Local Correlation Methods Research Articles

Symmetry-adapted Localized Wannier Functions Suitable for Periodic Local Correlation Methods

A scheme for the a posteriori symmetrization of a set of localized Wannier functions is presented and illustrated with some examples. The method has been implemented in the periodic code CRYSTAL using the LCAO approach and it is shown that they can be useful in the computational implementation of periodic local correlation methods. The resulting functions feature very accurate symmetry correspondences without a significant loss of the original localization properties.

Nonorthogonal ultralocalized functions and fitted Wannier functions for local electron correlation methods for solids

Despite the formal exponential decay behavior of Wannier functions (WFs), their spatial extent, which is a key parameter determining the computational cost of local correlation calculations for solids, is still rather large. The problems with the localization of the WFs can partly be attributed to their mutual orthogonality. Possibilities of reduction of the spatial extent of the WFs without losing the accuracy of the calculations are investigated. A method for generation of nonorthogonal ultralocalized functions based on maximization of their Lowdin populations is developed. A scheme for fitting of the WFs and nonorthogonal localized functions with a limited support is proposed. The calculations show that by combining both techniques one can obtain quite compact linearly independent localized functions, which may significantly decrease the computational cost in post-HF calculations.

Implicit infinite lattice summations for real space ab initio correlation methods

We suggest a local wave function-based ab initio correlation method for infinite periodic systems, which can describe both the near-range as well as the long-range correlation effects coherently in the same scheme. Specifically, this work introduces a formalism which allows to describe the long-range polarization cloud around a quasi particle in a solid explicitly in the formalism of local wave function-based ab initio descriptions. To this end we reformulate the infinite lattice summation underlying the quantum chemistry formula to second order in a closed analytic form employing the elliptic theta function of the third kind. All formulas and manipulations are developed explicitly in full detail and a first numeric example demonstrates the principle idea. Good results for the long-range polarization effects in LiH and LiF are found in agreement with earlier estimates.

Fast localized orthonormal virtual orbitals which depend smoothly on nuclear coordinates

We present here an algorithm for computing stable, well-defined localized orthonormal virtual orbitals which depend smoothly on nuclear coordinates. The algorithm is very fast, limited only by diagonalization of two matrices with dimension the size of the number of virtual orbitals. Furthermore, we require no more than quadratic (in the number of electrons) storage. The basic premise behind our algorithm is that one can decompose any given atomic-orbital (AO) vector space as a minimal basis space (which includes the occupied and valence virtual spaces) and a hard-virtual (HV) space (which includes everything else). The valence virtual space localizes easily with standard methods, while the hard-virtual space is constructed to be atom centered and automatically local. The orbitals presented here may be computed almost as quickly as projecting the AO basis onto the virtual space and are almost as local (according to orbital variance), while our orbitals are orthonormal (rather than redundant and nonorthogonal). We expect this algorithm to find use in local-correlation methods.

Fast electronic structure methods for strongly correlated molecular systems

A short review is given of newly developed fast electronic structure methods that are designed to treat molecular systems with strong electron correlations, such as diradicaloid molecules, for which standard electronic structure methods such as density functional theory are inadequate. These new local correlation methods are based on coupled cluster theory within a perfect pairing active space, containing either a linear or quadratic number of pair correlation amplitudes, to yield the perfect pairing (PP) and imperfect pairing (IP) models. This reduces the scaling of the coupled cluster iterations to no worse than cubic, relative to the sixth power dependence of the usual (untruncated) coupled cluster doubles model. A second order perturbation correction, PP(2), to treat the neglected (weaker) correlations is formulated for the PP model. To ensure minimal prefactors, in addition to favorable size-scaling, highly efficient implementations of PP, IP and PP(2) have been completed, using auxiliary basis expansions. This yields speedups of almost an order of magnitude over the best alternatives using 4-center 2-electron integrals. A short discussion of the scope of accessible chemical applications is given.

How mutual geometrical distortions affect the possibility of spatially combining images by the local-correlation method*

This paper discusses how the mutual spatial transformation of local segments of images affects the result of their phase correlation. The equations obtained for the noise of the cross-correlation field caused by scale mismatch of segments of the images or their relative rotation make it possible to estimate the allowable values of the coefficients of a mutual affine transformation for which the relative displacement of the segments can be reliably determined from the correlation peak. An estimate is given of the accuracy with which this displacement is determined. Based on this analysis, filtering of the spectra of the images is proposed that makes it possible to broaden the allowable range of parameters of the mutual spatial transformation. The results were applied in practice in the local-correlation method described earlier. © 2004 Optical Society of America

Potential energy surface discontinuities in local correlation methods.

We have examined the occurrence of discontinuities in bond-breaking potential energy surfaces given by local correlation methods based on the Pulay-Saebø orbital domain approach. Our analysis focuses on three prototypical dissociating systems: the C-F bond in fluoromethane, the C-C bond in singlet, ketene, and the central C-C bond in propadienone. We find that such discontinuities do not occur in cases of homolytic bond cleavage due to the inability of the Pipek-Mezey orbital localization method to separate singlet-coupled charges on distant fragments. However, for heterolytic bond cleavage, such as that observed in singlet ketene and propadienone, discontinuities occur both at stretched geometries and near equilibrium. These discontinuities are usually small, but may be of the same order of magnitude as the localization error in some cases.

Bringing aerospace images into coincidence with subpixel accuracy by the local-correlation method

This paper proposes a local-correlation method that makes it possible to bring aerospace images into coincidence with subpixel accuracy after preliminary rough juxtaposition by transforming them relative to each other by uniform projective transformation and by additional mutual local displacements. The basis of the method is to establish a correspondence between the points of a pair of images by means of a phase correlation of individual segments of the images. The dissemination of information concerning measured shifts from reference points for which the correspondence has been found on a pair of images, as well as the use of analysis with variable spatial resolution, makes the method workable when the errors of the preliminary superposition are greater than the size of the correlation window. This paper presents the results of an experimental verification of the approach, using actual pairs of aerospace images as an example.© 2004 Optical Society of America

The effect of local approximations in coupled-cluster wave functions on dipole moments and static dipole polarisabilities

The influence of local approximations in electron correlation treatments on electric dipole moments and static dipole polarisabilities is examined for a test set of 16 molecules for the case of coupled-cluster singles and doubles (CCSD) theory. Utilising standard local approximations as originally proposed by Pulay and used in our previous work, the average errors relative to the corresponding conventional CCSD calculations amount to 0.48% for mean dipole polarisabilities and to 1.61% for dipole moments. The accuracy of the computed mean dipole polarisabilities can be reduced to 0.32% by extending the domains of the strong pairs, while for the dipole moments the domain extension reduces the error to 0.91%. It is found that orbital relaxation effects are much more important in the local than in the conventional case. Weak pairs contribute substantially to the computed dipole polarisabilities, but the effect of distant pairs is small and can be neglected. This means that linear scaling local electron correlation methods can be faithfully used to compute these molecular properties.

Singular value decomposition approach for the approximate coupled-cluster method

We present a method for the approximate solution of the coupled-cluster (CC) equation, based upon the singular value decomposition (SVD). The key idea of this method is that we use SVD for the cluster amplitudes to exploit the physically important states in the CC equation. This method enables us to significantly reduce the computational requirements for CC calculations without losing the essence of the method. Relationships to the density matrix renormalization group theory and the local correlation methods are mentioned. We perform pilot calculations on some atoms and molecules to investigate the applicability of the method.

Local correlation in the virtual space in multireference singles and doubles configuration interaction

We describe a multireference configuration interaction method that takes advantage of local correlation methods in both the internal (originally occupied) and external (originally unoccupied or virtual) orbital spaces. In the internal space, implementation of local correlation is trivial and involves neglecting configurations having simultaneous excitations out of widely separated orbitals. In the external space, the method involves restricting the space of allowed correlating orbitals to those localized near the hole orbitals. Of course, this necessitates the use of localized virtual orbitals which in turn requires one to sacrifice the orthogonality of the virtual space. This complicates the formalism substantially, and we discuss the necessary changes to the traditional expressions in detail. The scaling of the method with system size, basis set size, and the average number of allowed virtual orbitals is explored. An examination of systems having up to 8 heavy atoms reveals that the computational costs of the method scales somewhere between the third and fourth power of the size of the system. Furthermore, this reduced scaling method is capable of recovering greater than 97% of the correlation energy. Additionally, we demonstrate that the method can produce smooth potential energy surfaces and recover bond dissociation energies in organic molecules at a fraction of the cost (⩾tenfold less expensive) while retaining accuracy. We go on to use this new reduced scaling approach to predict bond energies in several large organic molecules for which no experimental data are available.

The vibrational spectra of furoxan and dichlorofuroxan: A comparative theoretical study using density functional theory and local electron correlation methods

The vibrational spectra of furoxan and dichlorofuroxan have been studied using the local quadratic configuration interaction method including single and double excitations (LQCISD) and two density exchange-correlation functionals (B3LYP and B97r). The vibrational spectra of these molecules are very sensitive to electron correlation effects and are therefore well suited as benchmark systems for investigating the impact of local approximations at levels beyond second-order Moller–Plesset perturbation theory (MP2). Effects due to anharmonicity, basis set deficiencies, and lacking electron-correlation contributions are accounted for by scaling the force constants in internal coordinates with empirical factors, using the the scaled quantum mechanical (SQM) force fields procedure. For comparison, also the correlation-corrected vibrational SCF approach (cc-VSCF) has been applied to account for anharmonicity effects. Using the DFT and LQCISD results the experimental vibrational spectra are analyzed and assigned. The impact of the local approximations in the LQCISD method on the vibrational spectra has also been tested for related heterocyclic systems. As found earlier for local MP2, the effect of these approximations is in general small.

Local techniques for the ab initio quantum-mechanical description of the chemical properties of crystalline materials

Very reliable computational tools are presently available for the ab initio quantum-mechanical treatment of crystalline systems. In order to exploit translational periodicity, they adopt as a rule one-electron Hamiltonians, either based on density functional theory (DFT), or on the Hartree–Fock (HF) equations as a zero-level approximation for the solution of the Schrödinger equation. As concerns the former approach, it is almost straightforward to implement DFT at the same level of accuracy in crystalline as in molecular applications. On the contrary, the problem is still open of how to go beyond the HF approximation in the case of periodic systems, while a variety of powerful post-HF schemes have been developed and are currently used in the field of molecular quantum chemistry. A way out of this difficulty could now be offered by the implementation of efficient techniques of localization of the HF manifold of crystals. We describe here briefly one of such techniques which has recently been developed to generate well-localized Wannier functions (WF) for non-metallic periodic structures, starting from canonical crystalline orbitals in an atomic orbital representation. WFs are next shown to be ideally suited for transferring to the periodic case the local-correlation methods developed by Pulay, Werner et al. for molecules. Finally, it is proposed to use WFs for formulating new embedding schemes which could allow both the HF and the post-HF solution of the problem of local defects in crystals to be obtained in a simpler way than presently possible.

Particle finding in electron micrographs using a fast local correlation algorithm

A versatile tool for selecting particles from electron micrographs, intended for single particle analysis and three-dimensional reconstruction, is presented. It is based on a local real-space correlation method. Real-space correlations calculated over a local area are suitable for finding small objects or patterns in a larger field. They provide a very sensitive measure-of-fit, partly due to local optimisation of the numerical scaling. It is equivalent to least squares with optimised scaling between the two objects being correlated. The only disadvantage of real-space methods is that they are slow to compute. A fast local correlation algorithm based on Fourier transforms has been developed, which is approximately two orders of magnitude faster than the explicit real-space formulation. The algorithm is demonstrated by application to the problem of locating images of macromolecules in transmission electron micrographs of unstained frozen hydrated specimens. This is a challenging computational problem because these images have low contrast and a low signal-to-noise ratio. Picking particles by hand is very time consuming and can be less accurate. The automated procedure gives a significant increase in speed, which is important if large numbers of particles have to be picked.

Local weak-pairs pseudospectral multireference configuration interaction

We present a new reduced scaling multireference singles and doubles configuration interaction (MRSDCI) algorithm based upon the combination of local correlation and pseudospectral methods. Taking advantage of the locality of the Coulomb potential, the weak-pairs approximation of Saebo/ and Pulay is employed to eliminate configurations having simultaneous excitations out of pairs of distant, weakly interacting orbitals. In conjunction with this, the pseudospectral approximation is used to break down the most time-consuming two-electron integrals into a product of intermediate quantities depending on no more than two orbital indices. The resulting intermediate quantities are then used directly in the CI equations to substantially reduce the number of floating point operations required for diagonalization of the Hamiltonian. Additionally, our CI algorithm is based upon the symmetric group graphical approach CI (SGGA-CI) of Duch and Karwowski. For the purpose of developing reduced scaling CI algorithms, this approach has some important advantages. The most important of these advantages are the on-the-fly calculation of integral coupling coefficients and the separation of the spin and spatial parts of the wave function, which simplifies implementation of local correlation approximations. We apply the method to determine a series of binding energies in hydrocarbons and show that the approximate method predicts binding energies that are within a few kcal/mol of those predicted by the analytic nonlocal method. For large molecules, the local pseudospectral method was shown to be over 7 times as fast as the analytic nonlocal method. We also carry out a systematic study on the performance of different basis sets in the weak-pairs method. It was determined that triple-ζ basis sets were capable of recovering only 99.0% of the correlation energy, whereas double-ζ basis sets recovered 99.9% of the correlation energy.

Low-order scaling local electron correlation methods. V. Connected triples beyond (T): Linear scaling local CCSDT-1b

A new O(N ) method for the iterative treatment of connected triple substitutions in the framework of local coupled cluster theory is introduced here, which is the local equivalent of the canonical CCSDT-1b method. The effect of the triple substitutions is treated in a self-consistent manner in each coupled cluster iteration. As for the local (T) method presented earlier in this series the computational cost of the method scales asymptotically linear with molecular size. The additional computational burden due to the involvement of triples in each coupled cluster iteration hence is not nearly as dramatic as for the parental canonical method, where it implies an increase in the computational complexity of the coupled cluster iteration from O(N6) to O(N7). The method has certain advantages in comparison to the perturbative a posteriori treatment of connected triples (T) for cases where static correlation effects start to play a role. It is demonstrated that molecules with about 100 atoms and 1000 basis functions can be treated with the local CCSDT-1b method, i.e., at a level beyond local CCSD(T). The new local coupled cluster methods introduced here and in previous papers of this series are applied in a study on the energetics of the Bergman auto-cyclization and retro-Bergman ring opening of an azaenediyne derivate, which was recently proposed as a promising candidate for anti-cancer drug development.

New challenges in quantum chemistry: Quests for accurate calculations for large molecular systems.

As quantum chemistry plays a more and more central role in many complicated chemical problems, it has become necessary to obtain accurate results for large molecular systems. Conventional quantum chemistry methods are either too expensive to apply to large systems or too approximate for the results to be reliable, and they fail to satisfy this requirement. A variety of different approaches is being developed with the aim of achieving this goal: local correlation methods; divide-and-conquer methods; linear-scaling density functional methods based on the fast multipole and other approximations; effective potential methods; and hybrid methods. ONIOM (our N-layered integrated molecular orbital plus molecular mechanics method), developed by the authors, is a hybrid method in which a large molecular system is divided into onion-skin-like layers, and different quantum chemistry/molecular mechanics methods are used for different parts of the system; the results are combined to extrapolatively estimate the results of high-level calculation for the real system. Several applications of ONIOM will be discussed.

Extracting polarized atomic orbitals from molecular orbital calculations

We present a class of methods for extracting a polarized atomic orbital (EPAO) minimal basis set from a converged molecular orbital (MO) calculation. Unlike minimal basis sets obtained from previous approaches, EPAOs rigorously contain the occupied molecular orbital space. EPAOs achieve this exactness because their spatial extent is not restricted. Nonetheless, EPAOs are optimally localized with respect to a localization criterion and are essentially single-centered. EPAOs provide an alternative scheme for partitioning the electron density into atomic subspaces. Therefore, they can be used to determine atomic and chemical group properties such as charge populations. Since EPAOs provide a compact description to the occupied space, they may have other computational applications such as in local correlation methods. Additionally, the EPAOs give a description of valence antibonding orbitals that may be appropriate for nondynamical electron correlation. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 76: 169–184, 2000

The aurophilic attraction as interpreted by local correlation methods

The nature of the aurophilic interaction is studied by applying local second-order Mo/ller–Plesset perturbation theory (LMP2) on model dimers of [X–Au–PH3]2 (X=H, Cl) type. The possibility to decompose the correlation contribution of the interaction energy in the dimer (A–B) into different excitation classes reveals that the dispersion contribution (A→A′,B→B′) is accompanied by an almost equally important ionic component (A→A′,B→A′), at shorter distances. Double excitations where at least one electron originates from the gold 5d orbitals account for almost 90% of the attraction. Relativistic effects amount to 28% of the binding energy and can be traced to arise almost exclusively from the relativistic expansion of gold d-shells.

Noniterative local second order Mo/ller–Plesset theory: Convergence with local correlation space

We extend our noniterative local correlation method [P. E. Maslen and M. Head-Gordon, Chem. Phys. Lett., 283, 102 (1998)] by defining a hierarchy of local spaces, ranging from small to large. The accuracy of the local method is then examined as a function of the size of the local space. A medium size local space recovers 98% of the MP2 correlation energy, and reproduces fine details of the potential energy surface such as rotational barriers with an RMS error of 0.2 kcal/mol and a maximum error of 0.4 kcal/mol. A large local space recovers 99.5% of the correlation energy and yields rotational barriers with a RMS error of 0.05 kcal/mol and a maximum error of 0.1 kcal/mol, at significantly increased computational cost.