Linear scaling computation of the Fock matrix. III. Formation of the exchange matrix with permutational symmetry

 A direct comparison is made between two recently proposed methods for linear scaling computation of the Hartree–Fock exchange matrix to investigate the importance of exploiting two-electron integral permutational symmetry. Calculations on three-dimensional water clusters and graphitic sheets with different basis sets and levels of accuracy are presented to identify specific cases where permutational symmetry may or may not be useful. We conclude that a reduction in integrals via permutational symmetry does not necessarily translate into a reduction in computation times. For large insulating systems and weakly contracted basis sets the advantage of permutational symmetry is found to be negligible, while for noninsulating systems and highly contracted basis sets a fourfold speedup is approached. Received: 8 October 1999 / Accepted: 3 January 2000 / Published online: 21 June 2000     

The search for stationary points on a quantum mechanical/molecular mechanical potential-energy surface

 We propose a methodology to locate stationary points on a quantum mechanical/molecular mechanical potential-energy surface. This algorithm is based on a suitable approximation of an initial full Hessian matrix, either a modified Broyden–Fletcher–Goldfarg–Shanno or a Powell update formula for the location of, respectively, a minimum or a transition state, and the so-called rational function optimization. The latter avoids the Hessian matrix inversion required by a quasi-Newton–Raphson method. Some examples are presented and analyzed. Received: 16 July 2001 / Accepted: 9 October 2001 / Published online: 9 January 2002     

The alternant hydrocarbon pairing theorem and all-valence electrons theory. An approximate LCOAO theory for the electronic absorption and MCD spectra of conjugated organic compounds, part 2

An approximate linear combination of orthogonalized atomic orbitals (LCOAO) all-valence electrons theory is described, based on a previously suggested partitioning of the Fock operator. Kinetic energy and penetration terms are evaluated explicitly in a L?wdin OAO basis, while two-electron repulsion terms are treated according to the conventional neglect of differential overlap (NDO) approximation. One-electron and penetration integrals are parameterized explicitly to predict approximate alternant pairing symmetry for the π-systems of benzene and napthalene. Application of the resulting LCOAO theory to a variety of alternant and non-alternant hydrocarbons demonstrates significant improvements in the prediction of MCD B-terms and transition moment directions, particularly for alternant (4N+2)- or 4N-perimeter π-systems for which traditional NDO procedures fail. Received: 27 May 1997 / Accepted: 28 August 1997     

Key requirements for introducing quality assurance in measurement laboratories

 The implementation of a quality assurance system is fraught with difficulties. However, these difficulties may be overcome if the laboratory uses suitable means to facilitate the process. It is necessary to mobilise the intelligence and energy of all members of the laboratory. In order to command adherence, the project must be shared, and this necessitates a major effort by all concerned. Communication is a major factor in obtaining the support of all parties. Six important steps must be distinguished: – Defining quality policy – Creating awareness, information, training – Creating a quality structure – Establishing a deadline for obtaining accreditation – Progressive implementation – Experimentation and validation. Even if the task of obtaining and maintaining accreditation remains difficult, it clearly promotes a minimum level of organisation and stepwise progress in quality assurance. The laboratory must keep improving its quality system, using European Standard EN 45001 as an effective management model. Received: 9 April 1997 · Accepted: 11 September 1997     

Extremely localized molecular orbitals (ELMO): a non-orthogonal Hartree-Fock method

A new optimization method for extremely localized molecular orbitals (ELMO) is derived in a non-orthogonal formalism. The method is based on a quasi Newton-Raphson algorithm in which an approximate diagonal-blocked Hessian matrix is calculated through the Fock matrix. The Hessian matrix inverse is updated at each iteration by a variable metric updating procedure to account for the intrinsically small coupling between the orbitals. The updated orbitals are obtained with approximately n ² operations. No n ³ processes such as matrix diagonalization, matrix multiplication or orbital orthogonalization are employed. The use of localized orbitals allows for the creation of high-quality initial “guess” orbitals from optimized molecular orbitals of small systems and thus reduces the number of iterations to converge. The delocalization effects are included by a Jacobi correction (JC) which allows the accurate calculation of the total energy with a limited number of operations. This extension, referred to as ELMO(JC), is a variational method that reproduces the Hartree-Fock (HF) energy with an error of less than 2 kcal/mol for a reduced total cost compared to standard HF methods. The small number of variables, even for a very large system, and the limited number of operations potentially makes ELMO a method of choice to study large systems. Received: 30 December 1996 / Accepted: 5 June 1997     

Spin-orbit effects on structures of closed-shell polyatomic molecules containing heavy atoms calculated by two-component Hartree–Fock method

We have implemented geometry optimization using an analytic gradient to a two-component Kramers' restricted Hartree–Fock (KRHF) method for polyatomic molecules with closed-shell configurations. The KRHF method is a Hartree–Fock method based on relativistic effective core potentials with effective spin-orbit operators. The derivatives of spin-orbit integrals are obtained by numerical differentiation. Geometries for the various forms of polyatomic hydrides containing row 6 p-block elements are optimized with and without spin-orbit interactions. The structural changes due to spin-orbit interactions are small, but show definite trends, which correlate well with the p_1/2 spinor population. Atomization energies are reduced significantly by incorporating spin-orbit interactions for all molecules considered. The KRHF calculations of several methylhalides demonstrate that the spinor energies from the KRHF method can be useful for the interpretation of experimental photoelectron spectra of molecules exhibiting spin-orbit splittings. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1526–1533, 1998     

Development of spin‐dependent relativistic open‐shell Hartree–Fock theory with time‐reversal symmetry (II): The restricted open‐shell approach

An open‐shell Hartree–Fock (HF) theory for spin‐dependent two‐component relativistic calculations, termed the Kramers‐restricted open‐shell HF (KROHF) method, is developed. The present KROHF method is defined as a relativistic analogue of ROHF using time‐reversal symmetry and quaternion algebra, based on the Kramers‐unrestricted HF (KUHF) theory reported in our previous study (Int. J. Quantum Chem., doi: 10.1002/qua.25356 ). As seen in the nonrelativistic ROHF theory, the ambiguity of the KROHF Fock operator gives physically meaningless spinor energies. To avoid this problem, the canonical parametrization of KROHF to satisfy Koopmans' theorem is also discussed based on the procedure proposed by Plakhutin et al. (J. Chem. Phys. 2006 , 125, 204110). Numerical assessments confirmed that KROHF using Plakhutin's canonicalization procedure correctly gives physical spinor energies within the frozen‐orbital approximation under spin–orbit interactions.     

A pilot study for the preparation of a new Reference Material based on antarctic krill

The preparation of new Certified Reference Materials (CRMs) of antarctic matrices forms the backbone of an ongoing project in the framework of the Italian National Program for Antarctic Research. The first CRM of this kind (MURST-ISS-A1 Antarctic Marine Sediment) is already available. The second phase focuses on the certification of antarctic krill, a small shrimp extremely abundant in the Southern Ocean. The total mass of krill available for this purpose is approximately 44 kg and results from the combination of three different catches (Ross Sea, Marguerite Bay and Livingston Island, respectively). The quantification of the following elements in the raw mass appears to be affordable by current analytical techniques, values being in the range of (in μg/g) 0.11–0.30 for As, 0.03–0.12 for Cd, 0.06–0.23 for Cr, 6.1–21 for Cu, 5.7–7.6 for Fe, 0.005–0.008 for Hg, 0.7–1.2 for Mn, 0.013– 0.077 for Ni, 0.04–0.57 for Pb and 12–16 for Zn. On the other hand, the average values ascertained in freeze-dried krill are as a rule one order of magnitude higher, i.e., (in μg/g), 3.2 for As, 0.6 for Cd, 1.8 for Cr, 75 for Cu, 61 for Fe, 0.025 for Hg, 4.6 for Mn, 0.7 for Ni, 2.1 for Pb and 81 for Zn. Information on the pretreatment of krill and details on the planned certification campaign are also given. Received: 10 June 1997 / Revised: 24 September 1997 / Accepted: 28 September 1997     

Fock space multireference coupled cluster theory: noniterative inclusion of triples for excitation energies

In this paper we propose and numerically implement a specific scheme for calculating the excitation energies (EEs) within the Fock space multireference coupled cluster framework, which includes the contributions from noniterative triples cluster amplitudes. These contribute to the EEs at the third order. We present results for CH⁺ and N2, and study the effects of these noniterative triples on EEs. Received: 28 July 1997 / Accepted: 8 December 1997     

A survey of optimization procedures for stable structures and transition states

We examine a variety of methods for obtaining the stable geometry of molecules and the transition states of simple systems and summarize some of our findings. We find the most efficient methods for optimizing structure to be those based on calculated gradients and estimated second derivative (Hessian) matrices, the later obtained either from the Broyden–Fletcher–Goldfarb–Shanno (BFGS ) quasi-Newton update method or from approximations to the coupled perturbed Hartree–Fock method. For uncovering transition states we find particularly useful a variety of the augmented Hessian theory used to uncover regions of the potential energy hypersurface with one and only one negative eigenvalue of the Hessian matrix characterizing the catchment region of the transition state. Once this region is found we minimize the norm of the gradient vector to catch the nearest extreme point of the surface. Examples of these procedures are given.     

Virtual geometrical symmetry group of a nonrigid molecule with energetically nonequivalent equilibrium configurations

The notion of a virtual geometrical symmetry group of a nonrigid molecule with energetically nonequivalent equilibrium configurations is introduced. Due to this, methods of qualitative analysis of intramolecular effects for such molecules using the concept of symmetry group chains are significantly generalized. Insitute of Applied Physics, Novgorod. Translated fromZhumal Struckturmoi Khimii, Vol. 37, No. 3, pp. 432–439, May–June, 1996.     

Normal mode analysis of oligomeric proteins: Reduction of the memory requirement by consideration of rigid geometry and molecular symmetry

A method is presented to reduce the memory requirement of normal mode analysis applied to systems containing two or more large proteins when these systems exhibit symmetry properties. We use a rigid geometry model (i.e., only the dihedral angles of the polypeptide chain are considered as variables). This model allows a reduction by a factor of 8 on average of the number of variables with a concomitant freezing of the high-frequency modes. The symmetry properties of the system are used to reduce further the number of variables that must be considered in the computation. Application of group theory leads to a factorization of the matrices of interest (the coefficient and the Hessian matrices) into independent blocks along the diagonal. The initial, reducible representation is thus transformed into a number of irreducible representations of smaller dimensions. In the case of the C₂ symmetry group, the method leads to a reduction of the size of the matrices that must be manipulated during the computation (coefficient matrix, Hessian matrix, and eigenvectors matrix) by a factor of 256 compared with the usual normal mode analysis in Cartesian coordinate space. The method is particularly well adapted to the study of the dynamics of oligomeric proteins because these proteins often display symmetry properties (e.g., virus coat proteins, immunoglobulins, hemoglobin, etc.). In favorable cases, in conjunction with X-ray diffuse scattering data, the study of systems showing allosteric properties might be considered. © 1994 by John Wiley & Sons, Inc.     

Adiabatic integration formula for the correlation energy functional of the Hartree–Fock density

An adiabatic integration formula for the quantum chemistry correlation energy functional of the Hartree–Fock density, E _c ^QC[n], is presented. The functional E _c ^QC[n] is meant to be added to the completed Hartree–Fock energy to produce the exact ground-state energy of the system under consideration. The initial slope of the integrand in this connection formula is identified as a second-order energy and an explicit expression for the initial slope of the integrand is presented. Our expression should be useful for arriving at new improved approximations to E _c ^QC[n]. Previous numerical results by Huang and Umrigar (1997) Phys Rev A 56:290, for two-electron densities are proved, and a generalization to more than two electrons is presented. Results obtained by means of the present density functional theory correlation energy functionals, when used to approximate the initial slope in our adiabatic integration formula for E _c ^QC[n], are compared against exact numbers. Received: 10 September 1998 / Accepted: 3 February 1999 / Published online: 21 June 1999     

Perturbation calculation of the interaction energy using orthogonalized orbitals

The diagrammatic Rayleigh-Schr?dinger perturbation theory for the interaction of two closed-shell systems is developed up to the third order of pertur-bation using orthogonalized orbitals. The interaction energy is expressed by the Rayleigh-Schr?dinger perturbation expansion. A simple approach for the estimation of basis set superposition error is introduced. The preliminary calculations of the intermolecular interactions for the He dimer within the augmented cc-pVTZ basis set are compared with the supermolecular approach, perturbation calculation in biorthogonal basis sets and symmetry adapted perturbation theory results. Received: 17 December 1996 / Accepted: 5 November 1997     

Optimization of virtual orbitals in the framework of a multiconfiguration spin-coupled wave function

A new method is introduced for the optimization of nonorthogonal virtual orbitals for use in general multiconfiguration spin-coupled wave functions. The use of a number of highly effective approximations greatly reduces the computational effort involved, the most important being the use of a second-order perturbation expression for the energy and an approximate expression for the elements of the Hessian. As a result, the overall scheme scales very favourably with respect to the numbers of active electrons and of basis functions, making it suitable for the accurate study of large systems. Benchmark calculations are presented for the dissociation of LiH(X¹Σ⁺) and Li₂(X¹Σ⁺ _g ) using a highly compact four-configuration wave function. Standard spin-coupled valence bond expansions in the same virtual space are required to be significantly larger before equivalent results are obtained. The results are shown to compare very favourably with full valence complete active space self-consistent field calculations using an identical basis, and binding energies are within 4% of the values obtained from full configuration interaction calculations in the same basis set. Received: 10 June 1997 / Accepted: 7 October 1997     

Efficient electronic structure calculations for systems of one-dimensional periodicity with the restricted Hartree–Fock–linear combination of atomic orbitals method implemented in Fourier space

 Formulas are presented for restricted Hartree–Fock (RHF) calculations on systems with periodicity in one dimension using a basis set of contracted spherical Gaussians. Applying Fourier-space and Ewald-type methods, all lattice sums appearing in the formulation have been brought to forms exhibiting accelerated convergence. Calculations have been carried out for infinite chains of Li₂ molecules and a poly(oxymethylene) chain. The methods used here yield results that are far more precise than corresponding direct-space calculations and for the first time show the vanishing of the RHF density of states at the Fermi level for situations of partial band occupancy. Our initial computational implementation was about 5 times slower than the fastest direct-space RHF code, but improvements in special-function evaluations and numerical integrations over the Brillouin zone are shown to remove this disparity in computing speed. Received: 20 August 1999 / Accepted: 17 January 2000 / Published online: 5 June 2000     

Ab initio calculations with effective core potentials on trivalent lanthanide–terpyridine complexes

 The electronic structures of complexes of terpyridine (tpy) with trivalent lanthanides (Ln) were calculated using ab initio methods with effective core potentials at Hartree–Fock and post-Hartree–Fock levels of theory. The quasirelativistic large-core (with 4f electrons included in the core) pseudopotentials of the Stuttgart group were chosen for the Ln atoms. The variation of several properties of the Ln(tpy)³⁺ complexes was studied for the whole Ln series. It was shown that there was a monotonous variation for all properties (geometrical and energetic) along the Ln series, except for Mulliken charges on the metal atom. Calculations were performed on three complexes of known solid-state structure. The difference between experimental and calculated geometries is discussed; for all structures, it is found to be lower than 0.2 ?. In all cases, the relative order from one complex to another is conserved. Received: 13 September 1999 / Accepted: 3 February 2000 / Published online: 19 April 2000     

The infrared spectra of polycyclic aromatic hydrocarbons containing a five-membered ring: symmetry breaking and the B3LYP functional

The infrared spectra of six molecules, each of which contains a five-membered ring, and their cations are determined using density functional theory; both the B3LYP and BP86 functionals are used. The computed results are compared with the experimental spectra. For the neutral molecules, both methods are in good agreement with experiment. Even the Hartree–Fock (HF) approach is qualitatively correct for the neutral species. For the cations, the HF approach fails, as found for other organic ring systems. The B3LYP and BP86 approaches are in good mutual agreement for five of the six cation spectra, and are in good agreement with experiment for four of the five cations where the experimental spectra are available. It is only for the fluoranthene cation where the BP86 and B3LYP functionals yield different results; the BP86 approach yields the expected C _{2 v} symmetry, while the B3LYP approach breaks symmetry. The experimental spectra support the BP86 spectra over the B3LYP spectra, but the quality of the experimental spectra does not allow a critical evaluation of the accuracy of the BP86 approach for this difficult system. Received: 9 February 1999 / Accepted: 31 March 1999 / Published online: 14 July 1999     

Local-orbital-based correlated ab initio band structure calculations in insulating solids: LiF

 A local-orbital-based ab initio approach to calculate correlation effects on quasi-particle energies in insulating solids is presented. The use of localized Wannier-type Hartree–Fock orbitals allows correlation effects to be efficiently assessed. First a Green's function approach based on exact diagonalization is introduced and this is combined with an incremental scheme, while subsequently different levels of perturbative approximations are derived from the general procedure. With these methods the band structure of LiF is calculated and good agreement with experiment is found. By comparing the different approximations proposed, including the exact diagonalization procedure, their relative quality is established. Received: 25 June 2001 / Accepted: 31 August 2001 / Published online: 19 December 2001     

Assessment of the Van Voorhis–Scuseria exchange–correlation functional for predicting excitation energies using time-dependent density functional theory

We assess the performance of the Van Voorhis–Scuseria exchange–correlation functional (VSXC), a kinetic-energy-density-dependent exchange–correlation functional recently developed in our group, for calculating vertical excitation energies using time-dependent density functional theory in a benchmark set of molecules. Overall, VSXC performs very well, with accuracy similar to that of hybrid functionals such as the hybrid Perdew–Burke–Ernzerhof functional and Becke's three parameter hybrid method with the Lee, Yang, and Parr correlation functional, which contain a portion of Hartree–Fock exchange. Received: 29 December 1999 / Accepted: 5 June 2000 / Published online: 11 September 2000