Description of functional groups by means of domain-restricted reduced density matrices

This report constitutes an application of our previous theoretical works on partitionings of the first-order reduced density matrix according to the atomic domains defined in the theory of atoms in molecules. The numerical determinations obtained reveal that the domain-restricted reduced density matrices, which are the tools resulting from the former treatments, are suitable devices to describe chemical features of molecular fragments. We have focused attention on a study of functional groups in several series of organic compounds confirming the usefulness of these tools.     

Kohn-Sham Density Matrix and the Kernel Energy MethodSCI北大核心CSCD

The kernel energy method(KEM) has been shown to provide fast and accurate molecular energy calculations for molecules at their equilibrium geometries.KEM breaks a molecule into smaller subsets,called kernels,for the purposes of calculation.The results from the kernels are summed according to an expression characteristic of KEM to obtain the full molecule energy.A generalization of the kernel expansion to density matrices provides the full molecule density matrix and orbitals.In this study,the kernel expansion for the density matrix is examined in the context of density functional theory(DFT) Kohn-Sham(KS) calculations.A kernel expansion for the one-body density matrix analogous to the kernel expansion for energy is defined,and is then converted into a normalizedprojector by using the Clinton algorithm.Such normalized projectors are factorizable into linear combination of atomic orbitals(LCAO) matrices that deliver full-molecule Kohn-Sham molecular orbitals in the atomic orbital basis.Both straightforward KEM energies and energies from a normalized,idempotent density matrix obtained from a density matrix kernel expansion to which the Clinton algorithm has been applied are compared to reference energies obtained from calculations on the full system without any kernel expansion.Calculations were performed both for a simple proof-of-concept system consisting of three atoms in a linear configuration and for a water cluster consisting of twelve water molecules.In the case of the proof-of-concept system,calculations were performed using the STO-3 G and6-31 G(d,p) bases over a range of atomic separations,some very far from equilibrium.The water cluster was calculated in the 6-31 G(d,p) basis at an equilibrium geometry.The normalized projector density energies are more accurate than the straightforward KEM energy results in nearly all cases.In the case of the water cluster,the energy of the normalized projector is approximately four times more accurate than the straightforward KEM energy result.The KS density matrices of this study are applicable to quantum crystallography.     

Decomposition of the first-order reduced density matrix: an isopycnic localization treatment

In this work, we propose a partitioning of the first-order reduced density matrix corresponding to an N-electron system into first-order reduced density matrices associated with regions defined in the real space (regional matrices). The treatment is based on an isopycnic orbital localization transformation that provides regional matrices that are diagonalized by identical localized orbitals, having many attributes associated with chemical concepts (appropriate localization in space, high transferability, etc.). Although the obtained numerical values are similar to those arising from previous studies, their interpretation is more rigorous and the computational cost is much lower.     

Performance of the density matrix functional theory in the quantum theory of atoms in molecules

The generalization to arbitrary molecular geometries of the energetic partitioning provided by the atomic virial theorem of the quantum theory of atoms in molecules (QTAIM) leads to an exact and chemically intuitive energy partitioning scheme, the interacting quantum atoms (IQA) approach, that depends on the availability of second-order reduced density matrices (2-RDMs). This work explores the performance of this approach in particular and of the QTAIM in general with approximate 2-RDMs obtained from the density matrix functional theory (DMFT), which rests on the natural expansion (natural orbitals and their corresponding occupation numbers) of the first-order reduced density matrix (1-RDM). A number of these functionals have been implemented in the promolden code and used to perform QTAIM and IQA analyses on several representative molecules and model chemical reactions. Total energies, covalent intra- and interbasin exchange-correlation interactions, as well as localization and delocalization indices have been determined with these functionals from 1-RDMs obtained at different levels of theory. Results are compared to the values computed from the exact 2-RDMs, whenever possible.     

Determination of energies and electronic densities of functional groups according to partitionings in the physical space

This work formulates our previously reported partitionings of the first-order reduced density matrix and the molecular electronic energy using for both quantities an identical mathematical framework. The procedure provides a consistent and rigorous scheme for extending our algorithms to unions of atomic domains, in order to describe molecular fragments which can be identified as functional groups. Numerical determinations, performed in several series of organic compounds and clusters, support the reliability of our methodology to describe properties of atomic groups.     

Application of minimization methods in calculating atoms with several open shells

Basing on minimization methods, earlier suggested algorithms for the solution of a many-electron problem in Hartree-Fock-Roothaan approximation for systems with close and open shells extend over Roothaan-Hartree-Fock atomic theory (Roothaan-Bagus method). In present work the expressions for energy derivatives with respect to elements of density matrices and nonlinear parameters of atomic orbitals — orbital exponents — have been obtained to solve Hartree-Fock (HF) equations in algebraic approximation. It is possible to create an algorithm of the first-order minimization or quasi-Newton method on their basis. Calculations of atoms and ions with several open shells were carried out by minimization methods. The energy values, close to the results of numerical solution of HF equations with high accuracy of virial relation, were gained using a sufficiently narrow basic set of Slater-type AO.     

Local and nonlocal contributions to molecular first-order hyperpolarizability: a Hirshfeld partitioning analysis

Based on first-principles calculations, a decomposition scheme is proposed to investigate the molecular site-specific first-order hyperpolarizability (β) responses by means of Hirshfeld population analysis and finite field method. For a molecule, its β is decomposed into local and nonlocal contributions of individual atoms or groups. The former describes the response within the atomic sphere, while the latter describes the contributions from interatomic charge transfer. This scheme is then applied to six prototypical donor-acceptor (D-A) or D-π-A molecules for which the local and nonlocal hyperpolarizabilities are evaluated based on their MP2 density. Both the local and nonlocal parts exhibit site-specific characteristics, but vary differently with molecular structures. The local part depends mainly on the atomic attributes such as electronegativity and charge state, as well as its location in the molecule, while the nonlocal part relates to the ability and distance of charge delocalization within the molecule, increasing rapidly with molecular size. The proposed decomposition scheme provides a way to distinguish atomic or group contributions to molecular hyperpolarizabilities, which is useful in the molecular design for organic nonlinear optical materials.     

Communication: Hilbert-space partitioning of the molecular one-electron density matrix with orthogonal projectors

A double-atom partitioning of the molecular one-electron density matrix is used to describe atoms and bonds. All calculations are performed in Hilbert space. The concept of atomic weight functions (familiar from Hirshfeld analysis of the electron density) is extended to atomic weight matrices. These are constructed to be orthogonal projection operators on atomic subspaces, which has significant advantages in the interpretation of the bond contributions. In close analogy to the iterative Hirshfeld procedure, self-consistency is built in at the level of atomic charges and occupancies. The method is applied to a test set of about 67 molecules, representing various types of chemical binding. A close correlation is observed between the atomic charges and the Hirshfeld-I atomic charges.     

Two-dimensional facets in Langmuir monolayers of 1-O-hexadecyl-rac-glycerol at the air-water interface

We study the surface phase behavior in Langmuir monolayers of 1-O-hexadecyl-rac-glycerol (C16G) by film balance and Brewster angle microscopy over a wide range of temperatures. A cusp point followed by a pronounced plateau region in the pressure-area (pi-A) isotherm indicates a first-order phase transition between a lower density liquid expanded (LE) phase and a higher density liquid condensed (LC) phase at the air-water interface. A wide variety of condensed domains are found to form just after the appearance of the cusp point. The observed surface morphology was compared with that of ethylene glycol mono-n-hexadecyl ether (C16E1) that bears an ethylene oxide (EO) unit in the head-group. As usually observed, the domains of C16E1 are found to be circular at lower temperatures and fractal at higher temperatures. Contrary to this usual behavior, the domains of C16G are found to be strip-like structures at lower temperatures, which attain increasingly compact shape as the temperature increases and finally attain faceted structures at > or = 25 degrees C. It is concluded that a higher degree of dehydration around the head-group region of C16G appreciably reduces the hydration-induced repulsive interactions between the head-groups and imparts to the molecules an increase in hydrophobicity, thereby a closer molecular packing. As a result, the molecules form increasingly compact domains as the temperature increases. Since the head-group of C16E1 is much smaller than that of C16G, dehydration effect cannot appreciably increase its hydrophobic character. Rather, increases in subphase temperature result in a decrease in the line tension of the interface giving fractal structures at higher temperatures. In addition, the changes in enthalpy (deltaH) and entropy (deltaS) values were also calculated to understand the thermodynamic nature of condensation of the molecules in the LE-LC transition region.     

Surface phase behavior of di-n-tetradecyl hydrogen phosphate in Langmuir monolayers at the air-water interface

Surface phase behavior of di-n-tetradecyl hydrogen phosphate, DTP, has been studied by measuring pi-A isotherms with a film balance and observing monolayer morphology with a Brewster angle microscopy (BAM) at different temperatures. A generalized phase diagram, which shows a triple point for gas (G), liquid-expanded (LE) and liquid-condensed (LC) phases at about 32 degrees C, is constructed for the amphiphile. Below the triple point, a first-order G-LC phase transition has been shown to occur, whereas a first-order G-LE phase transition followed by another first-order LE-LC transition has been found to take place at a temperature above the triple point. The amphiphile shows the fingering LC domains with uniform brightness indicating the presence of untilted molecules. The domain shapes are independent of the change in temperature and compression rate. The existence of similar fingering domains over a wide range of temperature is rather uncommon in the monolayer systems and is considered to be due to the restricted movement of the molecules incorporating into the LC phase. Because the two-alkyl chains are directly attached to two covalent bonds of the phosphate head group, the rearrangement of the molecules, which is an essential condition for the circular domain formation, needs the movement of the whole molecules including the hydration sphere. The difficulty related to such a movement of the molecules causes fingering domains, which are independent of external variables.     

Second-order atomic Fukui indices from the electron-pair density in the framework of the atoms in molecules theory

Atomic Fukui indices, which are obtained from the electron density, have been previously shown to be useful in predicting which atoms in a molecule are most likely to suffer nucleophilic, electrophilic, or radicalary attacks. Here, we present a second-order generalization of these indices based on the electron pair density. We show how second-order atomic Fukui indices can be used to analyze the effects of electron loss or gain in several molecules from an electron pair point of view. Further, these indices also highlight which atoms or pairs of atoms are more likely to suffer nucleophilic, electrophilic, or radical attacks. In conclusion, second-order indices can complement first-order ones by affording relevant information on molecular reactivity from an electron pair perspective.     

Atomic charges,dipole moments,and Fukui functions using the Hirshfeld partitioning of the electron density

In the Hirshfeld partitioning of the electron density, the molecular electron density is decomposed in atomic contributions, proportional to the weight of the isolated atom density in the promolecule density, constructed by superimposing the isolated atom electron densities placed on the positions the atoms have in the molecule. A maximal conservation of the information of the isolated atoms in the atoms-in-molecules is thereby secured. Atomic charges, atomic dipole moments, and Fukui functions resulting from the Hirshfeld partitioning of the electron density are computed for a large series of molecules. In a representative set of organic and hypervalent molecules, they are compared with other commonly used population analysis methods. The expected bond polarities are recovered, but the charges are much smaller compared to other methods. Condensed Fukui functions for a large number of molecules, undergoing an electrophilic or a nucleophilic attack, are computed and compared with the HOMO and LUMO densities, integrated over the Hirshfeld atoms in molecules.     

First-order semidefinite programming for the direct determination of two-electron reduced density matrices with application to many-electron atoms and molecules

Direct variational calculation of two-electron reduced density matrices (2-RDMs) for many-electron atoms and molecules in nonminimal basis sets has recently been achieved through the use of first-order semidefinite programming [D. A. Mazziotti, Phys. Rev. Lett. (in press)]. With semidefinite programming, the electronic ground-state energy of a molecule is minimized with respect to the 2-RDM subject to N-representability constraints known as positivity conditions. Here we present a detailed account of the first-order algorithm for semidefinite programming and its comparison with the primal-dual interior-point algorithms employed in earlier variational 2-RDM calculations. The first-order semidefinite-programming algorithm, computations show, offers an orders-of-magnitude reduction in floating-point operations and storage in comparison with previous implementations. We also examine the ability of the positivity conditions to treat strong correlation and multireference effects through an analysis of the Hamiltonians for which the conditions are exact. Calculations are performed in nonminimal basis sets for a variety of atoms and molecules and the potential-energy curves for CO and H(2)O.     

Localized orbitals for optimal decomposition of molecular properties

We present the procedure for transforming delocalized molecular orbitals into the localized property-optimized orbitals (LPOs) designed for building the most accurate, in the Frobenius norm sense, approximation to the first-order reduced density matrix in form of the sum of localized monoatomic and diatomic terms. In this way, a decomposition of molecular properties into contributions associated with individual atoms and the pairs of atoms is obtained with the a priori known upper bound for the decomposition accuracy. Additional algorithm is proposed for obtaining the set of “the Chemist's LPOs” (CLPOs) containing a single localized orbital, with nearly double occupancy, per a pair of electrons. CLPOs form an idealized Lewis structure optimized for the closest possible reproduction of one-electron properties derived from the original many-electron wavefunction. The computational algorithms for constructing LPOs and CLPOs from a general wavefunction are presented and their implementation within the open-source freeware program JANPA ( http://janpa.sourceforge.net /) is discussed. The performance of the proposed procedures is assessed using the test set of density matrices of 33 432 small molecules obtained at both Hartree-Fock and second-order Moller-Plesset theory levels and excellent agreement with the chemist's Lewis-structure picture is found.     

A comparison of geometric methods with other methods for the characterization of density matrices

Some recently developed geometric methods for characterizing the subset of density matrices within the space of Hermitian matrices are compared with methods commonly used for the approximate characterization of reduced density matrices. The decomposition of a density matrix into components in terms of the reducing basis set is compared with decomposition in terms of representations of U(r).     

First-order exchange energy of intermolecular interactions from coupled cluster density matrices and their cumulants

A new method for the calculation of the first-order intermolecular exchange energy is proposed. It is based on the partition of two-particle density matrices of monomers into the antisymmetrized product of one-particle density matrices and the remaining cumulant part. This partition is used to modify the formula for the first-order exchange energy developed by Moszynski et al. [J. Chem. Phys. 100, 5080 (1994)]. The new expression has been applied for the case of monomer density matrices derived from the expectation value expression for the coupled cluster singles and doubles wave function. In this way an accurate method of calculation of the first-order exchange energy for many-electron systems has been obtained, where both monomers are described on the coupled cluster level. Numerical results are presented for several benchmark van der Waals systems to illustrate the performance of the new approach.     

Topological population analysis from higher order densities. I. Hartree–Fock level

In this work we present the formulation of a topological population analysis derived from reduced density matrices of arbitrary order. We describe the construction of a mathematical framework which is suitable for handling any number of physical atomic regions. The procedure provides appropriate tools for enabling the detection and direct localization of multicenter bondings in molecules.     

Hierarchy equations for reduced density matrices in different representations

We derive exact relationships for the reduced density matrices in representations where the transformation matrix is a product of one-body transformation matrices. We specialize to the momentum and onebody energy representations. By decoupling the equations we are able to write the Hartree-Fock equation in terms of the first-order density matrix in an arbitrary representation. Applications to reduced local energy and the correlation problem are discussed.