Towards a “Chemical” Hamiltonian

An analysis of the LCAO Hamiltonian is performed in terms of a “mixed” formulation of the second quantization for nonorthogonal orbitals, compressing the different interactions to one- and two-center terms as far as possible by performing appropriate projections. For this purpose an operator of atomic charge is also introduced, the expectation values of which are the Mulliken gross atomic populations on the individual atoms. The LCAO Hamiltonian is decomposed into terms having different physical meaning and significance: (i) sum of effective atomic Hamiltonians; (ii) the electrostatic interactions in the point-charge approximation; (iii) the electrostatic effects connected with the deviation of the actual charge distribution from the pointlike one; (iv) two-center overlap effects; (v) finite basis (“counterpoise”) correction terms related to the individual atoms; and (vi) similar finite basis correction terms with respect to the two-center interactions. Only terms of types (i) to (iv), containing no three- or four-center integrals, are considered as having physical significance. Based on the analysis of the Hamiltonian, an energy partitioning scheme is developed, and explicit expressions are given for one- and two-center (and basis extension) components of the SCF energy. The approach is also applied to the problem of intermolecular interactions, and an explicit formula is given permitting calculation of the “counterpoise” part of the supermolecule energy by properly taking into account that it depends not only on the extension of the basis, but also on the occupation of the additional orbitals in the intervening molecule—a factor completely overlooked in the usual scheme of calculations.     

Fast density matrix-based partitioning of the energy over the atoms in a molecule consistent with the Hirshfeld-I partitioning of the electron density

For the Hirshfeld-I atom in the molecule (AIM) model, associated single-atom energies and interaction energies at the Hartree-Fock level are efficiently determined in one-electron Hilbert space. In contrast to most other approaches, the energy terms are fully consistent with the partitioning of the underlying one-electron density matrix (1DM). Starting from the Hirshfeld-I AIM model for the electron density, the molecular 1DM is partitioned with a previously introduced double-atom scheme (Vanfleteren et al., J Chem Phys 2010, 132, 164111). Single-atom density matrices are constructed from the atomic and bond contributions of the double-atom scheme. As the Hartree-Fock energy can be expressed solely in terms of the 1DM, the partitioning of the latter over the AIM naturally leads to a corresponding partitioning of the Hartree-Fock energy. When the size of the molecule or the molecular basis set does not grow too large, the method shows considerable computational advantages compared with other approaches that require cumbersome numerical integration of the molecular energy integrals weighted by atomic weight functions.     

Diatomic interaction energies in the topological theory of atoms in molecules

Summary A partitioning of theab initio total energy into one-center and two-center terms is proposed. The partitioning scheme is developed using the auxiliary function (2, 1; 1, 2) = γ(2, 1)γ(1, 2) and the topological theory of atoms in molecules. It is shown that this scheme can be used at theoretical levels beyond Hartree-Fock. The numerical results indicate that the two-center terms follow the experimental trend of the dissociation energies for a series of related compounds.     

Reduced-diabatic vibrational close coupled treatment of molecular dissociation dynamics

A close coupled treatment in a vibrational adiabatic representation is applied to the study of molecular photodissociation dynamics. The procedure which is developed here involves three steps: transformation from a diabatic to an adiabatic basis set, truncation of the adiabatic basis set, back transformation to a reduced–diabatic basis set. In the two model cases which are studied, dissociation spectra show complicated peaks and dips, patterns interpreted in terms of shape and Feshbach resonances associated to vibrational predissociation with a relatively high potential barrier in the excited state. An important reduction in the number of channels required for a given final accuracy can be reached by using the reduced–diabatic basis set instead of the usual diabatic one. This is very promising for studying energy partitioning in molecular systems with several internal degrees of freedom taking part in the dynamics.     

Diatomic interaction energies in the topological theory of atoms in molecules

Summary.  A partitioning of the ab initio total energy into one-center and two-center terms is proposed. The partitioning scheme is developed using the auxiliary function L˜(2, 1; 1, 2)=γ(2, 1)γ(1, 2) and the topological theory of atoms in molecules. It is shown that this scheme can be used at theoretical levels beyond Hartree–Fock. The numerical results indicate that the two-center terms follow the experimental trend of the dissociation energies for a series of related compounds. Received March 5, 1996/Final revision received August 19, 1996/Accepted August 29, 1996     

Reduced hamiltonian orbitals. III. Unitarily invariant decomposition of hermitian operators

A decomposition of an N-particle operator as a sum of N + 1 components is defined such that, in the case of a model system employing a finite one-particle basis set, the decomposition is invariant under unitary transformations of the basis set. Applied to a two-particle Hamiltonian, this decomposition gives rise to the distinction between the independent-particle energy and the coupling energy defined in previous papers. Applied to the reduced density operator for a quantum state, the decomposition corresponds to partitioning the density into irreducible components. This partitioning is illustrated by graphs of electron density for the water molecule.     

Hierarchies of Intermolecular Potentials and Forces: Progress Towards a Quantitative Evaluation

Conventional views of intermolecular cohesion, based on the traditional categories of hydrogen bonding, of aromatic interactions, of dipolar or quadrupolar contacts, and of the broad, gray zone of ‘van der Waals’ liaison, often define strength hierarchies on the basis of qualitative categories like approximate molecular orientations or distances between atomic nuclei in molecules. When interaction energies are quantitatively evaluated between molecular pairs, in a more justifiable partitioning scheme, often a completely different picture emerges. Examples are given for selected molecular dimers and organic crystals, using a new semiempirical scheme, the PIXEL method, which also allows a separate evaluation of coulombic, polarization, dispersion and repulsion energy terms.     

Comparison of high-order many-body perturbation theory and configuration interaction for H2O

Diagrammatic many-body perturbation theory, coupled with a recursive computational procedure, is employed to obtain the correlation energy of H₂O within a 39-STO basis set by evaluating all double-excitation diagrams through twelfth order without any approximations. This provides, for the first time, the complete double-excitation diagrams contributions to the correlation energy, which is ?0.28826 hartree, compared with a correlation energy of ?0.27402 hartree obtained from a configuration interaction calculation which includes all double excitations. The difference of 0.0142 hartree includes the “size consistency” correction to the all-double-excitations CI energy, due to the “pathological” unliked-diagram terms remaining in that result, but also involves certain fourth- and higher-order rearrangement diagrams. Contrary to conventional belief, the unshifted, or Møller-Plesset partitioning of the hamiltonian provides a much more rapid convergence of the perturbation series that does the shifted, or Epstein-Nesbet partitioning. In both cases. Padé approximants enhance the convergence of the series considerably. A simple variation-perturbation scheme based on the first-order MBPT wavefunction is sufficient to provide 97.5% of the all-doubles CI correlation energy.     

Analysis of the chemical bond

The energy expression of the MO-LCAO scheme is corrected approximately for the left-right correlation such that it leads to the correct dissociation limit. Together with the correlation correction a correction is applied to the interference term, whereas the sharing penetration effects are neglected. The derivation of this corrected approximate energy formula is suggested from an analysis of binding in H₂ ⁺ and H₂. The binding energy consists mainly of three contributions: interference, quasiclassical interaction, promotion. Two-electron interference contributions are absorbed into the one-electron terms. The basis dependence of the fragmentation of the binding energy is discussed and an appropriate hybrid basis is constructed. Rotational invariance is found to a high degree of accuracy. In terms of the proposed scheme the binding in several diatomic and polyatomic molecules is analysed. The individual contributions to the binding energy turn out to be physically meaningful.     

Simple ab initio calculations using a floating basis

Calculations are described on three rotamers of hydrogen disulphide (transgauche- and cis-HSSH) using an ab initio Floating Gaussian Orbital model. The optimised geometrical and electronic structures of each rotamer are discussed in terms of several electronic properties, a population and orbital analysis and an extensive partitioning of the electronic energy amongst the orbitals. The so-called Gauche Effect in HSSH is discussed in connexion with the various models proposed to account for this particular structural feature.     

Kinetic energy decomposition scheme based on information theory

We proposed a novel kinetic energy decomposition analysis based on information theory. Since the Hirshfeld partitioning for electron densities can be formulated in terms of Kullback–Leibler information deficiency in information theory, a similar partitioning for kinetic energy densities was newly proposed. The numerical assessments confirm that the current kinetic energy decomposition scheme provides reasonable chemical pictures for ionic and covalent molecules, and can also estimate atomic energies using a correction with viral ratios.     

SCF,MP2, and CEPA-1 calculations on the OH ‥ O hydrogen bonded complexes (H2O)2 and (H2O-H2CO)

Counterpoise corrected ab initio calculations are reported for (H₂O)₂ and H₂O-H₂CO. Geometry searches were done in the moment-optimized basis DZP' at the SCF, MP2, and CEPA-1 levels of theory, followed by more accurate single-point calculations in basis ESPB, which includes bondfunctions to saturate the dispersion energy. The final equilibrium binding energies obtained are ?4.7 ±0.3 kcal/mol for a near-linear (H₂O)₂ structure and ?4.6 ±0.3 kcal/mol for a strongly bent HOH ‥ OCH₂ structure. The energy difference between these systems is much smaller than in all previous ab initio work. Cyclic (C_2h) and bifurcated (C_2v) transition structures for (H₂O)₂ are located at 1.0 ±0.1 kcal/mol and 1.9 ±0.3 kcal/mol above the global minimum, respectively. A new partitioning scheme is presented that rigorously partitions the MP2 correlation interaction energy in intra and intermolecular (dispersion) contributions. These terms are large (up to 2 kcal/mol) but of opposite sign for most geometries studied and hence their overall effect upon the final structures is relatively small. The relative merits of the MP2 and CEPA-1 approaches are discussed are discussed and it is concluded that for economical reasons MP2 is to be preferred, especially for larger systems.     

On a new partitioning of the Hamiltonian in many-body calculation of pair-correlation energies in closed-shell systems

We introduce here a new partitioning of the Hamiltonian in calculating pair-correlation energies using many-body perturbation theory, by which we are able to eliminate the off-diagonal particle–hole (p–h) ladders exactly to all orders in the perturbation expansion. In this formulation, the particle states turn out to be different for each distinct pair of hole states in the correlation energy calculation. We have also included the contributions of the diagonal particle–particle (p–p) and hole–hole ladders exactly to all orders. The effect of the off-diagonal p–p ladders has been estimated for each pair by computing the third-, foruth- and fifth-order energies. For highly symmetric systems the present partitioning yields in general symmetry-broken orbitals. Here one may use an average kind of partitioning for all the partners of the degenerate sets, which restores the symmetry and at the same time ensures cancellation of the p–h ladders exactly at the lowest order and approximately at the higher orders. Results are presented for a selection of 6π-electron conjugated systems. The correlation energy for each pair is in excellent agreement with that obtained from a partial CI calculation involving all double excitations from this pair. The advantages of implementing the present scheme in larger systems has been discussed.     

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     

Charge distributions and chemical effects. XLII. Atomic charges and their role in energy calculations

A physically meaningful charge partitioning rooted in Mulliken's scheme meets with success, provided the usual equipartitioning of overlap populations involving dissimilar atoms is abandoned in favor of a constraint rendering all alkane carbons as similar as possible to one another. Charge analyses are to be carried out after configuration interaction involving reasonably large optimized basis sets. At this level, both the relative ordering and the magnitude of theoretical atomic charges in hydrocarbons are precisely those required for energy calculations of alkanes and alkenes featuring atomic charges in an explicit manner.     

Hydrogen–hydrogen interaction in planar biphenyl: A theoretical study based on the interacting quantum atoms and Hirshfeld atomic energy partitioning methods

The nature of H‐H interaction between ortho‐hydrogen atoms in planar biphenyl is investigated by two different atomic energy partitioning methods, namely fractional occupation iterative Hirshfeld (FOHI) and interacting quantum atoms (IQA), and compared with the traditional virial‐based approach of quantum theory of atoms in molecules (QTAIM). In agreement with Bader's hypothesis of H? H bonding, partitioning the atomic energy into intra‐atomic and interatomic terms reveals that there is a net attractive interaction between the ortho‐hydrogens in the planar biphenyl. This falsifies the classical view of steric repulsion between the hydrogens. In addition, in contrast to the traditional QTAIM energy analysis, both FOHI and IQA show that the total atomic energy of the ortho‐hydrogens remains almost constant when they participate in the H‐H interaction. Although, the interatomic part of atomic energy of the hydrogens plays a stabilizing role during the formation of the H? H bond, it is almost compensated by the destabilizing effects of the intra‐atomic parts and consequently, the total energy of the hydrogens remains constant. The trends in the changes of intra‐atomic and interatomic energy terms of ortho‐hydrogens during H? H bond formation are very similar to those observed for the H₂ molecule. © 2014 Wiley Periodicals, Inc.     

The internal energy of product ions formed in mass spectral reactions

Major factors which determine the distribution of internal energy of a primary product ion A⁺, P(E)_A+, at formation are delineated in terms of the quasi-equilibrium theory and a variety of experimental evidence is offered to support these conclusions. These major factors include: the P(E) of the molecular ion, the rate constants as a function of energy, κ(E), for M⁺·→A⁺ + N⁰ and for competing reactions of M⁺·, and the partitioning of excess internal energy between A⁺ and N⁰. The ‘fluctuation effect’ on this partitioning makes P(E)_A+ relatively insensitive to many structural and energy changes.     

Detailed mapping of intramolecular energy transfer in field-effect single-molecule nanoelectronic devices

The electronic and vibrational atomic responses to external electric field (EF) are computed to detail intramolecular energy transfer in a proposed molecular nanoelectronic field-effect system. The parallel and perpendicular electronic and vibrational contributions to intramolecular energy transfer are computed using, respectively, quantum theory of atoms-in-molecule and an energy partitioning scheme based on normal modes vibrational analysis. The symmetrical and asymmetrical intramolecular energy transfers are interpreted, respectively, as Peltier-like and Joule-like effects and quantified in terms of appropriate coefficients. Dependencies of these coefficients on EF are investigated. In addition, a semiclassical temperature model is introduced to describe symmetrical and asymmetrical temperature distributions which are attributed, respectively, to the Joule-like and Peltier-like heatings. This procedure can be used to map out intramolecular energy distribution in molecular nanoelectronic systems.     

Generalized electronic diabatic approach to structural similarity in two‐dimensional potential energy surfaces of various topologies

Active site properties in some proteins can be affected by conformational fluctuations of neighbor residues, even when the latter are not involved directly in the binding process. A local environment thus appears to alter the relevant potential energy surface and its reaction paths. Here, some aspects of this phenomenon are simulated within a generalized electronic diabatic (GED) scheme to study the geometry and structural similarity for a class of two‐dimensional (2D) energy surfaces. The electronic quantum state is a linear superposition of diabatic basis functions, each of which is taken to represent a single (pure) electronic state for the isolated material system. Here, we describe a model reaction of isomerization by shifts in amplitudes for three diabatic species (reactant, product, and an open‐shell transition state) coupled in an external field. The “effective” 2D energy surface in the field is characterized in terms of critical points, and the amplitudes along the main reaction paths. A new feature is the introduction of a phase diagram where all possible potential‐energy‐surface topologies (consistent with three‐state systems in two linear coordinates) are matched with actual model parameters. By varying the coupling strengths between diabatic states, we classify regions of this phase diagram in terms of electronic and structural similarities; some regions comprise models whose reaction paths have geometries that belong to the catchment region of the reactant, yet are electronically akin to the diabatic transition state or product. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Analysis on solvated molecules with a new energy partitioning scheme for intra- and intermolecular interactions

A new partitioning scheme for the total energy of molecules is presented. In the scheme, the Hartree-Fock total energy of a molecular system is represented as the sum of one- and two-center terms exactly. The present method provides physically reasonable behavior for a wide range of interactions, and intermolecular interaction is treated equivalently with intramolecular interaction. The method is applied to analysis on the inter- and intramolecular interactions of molecular complexes both in gas phase and in aqueous solution. The results strongly indicate that the present method is a powerful tool to understand not only the bonding nature of molecules but also interaction between molecules.