Bibliography

• Analysis of the electron density:

1. Analysis of the molecular density

Fernández Rico, J.; López, R.; Ramírez, G.  J Chem Phys  1999, 110, 4213-4220.


The minimal deformation criterion, previously proposed for the partition of the molecular density into atomic contributions, is updated and
extended. For any Gaussian basis set, these atomic contributions are expanded in series of real spherical harmonics by radial factors.
The terms with $l = 0$ determine the spherical parts of the atomic clouds and the remaining ones, their deformations.
This detailed description is complemented with a simplifiedrepresentation of the molecular density in terms of atomic charges and multipoles.
Moreover, these descriptions give a simple way to calculatethe electrostatic potential of the molecule as well as the electrostatic interaction between molecules.



2. Analysis of the molecular density: STO densities

Fernández Rico, J.; López, R.; Ema, I.; Ramírez, G.  J Chem Phys 2002, 117, 533-540.


A partition of the molecular density for Slater basis sets (STO), which parallels one previously developed for Gaussian basis sets (GTO), is reported.
The atomic fragments are expanded in spherical harmonics times radial factors. Each fragment contains all the one-center charge distributions centered in the atom plus the part of every two-center distribution assigned to the atom by the partition criterion.
The performance of the procedure is analyzed concluding that the analysis gives highly accurate representations of the molecular density at a very low cost. Moreover, the results of the analysis are illustrated with the study of the densities in CO and H₂O and the comparison of the atomic densities obtained from STO and GTO molecular calculations.



3. Accuracy of the electrostatic theorem for high quality Slater and Gaussian basis sets

Fernández Rico,J.; López, R.;Ema, I.; Ramírez, G., Int. J. Quantum. Chem. 2004, 100, 221-230.



The fulfillment of the Hellmann-Feynman electrostatic theorem is examined for the sequences of cc-pVxZ and cc-pCVxZ Gaussian basis sets as well as for the VBx and CVBx basis sets of Slater type orbitals. The difference between the energy gradient and electrostatic forces is large in small Gaussian basis sets
of the two types, but decreases quickly as basis sets improve. In VBx Slater basis sets these differences are small but the improvement is irregular, whilst in CVBx basis sets the fulfillment of the electrostatic theorem is very satisfactory. For the high quality basis sets (cc-pV5Z, cc-pCVQZ, cc-pCV5Z, CVB2 and CVB3) the energy gradient can be replaced by the electrostatic force in most practical applications.



4. Chemical notions from density

Fernández Rico,J.; López, R.;Ema, I.; Ramírez, G.,  J. Chem. Theory Comput.; 2005; 1, 1083 - 1095



The study of the density and the role played by its atomic representation is proposed as a way for the rationalization of the
chemical behavior. As this behavior has been long rationalized in terms of the basic concepts of the empirical structural chemistry, a direct link between both approaches is searched by using the exact representation of the density provided by the deformed atoms in molecules method(J. Fernández Rico, R. López, I. Ema, G. Ramírez, E. Ludeña, J. Comput. Chem. 2004, 25, 1355-1363). Noting that the spherical terms of the pseudoatoms cannot be the main responsible for the chemical behavior, we study the small non-spherical deformations and find that they reflect and support all basic concepts of the empirical structural chemistry. Lone pairs, single, double and triple bonds, different classes of atoms, functional groups, etc, are paralleled by the density deformations in a neat manner. These facts are illustrated with several examples.

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• Analytical representation and fast evaluation of density, electronic potential and field and forces on the nuclei.

1. Analytical method for the representation of atoms-in-molecules densities
   

Fernández Rico, J.; López, R.; Ema, I.; Ramírez, G.; Ludeña, E. V.  J Comp Chem 2004, 25, 1355-1363.


We present analytic refinements and applications of the deformed atomic densities method [J. Fernández Rico, R. López and G. Ramírez, J Chem Phys 1999, 110,  4213-4220]. In this method the molecular electron density is partitioned into atomic contributions, using a minimal deformation criterion for every two-center distributions, and the atomic contributions are expanded in spherical harmonics times radial factors.
Recurrence relations are introduced for the partition of the two-center distributions and the final radial factors are expressed in terms of exponential functions multiplied by polynomials. Algorithms for the practical implementation are developed and tested, showing excellent performances.
The usefulness of the present approach is illustrated by examining its ability to describe the deformation of atoms in different molecular environments and the relationship between these atomic densities and some chemical properties of molecules.



2. Electrostatic potentials and fields from density expansions of deformed atoms in molecules

Fernández Rico, J.; López, R.; Ema, I.; Ramírez, G. J Comp Chem 2004, 25, 1347-1354.


The exact representation of the molecular density by means of atomic expansions, consisting in spherical harmonics times analytical radial factors, is employed for the calculation of electrostatic potentials, fields and forces. The resulting procedure is equivalent to an atomic multipolar expansion in the long-range regions, but works with similar efficiency and accuracy in the short-range region, where multipolar expansions are not valid. The performances
of this procedure are tested on the calculation of the electrostatic potential contour maps and electrostatic field flux lines of water and nitrobenzene, computed from high quality molecular electron densities obtained with Slater basis sets.

• Density, binding forces and bond energies

1. Density and Binding Forces in Diatomics

Fernández Rico, J.; López, R.; Ema, I.; Ramírez, G. J Chem Phys 2002, 116, 1788-1799.


In a recently reported method, the molecular density is partitioned in minimally deformed atomic contributions, which are expanded in spherical harmonics times radial factors. Here we use this representation to express the electrostatic potential of the molecule, the force on its nuclei and the conformational variations of energy in terms of some simple integrals of the atomic radial factors. As a first application, we analyze the relationship between the density and the binding forces (and the bonding energy) in the diatomic molecules of the first row atoms. Two types of forces act on each nucleus: the self-pulling exerted by its own cloud and the external force due to the remaining atoms. The self-pulling comes only from the dipole type term of the atomic density. The external force comes from the other clouds and nuclei and is dominated by the {\it effective charges} which depend on the outermost region of the charge term.
Analyzing the progressive deformations of the atoms when they approach each other, the forces associated with these deformations and their contributions to the energy, one has a detailed description of the chemical bond which is complementary, and in many aspects more appealing, than the conventional ones.



2. Density and Binding Forces: rotational barrier of ethane

Fernández Rico, J.; López, R.; Ema, I.; Ramírez, G. J Chem Phys 2003, 119, 12251-12256.


The possibility of extending the relationships between density, binding forces and bonding energies to fine chemical effects is tested taking as an example the rotational barrier of ethane. Electron densities that reasonably fulfill the electrostatic theorem were obtained for several conformations using a Slater basis set. The analysis of these densities shows that the barrier is due to the internal forces acting on the H nuclei. Out of the staggered and eclipsed conformations, the clouds of the hydrogen atoms have a non-skeletal distortion that pull their nuclei toward the staggered conformation.