Some analysis of the ω-Technique

The ω-Technique is one of a number of ways of improving on the Hückel model by introducing a dependence on atomic orbital populations into the matrix-elements of the effective Hamiltonian. It requires iterative solution of secular equations until the populations calculated from the solutions are consistent with the populations used in setting up the Hamiltonian matrix. We derive simple equations showing how thedeviations of the populations from their final self-consistent values change with successive iterations. The results of consideration of these equations in several special cases, imply that the populations oscillate about their final values on successive iterations, as has actually been found experimentally. This suggests a simple means of speeding up convergence.     

New relativistic ANO basis sets for transition metal atoms

New basis sets of the atomic natural orbital (ANO) type have been developed for the first, second, and third row transition metal atoms. The ANOs have been obtained from the average density matrix of the ground and lowest excited states of the atom, the positive and negative ions, and the atom in an electric field. Scalar relativistic effects are included through the use of a Douglas-Kroll-Hess Hamiltonian. Multiconfigurational wave functions have been used with dynamic correlation included using second order perturbation theory (CASSCF/CASPT2). The basis sets are applied in calculations of ionization energies, electron affinities, and excitation energies for all atoms and polarizabilities for spherically symmetric atoms. These calculations include spin-orbit coupling using a variation-perturbation approach. Computed ionization energies have an accuracy better than 0.2 eV in most cases. The accuracy of computed electron affinities is the same except in cases where the experimental values are smaller than 0.5 eV. Accurate results are obtained for the polarizabilities of atoms with spherical symmetry. Multiplet levels are presented for some of the third row transition metals.     

Semiempirical spin–orbit coupling calculations. I. Theory and method. Benzophenone as a test case

The molecular spin–orbit coupling operator is brought into a simplified form through a convenient choice of origin for the orbital angular momentum operator. The eigenvalue problem of the Hamiltonian that includes the spin–orbit (SOC ) operator as a perturbation is solved by means of a linear variational procedure in the basis of the spin-pure molecular eigenstates. Test calculations on benzophenone are presented and the results are compared to experiment. We discuss the minimal size of the spin-pure variational basis needed to achieve stable results as well as the amount of single-excitation configurational mixing needed to describe the spin-pure molecular eigenstates.     

New relativistic atomic natural orbital basis sets for lanthanide atoms with applications to the Ce diatom and LuF3

New basis sets of the atomic natural orbital (ANO) type have been developed for the lanthanide atoms La-Lu. The ANOs have been obtained from the average density matrix of the ground and lowest excited states of the atom, the positive ions, and the atom in an electric field. Scalar relativistic effects are included through the use of a Douglas-Kroll-Hess Hamiltonian. Multiconfigurational wave functions have been used with dynamic correlation included using second-order perturbation theory (CASSCF/CASPT2). The basis sets are applied in calculations of ionization energies and some excitation energies. Computed ionization energies have an accuracy better than 0.1 eV in most cases. Two molecular applications are included as illustration: the cerium diatom and the LuF3 molecule. In both cases it is shown that 4f orbitals are not involved in the chemical bond in contrast to an earlier claim for the latter molecule.     

Convergence optimization of restricted open-shell self-consistent field calculations

Different self-consistent field (SCF) iteration schemes for open-shell systems are discussed. After a brief summary of the well-known level shifting and damping procedure, we describe the quadratically convergent SCF (QCSCF) approach based on the gradient and the Hessian matrix in a space of orbital rotation parameters. An analytical expression for the latter is derived for the general many-shell case. Starting from the expression for the energy change obtained by the QCSCF method, we then present a simplified direct procedure avoiding matrix diagonalization but also the difficulties of the QCSCF method in handling the Hessian matrix. Numerical calculations on some open-shell systems involving transition-metal complexes show that this method leads to rapid and reliable convergence of the iteration process in cases where the usual SCF procedure of iterative diagonalization tends to diverge. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 62: 617–637, 1997     

An analysis of the zero differential overlap approximation. Towards an improved semiempirical MO method beyond it

Summary Some systematic errors of the zero differential overlap (ZDO) approximation in semiempirical molecular orbital (MO) methods are discussed. In electron methods, a power series expansion of the inverse square rootS ^–1/2 of the overlap matrix and application of the Mulliken approximation to the two-electron integrals show that the ZDO Hamiltonian coincides with the Hamiltonian obtained by explicit performance of the Löwdin transformation up to first-order terms of diatomic overlap densities. Higher than first-order terms lead to a systematic up-shift of the canonical MO energies. Although a power series expansion ofS ^–1/2 is no longer possible in all-valence-electron methods, the MO levels resulting from the ZDO approximation are also systematically placed at too low energies, especially the low-lying occupied and the virtual MOs. A method based on explicit performance of the Löwdin transformation and retaining the simplicity of the ZDO approach for the calculation of Fock matrix elements is developed. The parameters of this method are obtained by very simple manipulations of the original ZDO parameters. Numerical calculations show that a considerable improvement of the MO energy spectrum in the inner valence region can be obtained in this way     

Quadratically convergent calculation of localized molecular orbitals

Two iterative procedures for the transformation of canonical self-consistent field molecular orbitals to intrinsic localized molecular orbitals are proposed. A first-order method based on a series of (n × n) unitary transformations may be applied to orbitals which are far from convergence. The second method, based on Newton's method, yields quadratic convergence. Numerical results based on Boys' criterion are presented for water, carbon monoxide, boron fluoride, nitric oxide, and methylacetylene. A composite method may be used to obtain rapid convergence for large molecules for which it is not practical to calculate the entire hessian matrix. The performance of the composite method is demonstrated by application to the dinitrogen tetroxide molecule. Highly converged localized molecular orbitals may be obtained for most molecules with five to eight first-order iterations followed by three or four iterations based on either the second-order or composite method.     

Localization of molecular orbitals on fragments

A non‐iterative algorithm for the localization of molecular orbitals (MOs) from complete active space self consistent field (CASSCF) and for single‐determinantal wave functions on predefined moieties is given. The localized fragment orbitals can be used to analyze chemical reactions between fragments and also the binding of fragments in the product molecule with a fragments‐in‐molecules approach by using a valence bond expansion of the CASSCF wave function. The algorithm is an example of the orthogonal Procrustes problem, which is a matrix optimization problem using the singular value decomposition. It is based on the similarity of the set of MOs for the moieties to the localized MOs of the molecule; the similarity is expressed by overlap matrices between the original fragment MOs and the localized MOs. For CASSCF wave functions, localization is done independently in the space of occupied orbitals and active orbitals, whereas, the space of virtual orbitals is mostly uninteresting. Localization of Hartree–Fock or Kohn–Sham density functional theory orbitals is not straightforward; rather, it needs careful consideration, because in this case some virtual orbitals are needed but the space of virtual orbitals depends on the basis sets used and causes considerable problems due to the diffuse character of most virtual orbitals. © 2012 Wiley Periodicals, Inc.     

Elimination of the diagonalization bottleneck in parallel Direct-SCF methods

Summary It is shown that the matrix diagonalization bottleneck associated with thesequential O(N _BFN ³ ) diagonalization of the fock matrix within each iteration of the Direct-SCF procedure may be eliminated, and replaced instead with a combination ofparallel O(N _BFN ^<4 ) andsequential O(N _Sub ³ ) steps. For large basis sets, the relation N_Sub N_BFN between the dimension of the expansion subspace and the number of basis functions leads to a method of wave-function optimization in which the sequential bottleneck is eliminated. As a side benefit, the second-order iterative procedure on which this method is based displays superior convergence properties, and provides greater insight into the behavior of the energy with respect to orbital variations, than the traditional first-order, fixed-point, iterative approaches. The implementation of this method may be incorporated into essentially any existing Direct-SCF program with only minimal, and localized, changes.     

A state-specific partially internally contracted multireference coupled cluster approach

A state-specific partially internally contracted multireference coupled cluster approach is presented for general complete active spaces with arbitrary number of active electrons. The dominant dynamical correlation is included via an exponential parametrization of internally contracted cluster operators ( ?T) which excite electrons from a multideterminantal reference function. The remaining dynamical correlation and relaxation effects are included via a diagonalization of the transformed Hamiltonian ?? =e(- ?T)H?e( ?T) in the multireference configuration interaction singles space in an uncontracted fashion. A new set of residual equations for determining the internally contracted cluster amplitudes is proposed. The second quantized matrix elements of ?? , expressed using the extended normal ordering of Kutzelnigg and Mukherjee, are used as the residual equations without projection onto the excited configurations. These residual equations, referred to as the many-body residuals, do not have any near-singularity and thus, should allow one to solve all the amplitudes without discarding any. There are some relatively minor remaining convergence issues that may arise from an attempt to solve all the amplitudes and an initial analysis is provided in this paper. Applications to the bond-stretching potential energy surfaces for N(2), CO, and the low-lying electronic states of C(2) indicate clear improvements of the results using the many-body residuals over the conventional projected residual equations.     

Large systems at ab initio multireference level: a cheap treatment thanks to a division into fragments

Thanks to the use of localized orbitals and the subsequent possibility of neglecting long-range interactions, the linear-scaling methods have allowed to treat large systems at ab initio level. However, the limitation of the number of active orbitals in a complete active space self consistent-field (CASSCF) calculation remains unchanged. The method presented in this paper suggests to divide the system into fragments containing only a small number of active orbitals. Starting from a guess wave function, each orbital is optimized in its corresponding fragment, in the presence of the other fragments. Once all the fragments have been treated, a new set of orbitals is obtained. The process is iterated until convergence. At the end of the calculation, a set of active orbitals is obtained, which is close to the exact CASSCF solution, and an accurate CASSCF energy can be estimated.     

Iterative solution of Bloch-type equations: stability conditions and chaotic behavior

Two different form of nonperturbative Bloch-type equations are studied: one for the wave operator of the N-electron Schr?dinger equation, another one for obtaining first-order density matrix P in one-electron theories (Hartree–Fock or Kohn–Sham). In both cases, we investigate the possibility of an iterative solution of the nonlinear Bloch equation. To have a closer view on convergence features, we determine the stability matrix of the iterative procedures and determine the Ljapunov exponents from its eigenvalues. For some of the cases when not every exponents are negative, chaotic solutions can be identified, which should of course be carefully avoided in practical iterations.     

A simple approach for improving the hybrid MMVB force field: application to the photoisomerization of s-cis butadiene

MMVB is a QM/MM hybrid method, consisting of a molecular mechanics force field coupled to a valence bond Heisenberg Hamiltonian parametrized from ab initio CASSCF calculations on several prototype molecules. The Heisenberg Hamiltonian matrix elements Q(ij) and K(ij), whose expressions are partitioned here into a primary contribution and second-order correction terms, are calculated analytically in MMVB. When the original MMVB force field fails to produce potential energy surfaces accurate enough for dynamics calculations, we show that significant improvements can be made by refitting the second-order correction terms for the particular molecule(s) being studied. This "local" reparametrization is based on values of K(ij) extracted (using effective Hamiltonian techniques) from CASSCF calculations on the same molecule(s). The method is demonstrated for the photoisomerization of s-cis butadiene, and we explain how the correction terms that enabled a successful MMVB dynamics study [Garavelli, M.; Bernardi, F.; Olivucci, M.; Bearpark, M. J.; Klein, S.; Robb, M. A. J Phys Chem A 2001, 105, 11496] were refitted.     

CCSD(T) expectation value calculations of first-order properties

An expectation value approach to calculations of first-order properties using the non-iterative, triple-excitation amplitudes in the coupled cluster wave function is exploited. Three methods are suggested and analysed using the many body perturbation theory (MBPT) expansion arguments. The first method, in which non-iterative triple-excitation amplitudes are used in the expression for the expectation values, makes the wave function accurate through the second order of MBPT. In the second method, which is an extension of the first, effects of triple-excitation amplitudes are coupled with single- and double-excitation amplitudes. The correlated density matrix equivalent through the fourth order to that obtained when CCSDT-la amplitudes are used is employed in the third method. The suggested methods are tested on dipole moment and polarizability calculations for several diatomic closed-shell molecules and are compared to other related approaches. Received: 15 May 1997 / Accepted: 5 June 1997     

Valence bond approach exploiting Clifford algebra realization of Rumer–Weyl basis

A detailed algorithm is described that enables an implementation of a general valence bond (VB ) method using the Clifford algebra unitary group approach (CAUGA ). In particular, a convenient scheme for the generation and labeling of classical Rumer–Weyl basis (up to a phase) is formulated, and simple rules are given for the evaluation of matrix elements of unitary group generators, and thus of any spin-independent operator, in this basis. The case of both orthogonal and nonrothogonal atomic orbital bases is considered, so that the proposed algorithm can also be exploited in molecular orbital configuration interaction calculations, if desired, enabling a greater flexibility for N-electron basis-set truncation than is possible with the standard Gel'fand–Tsetlin basis. Finally, an exploitation of this formalism for the VB method, based on semiempirical Pariser–Parr–Pople (PPP )-type Hamiltonian and nonorthogonal overlap-enhanced atomic orbital basis, and its computer implementation, enabling us to carry out arbitrarily truncated or full VB calculations, is described in detail.     

Improved convergence of self-consistence procedures in the MC SCF theory

To obtain optimized orbitals within the MC SCF theory, the energy surface near a chosen point is approximated by a quadratic function of independent matrix elements of a small orthogonal orbital transformation. The method of a second-order one-electron Hamiltonian (OEH) is developed on the basis of this approximation. A procedure is proposed to define step coordinates, insuring a rapid descent along an average-energy surface also in the cases when the matrix of second energy derivatives has eigenvalues negative or close to zero. The results obtained in applying the OEH method for the calculation of ground and triplet states of uracile in the π-electron approximation are discussed. When a complete matrix of the second energy derivatives is used, the self-consistence procedure is quadratically convergent. An exponential, yet rapid enough convergence is provided by a simplified computation scheme neglecting cross derivatives.     

Density matrix structure for saturated molecules

Conclusion The existence of a common Hamiltonian matrix structure for saturated systems results in common structural properties of the density matrices for the whole class of molecules, such as the zero occupation of AO in the first approximation, the density matrix perturbations due to a heteroatom, etc. This fact can be taken as a quantum-mechanical foundation for viewing saturated molecules as a separate class of compounds. The endowment of this system with the transferability of electronic structure properties, relative to atoms and bonds, to high accuracy, within the framework of the effective Hamiltonian method follows from an analysis of the general expressions for the density matrix elements. The transferability of the saturated system Hamiltonian matrix elements requisite for this is supported by a comparison among the self-consistent Fock matrix elements of various hydrocarbons in a localized orbital basis [9]. Independently of the detailed structure of the actual molecules, the influence of a heteroatom on the electron density distribution in saturated systems dies off quickly with distance from the heteroatom. From an analysis of expressions for the nondiagonal elements of the density matrix corresponding to nonneighboring AO we establish a connection between the degree of electron localization in saturated systems and the size or certain Hamiltonian matrix elements.There is a consequent analogy between saturated and alternatively conjugated hydrocarbons, which, starting from the common structure of the Hamiltonians, also leads to common properties of the density matrices [14]. However, the study of the influence of heteroatoms on the density matrices in these systems by means of perturbation theory is complicated by the dependence of the matrix N(4) on the molecular structure, which makes it necessary to introduce highly simplified approaches for the solution of Eq. (2) [8]. Therefore, for alternating hydrocarbons we have succeeded in establishing only the sign of the orbital — orbital polarizability (alternating polarity theorem [14]), while, as for saturated systems, the equality N=1 permits an analytic expression for the polarizability.V. Kapsukas Vilnius State University. Translated from Zhurnal Strukturnoi Khimii, Vol. 29, No. 5, pp. 3–8, September–October, 1988.     

Full S matrix calculation via a single real-symmetric Lanczos recursion: the Lanczos artificial boundary inhomogeneity method

We present an efficient and robust method for the calculation of all S matrix elements (elastic, inelastic, and reactive) over an arbitrary energy range from a single real-symmetric Lanczos recursion. Our new method transforms the fundamental equations associated with Light's artificial boundary inhomogeneity approach from the primary representation (original grid or basis representation of the Hamiltonian or its function) into a single tridiagonal Lanczos representation, thereby affording an iterative version of the original algorithm with greatly superior scaling properties. The method has important advantages over existing iterative quantum dynamical scattering methods: (a) the numerically intensive matrix propagation proceeds with real symmetric algebra, which is inherently more stable than its complex symmetric counterpart; (b) no complex absorbing potential or real damping operator is required, saving much of the exterior grid space which is commonly needed to support these operators and also removing the associated parameter dependence. Test calculations are presented for the collinear H+H(2) reaction, revealing excellent performance characteristics.     

Perturbational approach to the electron correlation cusp applied to helium‐like atoms

A recently proposed perturbational approach to the electron correlation cusp problem 1 is tested in the context of three spherically symmetrical two‐electron systems: helium atom, hydride anion, and a solvable model system. The interelectronic interaction is partitioned into long‐ and short‐range components. The long‐range interaction, lacking the singularities responsible for the electron correlation cusp, is included in the reference Hamiltonian. Accelerated convergence of orbital‐based methods for this smooth reference Hamiltonian is shown by a detailed partial wave analysis. Contracted orbital basis sets constructed from atomic natural orbitals are shown to be significantly better for the new Hamiltonian than standard basis sets of the same size. The short‐range component becomes the perturbation. The low‐order perturbation equations are solved variationally using basis sets of correlated Gaussian geminals. Variational energies and low‐order perturbation wave functions for the model system are shown to be in excellent agreement with highly accurate numerical solutions for that system. Approximations of the reference wave functions, described by fewer basis functions, are tested for use in the perturbation equations and shown to provide significant computational advantages with tolerable loss of accuracy. Lower bounds for the radius of convergence of the resulting perturbation expansions are estimated. The proposed method is capable of achieving sub‐μHartree accuracy for all systems considered here. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

Variational formulation of perturbative explicitly-correlated coupled-cluster methods

We present a variational formulation of the recently-proposed CCSD(2)(R12) method [Valeev, Phys. Chem. Chem. Phys., 2008, 10, 106]. The centerpiece of this approach is the CCSD(2)(R12) Lagrangian obtained via L?wdin partitioning of the coupled-cluster singles and doubles (CCSD) Hamiltonian. Extremization of the Lagrangian yields the second-order basis set incompleteness correction for the CCSD energy. We also developed a simpler Hylleraas-type functional that only depends on one set of geminal amplitudes by applying screening approximations. This functional is used to develop a diagonal orbital-invariant version of the method in which the geminal amplitudes are fixed at the values determined by the first-order cusp conditions. Extension of the variational method to include perturbatively the effect of connected triples produces the method that approximates the complete basis-set limit of the standard CCSD plus perturbative triples [CCSD(T)] method. For a set of 20 small closed-shell molecules, the method recovered at least 94.5/97.3% of the CBS CCSD(T) correlation energy with the aug-cc-pVDZ/aug-cc-pVTZ orbital basis set. For 12 isogyric reactions involving these molecules, combining the aug-cc-pVTZ correlation energies with the aug-cc-pVQZ Hartree-Fock energies produces the electronic reaction energies with a mean absolute deviation of 1.4 kJ mol(-1) from the experimental values. The method has the same number of optimized parameters as the corresponding CCSD(T) model, does not require any modification of the coupled-cluster computer program, and only needs a small triple-zeta basis to match the precision of the considerably more expensive standard quintuple-zeta CCSD(T) computation.