Quantum similarity superposition algorithm (QSSA): a consistent scheme for molecular alignment and molecular similarity based on quantum chemistry

The use of the molecular quantum similarity overlap measure for molecular alignment is investigated. A new algorithm is presented, the quantum similarity superposition algorithm (QSSA), expressing the relative positions of two molecules in terms of mutual translation in three Cartesian directions and three Euler angles. The quantum similarity overlap is then used to optimize the mutual positions of the molecules. A comparison is made with TGSA, a topogeometrical approach, and the influence of differences on molecular clustering is discussed.     

Toward a global maximization of the molecular similarity function: Superposition of two molecules

A quantum similarity measure between two molecules is normally identified with the maximum value of the overlap of the corresponding molecular electron densities. The electron density overlap is a function of the mutual positioning of the compared molecules, requiring the measurement of similarity, a solution of a multiple-maxima problem. Collapsing the molecular electron densities into the nuclei provides the essential information toward a global maximization of the overlap similarity function, the maximization of which, in this limit case, appears to be related to the so-called assignment problem. Three levels of approach are then proposed for a global search scanning of the similarity function. In addition, atom—atom similarity Lorentzian potential functions are defined for a rapid completion of the function scanning. Performance is tested among these three levels of simplification and the Monte Carlo and simplex methods. Results reveal the present algorithms as accurate, rapid, and unbiased techniques for density-based molecular alignments. © 1997 by John Wiley & Sons, Inc. J Comput Chem 18: 826–846, 1997     

An efficient and accurate molecular alignment and docking technique using ab initio quality scoring

An accurate and efficient molecular alignment technique is presented based on first principle electronic structure calculations. This new scheme maximizes quantum similarity matrices in the relative orientation of the molecules and uses Fourier transform techniques for two purposes. First, building up the numerical representation of true ab initio electronic densities and their Coulomb potentials is accelerated by the previously described Fourier transform Coulomb method. Second, the Fourier convolution technique is applied for accelerating optimizations in the translational coordinates. In order to avoid any interpolation error, the necessary analytical formulas are derived for the transformation of the ab initio wavefunctions in rotational coordinates. The results of our first implementation for a small test set are analyzed in detail and compared with published results of the literature. A new way of refinement of existing shape based alignments is also proposed by using Fourier convolutions of ab initio or other approximate electron densities. This new alignment technique is generally applicable for overlap, Coulomb, kinetic energy, etc., quantum similarity measures and can be extended to a genuine docking solution with ab initio scoring.     

Second-order quantum similarity measures from intracule and extracule densities

The calculation of quantum similarity measures from second-order density functions contracted to intracule and extracule densities obtained at the Hartree-Fock level is presented and applied to a series of atoms, (He, Li, Be, and Ne), isoelectronic molecules (C₂H₂, HCN, CNH, CO, and N₂), and model hydrogen-transfer processes (H₂/H⁺, H₂/Hot, H₂/H⁻). Second-order quantum similarity measures and indices are found to be suitable measures for quantitatively analyzing electron-pair density reorganizations in atoms, molecules, and chemical processes. For the molecular series, a comparative analysis between the topology of pairwise similarity functions as computed from one-electron, intracule, and extracule densities is carried out and the assignment of each particular local similarity maximum to a molecular alignment discussed. In the comparative study of the three hydrogen-transfer reactions considered, second-order quantum similarity indices are found to be more sensitive than first-order indices for analyzing the electron-density reorganization between the reactant complex and the transition state, thus providing additional insights for a better understanding of the mechanistic aspects of each process. Received: 7 July 1997 / Accepted: 29 October 1997     

Quantum algorithm for obtaining the energy spectrum of molecular systems

Simulating a quantum system is more efficient on a quantum computer than on a classical computer. The time required for solving the Schr?dinger equation to obtain molecular energies has been demonstrated to scale polynomially with system size on a quantum computer, in contrast to the well-known result of exponential scaling on a classical computer. In this paper, we present a quantum algorithm to obtain the energy spectrum of molecular systems based on the multiconfigurational self-consistent field (MCSCF) wave function. By using a MCSCF wave function as the initial guess, the excited states are accessible. Entire potential energy surfaces of molecules can be studied more efficiently than if the simpler Hartree-Fock guess was employed. We show that a small increase of the MCSCF space can dramatically increase the success probability of the quantum algorithm, even in regions of the potential energy surface that are far from the equilibrium geometry. For the treatment of larger systems, a multi-reference configuration interaction approach is suggested. We demonstrate that such an algorithm can be used to obtain the energy spectrum of the water molecule.     

Molecular electronic density fitting using elementary Jacobi rotations under atomic shell approximation

Fitted electron density functions constitute an important step in quantum similarity studies. This fact not only is presented in the published papers concerning quantum similarity measures (QSM), but also can be associated with the success of the developed fitting algorithms. As has been demonstrated in previous work, electronic density can be accurately fitted using the atomic shell approximation (ASA). This methodology expresses electron density functions as a linear combination of spherical functions, with the constraint that expansion coefficients must be positive definite, to preserve the statistical meaning of the density function as a probability distribution. Recently, an algorithm based on the elementary Jacobi rotations (EJR) technique was proven as an efficient electron density fitting procedure. In the preceding studies, the EJR algorithm was employed to fit atomic density functions, and subsequently molecular electron density was built in a promolecular way as a simple sum of atomic densities. Following previously established computational developments, in this paper the fitting methodology is applied to molecular systems. Although the promolecular approach is sufficiently accurate for quantum QSPR studies, some molecular properties, such as electrostatic potentials, cannot be described using such a level of approximation. The purpose of the present contribution is to demonstrate that using the promolecular ASA density function as the starting point, it is possible to fit ASA-type functions easily to the ab initio molecular electron density. A comparative study of promolecular and molecular ASA density functions for a large set of molecules using a fitted 6-311G atomic basis set is presented, and some application examples are also discussed.     

Study of molecular quantum similarity of enantiomers of amino acids

Molecular quantum similarity is evaluated for enantiomers in the case of molecules showing conformational flexibility, using our earlier proposed Boltzmann weighted similarity index. The conformers of the enantiomers of the amino acids alanine, asparagine, cysteine, leucine, serine, and valine were examined. Next to studying global indices, the evaluation of local similarity is carried out using our earlier proposed local similarity index based on the Hirshfeld partitioning, to further illustrate Mezey's holographic electron density theorem in chiral systems and to quantify dissimilarity of enantiomers.     

Nanosecond photofragment imaging of adiabatic molecular alignment

Adiabatic alignment of CH(3)I, induced by the anisotropic interaction of this symmetric top molecule with the intense field of a nonresonant infrared laser pulse, has been studied using velocity map imaging. We are using photodissociation imaging with pulsed nanosecond lasers to probe the distribution of the molecular axis in the laboratory space. In contrast to the commonly used probing with femtosecond laser pulses, this technique directly yields the degree of alignment over an extended space-time volume. This will be relevant for future reactive scattering experiments with laser-aligned molecules. The obtained degree of alignment, (cos?(2)θ), measured as a function of the infrared laser intensity, agrees well with a quantum calculation for rotationally cold methyl iodide. The strong infrared laser is also found to modify the photofragmentation dynamics and open up pathways to CH(3)I(+) formation and subsequent fragmentation.     

Similarity calculations using two-dimensional molecular representations

Molecular similarity calculations are important for rational drug design. Time constraints prevent these techniques being used on large data sets or on large molecules. By reducing the molecular representation to a two-dimensional form, the alignment of the molecules can be greatly speeded up. The accuracy of the resulting similarity values can be improved by using a neural network.     

On the calculation of ab initio quantum molecular similarities for large systems: Fitting the electron density

A set of procedures for rapid calculation of quantum molecular similarities from ab initio wave functions is discussed. In all cases a density fitting is carried out to eliminate the need of calculating costly four-centered integrals. It is proved that this methodology can be applied to large systems to reproduce exact quantum molecular similarity measures at an extremely low computational cost. © 1994 by John Wiley & Sons, Inc.     

SHAFTS: a hybrid approach for 3D molecular similarity calculation. 1. Method and assessment of virtual screening

We developed a novel approach called SHAFTS (SHApe-FeaTure Similarity) for 3D molecular similarity calculation and ligand-based virtual screening. SHAFTS adopts a hybrid similarity metric combined with molecular shape and colored (labeled) chemistry groups annotated by pharmacophore features for 3D similarity calculation and ranking, which is designed to integrate the strength of pharmacophore matching and volumetric overlay approaches. A feature triplet hashing method is used for fast molecular alignment poses enumeration, and the optimal superposition between the target and the query molecules can be prioritized by calculating corresponding "hybrid similarities". SHAFTS is suitable for large-scale virtual screening with single or multiple bioactive compounds as the query "templates" regardless of whether corresponding experimentally determined conformations are available. Two public test sets (DUD and Jain's sets) including active and decoy molecules from a panel of useful drug targets were adopted to evaluate the virtual screening performance. SHAFTS outperformed several other widely used virtual screening methods in terms of enrichment of known active compounds as well as novel chemotypes, thereby indicating its robustness in hit compounds identification and potential of scaffold hopping in virtual screening.     

Estimation of molecular similarity based on 4D-QSAR analysis: formalism and validation

The 4D-QSAR paradigm has been used to develop a formalism to estimate molecular similarity measures as a function of conformation, alignment, and atom type. It is possible to estimate the molecular similarity of pairs of molecules based upon the entire ensemble of conformational states each molecule can adopt at a given temperature, normally room temperature. Molecular similarity can be measured in terms of the types of atoms composing each molecule leading to multiple measures of molecular similarity. Multiple measures of molecular similarity can also arise from using different alignment rules to perform relative molecular similarity, RMS, analysis. An alignment independent method of determining molecular similarity measures, referred to as absolute molecular similarity, AMS, analysis has been developed. Various sets and libraries of compounds, including the amino acids, are analyzed using 4D-QSAR molecular similarity analysis to demonstrate the features of the formalism. Exploration of molecular similarity as a function of chirality, identification of common embedded 3D pharmacophores in compounds, and elucidation of 3D-isosteric compounds from structurally diverse libraries are carried out in the application studies.     

Subshell fitting of relativistic atomic core electron densities for use in QTAIM analyses of ECP-based wave functions

Scalar-relativistic, all-electron density functional theory (DFT) calculations were done for free, neutral atoms of all elements of the periodic table using the universal Gaussian basis set. Each core, closed-subshell contribution to a total atomic electron density distribution was separately fitted to a spherical electron density function: a linear combination of s-type Gaussian functions. The resulting core subshell electron densities are useful for systematically and compactly approximating total core electron densities of atoms in molecules, for any atomic core defined in terms of closed subshells. When used to augment the electron density from a wave function based on a calculation using effective core potentials (ECPs) in the Hamiltonian, the atomic core electron densities are sufficient to restore the otherwise-absent electron density maxima at the nuclear positions and eliminate spurious critical points in the neighborhood of the atom, thus enabling quantum theory of atoms in molecules (QTAIM) analyses to be done in the neighborhoods of atoms for which ECPs were used. Comparison of results from QTAIM analyses with all-electron, relativistic and nonrelativistic molecular wave functions validates the use of the atomic core electron densities for augmenting electron densities from ECP-based wave functions. For an atom in a molecule for which a small-core or medium-core ECPs is used, simply representing the core using a simplistic, tightly localized electron density function is actually sufficient to obtain a correct electron density topology and perform QTAIM analyses to obtain at least semiquantitatively meaningful results, but this is often not true when a large-core ECP is used. Comparison of QTAIM results from augmenting ECP-based molecular wave functions with the realistic atomic core electron densities presented here versus augmenting with the limiting case of tight core densities may be useful for diagnosing the reliability of large-core ECP models in particular cases. For molecules containing atoms of any elements of the periodic table, the production of extended wave function files that include the appropriate atomic core densities for ECP-based calculations, and the use of these wave functions for QTAIM analyses, has been automated.     

Quantum molecular similarity via momentum‐space indices

A method for assessing quantum molecular similarity has been developed based on the momentum‐space representation. The principles of the method are summarized and the results of two applications are presented. The approach emphasizes the variation of the outer valence electron density much more than the bonding topology, it avoids several problems associated with more conventional approaches based on the position‐space representation, and it can now be applied to extended series of large molecules. The first of the applications involves different sulfonylurea inhibitors of the enzyme acyl‐CoA : cholesterol acyltransferase (ACAT). The second is concerned with the relative toxicity of a number of anti‐HIV phospholipids. Further experimental work is suggested in both cases. This revised version was published online in July 2006 with corrections to the Cover Date.     

Further evaluation and development of charge similarity indices for choosing molecular analogues

Two molecular charge similarity index (CSI ) methods are further evaluated for practical application: one method based on a simple CNDO -type approximation to the electron density function and the other based on an ab initio pseudo total charge density function. The test system consists of isosteric analogues of dimethyl ether and methoxy acetic acid. The effects of differences in skeletal structure on the CSI measure of electron density similarity about corresponding atoms is estimated, and two new developments are presented for application of the ab initio-based method: (1) an INDO -type approximation which improves the efficiency of the CSI calculation; and (2) a FOCUS feature which enables comparisons of local molecule regions.     

Alignment of molecules by the Monte Carlo optimization of molecular similarity indices

3D-QSAR uses statistical techniques to correlate calculated structural properties with target properties like biological activity. The comparison of calculated structural properties is dependent upon the relative orientations of molecules in a given data set. Typically molecules are aligned by performing an overlap of common structural units. This “alignment rule” is adequate for a data set, that is closely related structurally, but is far more difficult to apply to either a diverse data set or on the basis of some structural property other than shape, even for sterically similar molecules. In this work we describe a new algorithm for molecular alignment based upon optimization of molecular similarity indices. We show that this Monte Carlo based algorithm is more effective and robust than other optimizers applied previously to the similarity based alignment problem. We show that QSARs derived using the alignments generated by our algorithm are superior to QSARs derived using the more common alignment of fitting of common structural units. © 1997 by John Wiley & Sons, Inc. J Comput Chem 18 : 1344–1353, 1997