complete active space wave functions or close approximations constitute the reference states. Simple field operators and transfer operators pertaining to the active space define the operator manifold. With the same basis sets, these methods produce ionization energies with accuracy that is comparable to that of the perturbative techniques. © 1996 John Wiley & Sons, Inc.     

The problem of hole localization in inner-shell states of N2 and CO2 revisited with complete active space self-consistent field approach

Potential energy curves for inner-shell states of nitrogen and carbon dioxide molecules are calculated by inner-shell complete active space self-consistent field (CASSCF) method, which is a protocol, recently proposed, to obtain specifically converged inner-shell states at multiconfigurational level. This is possible since the collapse of the wave function to a low-lying state is avoided by a sequence of constrained optimization in the orbital mixing step. The problem of localization of K-shell states is revisited by calculating their energies at CASSCF level based on both localized and delocalized orbitals. The localized basis presents the best results at this level of calculation. Transition energies are also calculated by perturbation theory, by taking the above mentioned MCSCF function as zeroth order wave function. Values for transition energy are in fairly good agreement with experimental ones. Bond dissociation energies for N(2) are considerably high, which means that these states are strongly bound. Potential curves along ground state normal modes of CO(2) indicate the occurrence of Renner-Teller effect in inner-shell states.     

Peroxynitrite in the hemoglobin composition

Electronic characteristics of trHbN hemoglobin whose composition contains the (ONOO) group with the structure close to the structure of 1) peroxynitrite and 2) a nitrate anion in the gas phase are calculated. Electron correlation is considered by the multiconfigurational self-consistent field (MCSCF) method during the optimization of the geometry of the whole structure. Localized molecular orbitals (MOs) are used as starting ones. In the wave function of the MCSCF method two complete active subspaces (CASs) are set. These are the subspace of iron atom 3d orbitals and the subspace describing chemical bonds in peroxynitrite (bonding and antibonding MOs plus the orbital of one lone pair on the O₂ moiety. The composition of the system involves two water molecules. The peroxynitrite structure is considered in two different spin states that correspond to the singlet and triplet states of this anion in the gas phase where the vibrational spectrum is characterized by frequencies of about (70-30) cm^?1. The protective reaction of the active center of the tubercule bacillus is discussed.     

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...     

Generalized-molecular-orbital theory: Simple multiconfiguration self-consistent-field method

Even after completing a multiconfiguration self-consistent-field (MCSCF ) calculation, one must often include additional configuration interaction (CI ) to obtain quantitative or semiquantitative results. There is some question of whether the prior MCSCF calculation is worthwhile, if additional CI is needed later. We have developed a new MCSCF computational method, which, because of our assumptions about the nature of the configurations, yields one Fock-like operator for all the “filled” orbitals (high occupation numbers) and a second Fock-like operator for all the “virtual” orbitals (low occupation numbers). Since there are only two matrices to build, our method is considerably faster than other MCSCF approaches. Because of these similarities to standard molecular-orbital (MO ) calculations, we have termed our approach generalized-molecular-orbital (GMO ) theory. However, the “virtual” orbitals, unlike those of standard MO theory, are optimized to correlate the “filled” ones and can he used in a subsequent CI calculation. Results are presented for the correlation energy of H₂O, the spectroscopic constants of N₂, the singlet–triplet energy separations in CH₂, and the nature of the chromium–chromium quadruple bond. Although these results are at a very low level of CI , the GMO approach appears to correct for the gross deficiencies of the single-determinant SCF procedure.     

Pathway analysis of super-exchange electronic couplings in electron transfer reactions using a multi-configuration self-consistent field method

We present a novel pathway analysis of super-exchange electronic couplings in electron transfer reactions using localized molecular orbitals from multi-configuration self-consistent field (MCSCF) calculations. In our analysis, the electronic coupling and the tunneling pathways can be calculated in terms of the configuration interaction (CI) Hamiltonian matrix obtained from the localized MCSCF wave function. Making use of the occupation restricted multiple active spaces (ORMAS) method can effectively produce the donor, acceptor, and intermediate configuration state functions (CSFs) and CIs among these CSFs. In order to express the electronic coupling as a sum of individual tunneling pathways contributions, we employed two perturbative methods: L?wdin projection-iteration method and higher-order super-exchange method. We applied them to anion couplings of butane-1,4-diyl and pentane-1,5-diyl. The results were (1) the electronic couplings calculated from the two perturbative methods were in reasonable agreement with those from a non-perturbative method (one-half value of the energy difference between the ground and first excited states), (2) the main tunneling pathways consisted of a small number of lower-order super-exchange pathways where bonding, anti-bonding, or extra-valence-shell orbitals were used once or twice, and (3) the interference among a huge number of higher-order super-exchange pathways significantly contributed to the overall electronic coupling, whereas each of them contributed only fractionally. Our method can adequately take into account both effects of non-dynamical electron correlation and orbital relaxation. Comparing with the analyses based on the Koopmans' theorem (ignoring both effects) and the ORMAS-CIs from frozen localized reference orbitals (ignoring the effect of orbital relaxation), we discuss these effects.