# Comparison of atomic charges, valencies and bond orders in some hydrogen-bonded complexes calculated from Mulliken and Löwdin SCF density matrices

TL;DR: In this paper, the atomic charges, valencies and bond orders in six H-bonded systems, (H2O)2, (HF)2 and HnFn+1− (n = 1 to 4) have been calculated using 4-31G, 6-31 G* and 6- 31G** (only in the case of (HF)/HF2−) basis sets and employing definitions based on Mulliken and Lowdin SCF density matrices.

Abstract: Atomic charges, valencies and bond orders in six H-bonded systems, (H2O)2, (HF)2 and HnFn+1− (n = 1 to 4) have been calculated using 4-31G, 6-31G* and 6-31G** (only in the case of (HF)2 and HF2−) basis sets and employing definitions based on Mulliken and Lowdin SCF density matrices. The definitions based on the former and the 6-31G* basis set are found to give results in conformity with the classical valence theory. The covalent bonding is highly exaggerated by the alternative definitions based on the Lowdin density matrix. The H-bond formation is found to be accompanied by decrease in total valency, and the valency and atomic charge of bridging hydrogen. A clear distinction between normal and strong H-bonding can be made on the basis of bond order and overlap population of the intermolecular HX (X = F or O) bonds and the amount of charge transferred from the proton acceptor to the donor molecule.

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TL;DR: The CM3 model is extended to semiempirical molecular orbital theory, in particular Austin Model 1 (AM1) and Parameterized Model 3 (PM3), and to the popular BLYP and B3LYP DFT and hybrid DFT methods, respectively.

Abstract: We have recently developed a new Class IV charge model for calculating partial atomic charges in molecules. The new model, called Charge Model 3 (CM3), was parameterized for calculations on molecules containing H, Li, C, N, O, F, Si, S, P, Cl, and Br by Hartree-Fock theory and by hybrid density functional theory (DFT) based on the modified Perdew-Wang density functional with several basis sets. In the present article we extend CM3 to semiempirical molecular orbital theory, in particular Austin Model 1 (AM1) and Parameterized Model 3 (PM3), and to the popular BLYP and B3LYP DFT and hybrid DFT methods, respectively. For the BLYP extension, we consider the 6-31G(d) basis set, and for the B3LYP extension, we consider three basis sets: 6-31G(d), 6-31+G(d), and MIDI!6D. We begin with the previous CM3 strategy, which involves 34 parameters for 30 pairs of elements. We then refine the model to improve the charges in compounds that contain N and O. This modification, involving two new parameters, leads to improved dipole moments for amides, bifunctional H, C, N, O compounds, aldehydes, ketones, esters, and carboxylic acids; the improvement for compounds not containing N results from obtaining more physical parameters for carbonyl groups when the O=C-N conjugation of amides is addressed in the parameterization. In addition, for the PM3 method, we added an additional parameter to improve dipole moments of compounds that contain bonds between C and N. This additional parameter leads to improved accuracy in the dipole moments of aromatic nitrogen heterocycles with five-membered rings.

99 citations

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TL;DR: In this paper, the authors presented a new continuum solvation model, called Solvation Model 5.43R, which is based on the generalized Born approximation for electrostatics augmented by terms that are proportional to the solvent-accessible surface areas (SASAs) of the atoms of the solute, and parametrized to predict the free energy of solutes containing H, C, N, O, F, P, S, Cl, and Br in water and organic solvents.

Abstract: We present a new continuum solvation model, called Solvation Model 5.43R (SM5.43R). The model is based on the generalized Born approximation for electrostatics augmented by terms that are proportional to the solvent-accessible surface areas (SASAs) of the atoms of the solute, and it is parametrized to predict the free energy of solvation of solutes containing H, C, N, O, F, P, S, Cl, and Br in water and organic solvents. The new model is an improvement over our previous solvation model, SM5.42R, in that it is based on CM3 charges rather than CM2 charges, it was trained over a larger and more diverse training set, and the choice of the value of the solvent radius, which is used to compute the SASA of the atoms of the solute, was made on a different basis than was used for SM5.42R. This paper presents parametrizations of SM5.43R using HF/6-31G(d), B3LYP/6-31G(d), mPW1PW91/6-31G(d), and mPW1PW91/6-31+G(d) to describe the electronic structure of the solute. For a data set of neutral solutes with known experim...

95 citations

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TL;DR: In this paper, two-center and three-center bond indices have been calculated using sp and spd basis sets for some hypervalent compounds (H−3, FHF−, ClHCl−, F−3 and ClF3, ClF2 and XeF4).

Abstract: Two-center and three-center bond indices have been calculated using sp and spd basis sets for some hypervalent compounds (H−3, FHF−, ClHCl−, F−3, Cl−3, ClF3, SF4, XeF2 and XeF4). The nature of bonding in these molecules depends on the population of the d orbitals (p orbitals in the case of H hypervalents) of the central atom. The three-center four-electron bonds are associated with negative 3c-bond indices, the reverse, however, is not true. The origin of the negative value is due to the negative charge of the terminal atoms. Finally, the two-center bond index has been decomposed in terms of molecular orbital contributions, the usefulness of which is discussed.

83 citations

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TL;DR: In this article, the applications of the quantitative definitions of bond index and valency for single determinant self-consistent field (SCF) wavefunctions are reviewed and a statistical interpretation of the bond index is also presented.

Abstract: Publisher Summary This chapter reviews the applications of the quantitative definitions of bond index and valency for single determinant self-consistent-field (SCF) wavefunctions. A statistical interpretation of bond index is also presented. The chapter discusses the definitions of bond index and valency for correlated wavefunctions. Later, the results corresponding to SCF and correlated wavefunctions are discussed. The results of the calculations reviewed indicate that at the ground state equilibrium geometry of a molecule the SCF wavefunction is adequate for the study of its nature of bonding on the basis of bond indices and valencies. The latter quantities are sensitive to basis sets and the population analysis schemes. Generally basis sets of double-zeta quality augmented with a set of polarization and a set of diffuse functions (only in the case of highly ionic and/or negatively charged species) on highly electronegative centers should be employed in these calculations. Of the two schemes of population analysis—Mulliken's population analysis (MPA) and Lowdin population analysis (LPA) that are currently in use, the MPA generally gives a more consistent picture of bonding.

71 citations

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TL;DR: In this article, a mathematically well-defined measure of localization is presented based on Mulliken's orbital populations, which is essentially intrinsic, i.e., no subdivision of the molecule is required.

Abstract: A mathematically well-defined measure of localization is presented based on Mulliken's orbital populations. It is shown that this quantity equals 1 for core- and lone-pair orbitals, 2 for two-atomic bonds, 6 for benzene rings, etc., and it is applicable for delocalized canonical HF orbitals as well. The definition of this quantity is general in the sense that ab initio MOS with overlapping AO expansion, and semiempirical wave functions using the ZDO approximation as well, can be treated. The localization quantity is essentially “intrinsic,” i.e., no subdivision of the molecule is required. For N-electron wave functions, mean delocalization can be defined. This measure is not invariant to unitary transformations of the one-electron orbitals, characterizing in this way the localized or extended representation of the N-electron wave function. It can be proven, however, that for unitary transformed wave functions a maximum delocalization exists which depends only on the physical (N-electron) properties of the molecule. It is shown that inhomogeneous charge distribution can cause strong electron localization in molecular systems. The delocalization of the canonical Hartree–Fock orbitals, the Parr–Chen circulant orbitals, and the optimum delocalized orbitals is studied by numerical calculations in extended systems.

43 citations

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TL;DR: In this paper, an analysis in quantitative form is given in terms of breakdowns of the electronic population into partial and total ''gross atomic populations'' and ''overlap populations'' for molecules.

Abstract: With increasing availability of good all‐electron LCAO MO (LCAO molecular orbital) wave functions for molecules, a systematic procedure for obtaining maximum insight from such data has become desirable. An analysis in quantitative form is given here in terms of breakdowns of the electronic population into partial and total ``gross atomic populations,'' or into partial and total ``net atomic populations'' together with ``overlap populations.'' ``Gross atomic populations'' distribute the electrons almost perfectly among the various AOs (atomic orbitals) of the various atoms in the molecule. From these numbers, a definite figure is obtained for the amount of promotion (e.g., from 2s to 2p) in each atom; and also for the gross charge Q on each atom if the bonds are polar. The total overlap population for any pair of atoms in a molecule is in general made up of positive and negative contributions. If the total overlap population between two atoms is positive, they are bonded; if negative, they are antibonded. Tables of gross atomic populations and overlap populations, also gross atomic charges Q, computed from SCF (self‐consistent field) LCAO‐MO data on CO and H2O, are given. The amount of s‐p promotion is found to be nearly the same for the O atom in CO and in H2O (0.14 electron in CO and 0.15e in H2O). For the C atom in CO it is 0.50e. For the N atom in N2 it is 0.26e according to calculations by Scherr. In spite of very strong polarity in the π bonds in CO, the σ and π overlap populations are very similar to those in N2. In CO the total overlap population for the π electrons is about twice that for the σ electrons. The most easily ionized electrons of CO are in an MO such that its gross atomic population is 94% localized on the carbon atom; these electrons account for the (weak) electron donor properties of CO. A comparison between changes of bond lengths observed on removal of an electron from one or another MO of CO and H2, and corresponding changes in computed overlap populations, shows good correlation. Several other points of interest are discussed.

9,238 citations

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TL;DR: In this paper, a method of "natural population analysis" was developed to calculate atomic charges and orbital populations of molecular wave functions in general atomic orbital basis sets, which seems to exhibit improved numerical stability and to better describe the electron distribution in compounds of high ionic character.

Abstract: A method of ‘‘natural population analysis’’ has been developed to calculate atomic charges and orbital populations of molecular wave functions in general atomic orbital basis sets. The natural analysis is an alternative to conventional Mulliken population analysis, and seems to exhibit improved numerical stability and to better describe the electron distribution in compounds of high ionic character, such as those containing metal atoms. We calculated ab initio SCF‐MO wave functions for compounds of type CH3X and LiX (X=F, OH, NH2, CH3, BH2, BeH, Li, H) in a variety of basis sets to illustrate the generality of the method, and to compare the natural populations with results of Mulliken analysis, density integration, and empirical measures of ionic character. Natural populations are found to give a satisfactory description of these molecules, providing a unified treatment of covalent and extreme ionic limits at modest computational cost.

8,332 citations

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Yale University

^{1}TL;DR: In this article, the CNDO method has been applied to the cyclopropylcarbinyl and cyclobutyl cations, and has given results which are in very good accord with experimental data.

Abstract: The CNDO method has been applied to the cyclopropylcarbinyl and cyclobutyl cations, and has given results which are in very good accord with experimental data. A cross-ring interaction is calculated to be of importance with cyclobutyl derivatives, and agrees with the large difference in rate observed with equatorial and axial leaving groups. Some properties of bicyclobutane as well as the relative energies for some models of the activated complex for the thermal rearrangement of bicyclobutane have also been calculated and compared with experimental data. The CNDO method appears to have considerable promise in the investigation of the organic chemical phenomena.

3,778 citations

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TL;DR: In this article, the authors define a set of generalized density matrices for the Hermitean density matrix of order $k, which is further antisymmetric in each set of these indices.

Abstract: In order to calculate the average value of a physical quantity containing also many-particle interactions in a system of $N$ antisymmetric particles, a set of generalized density matrices are defined. In order to permit the investigation of the same physical situation in two complementary spaces, the Hermitean density matrix of order $k$ has two sets of indices of each $k$ variables, and it is further antisymmetric in each set of these indices.Every normalizable antisymmetric wave function may be expanded in a series of determinants of order $N$ over all ordered configurations formed from a basic complete set of one-particle functions ${\ensuremath{\psi}}_{k}$, which gives a representation of the wave function and its density matrices also in the discrete $k$-space. The coefficients in an expansion of an eigenfunction to a particular operator may be determined by the variation principle, leading to the ordinary secular equation of the method of configurational interaction. It is shown that the first-order density matrix may be brought to diagonal form, which defines the "natural spin-orbitals" associated with the system. The situation is then partly characterized by the corresponding occupation numbers, which are shown to lie between 0 and 1 and to assume the value 1, only if the corresponding spin-orbital occurs in all configurations necessary for describing the situation. If the system has exactly $N$ spin-orbitals which are fully occupied, the total wave function may be reduced to a single Slater determinant. However, due to the mutual interaction between the particles, this limiting case is never physically realized, but the introduction of natural spin-orbitals leads then instead to a configurational expansion of most rapid convergence.In case the basic set is of finite order $M$, the best choice of this set is determined by a form of extended Hartree-Fock equations. It is shown that, in this case, the natural spin-orbitals approximately fulfill some equations previously proposed by Slater.

2,724 citations

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TL;DR: In this paper, a general approximate formula for covalent resonance energies is obtained in terms of partial overlap populations and Ī's, including one or two empirical coefficients, which indicates that forced hybridization due to inner shells should make important negative contributions to bond energies.

Abstract: LCAO molecular orbital overlap populations give in general much more flexible and widely useful measures of the non‐Coulombic parts of covalent bond energies than do LCAO bond orders. They are immediately applicable to both π and σ bonds, including bonds involving hybrid AOs of all kinds, and they take account directly of the effects of variations in bond length on bond strength. In the last section of this paper, a number of ways of defining LCAO bond orders are reviewed, and their advantages and disadvantages discussed.If all LCAO parameters β are assumed proportional to corresponding overlap integrals S times suitable mean atomic ionization energies Ī, a simple general approximate formula for covalent resonance energies is obtained in terms of partial overlap populations and Ī's, including one or two empirical coefficients. This formula indicates that forced hybridization (see III of this series) due to inner shells should make important negative contributions to bond energies. The application of the f...

1,844 citations