Potentials electrostatic

2 Electrostatic Potential. - An investigation of the topology of the MEP led to a chemical interpretation of the CPS in terms of lone pairs, n bonds, hybrid orbitals and other electronic structure elements. A Poincare-Hopf relationship for the MEP connects electronic structure elements and electrostatic reactivity. 

Martinez et investigated the topology of the electrostatic potential of NbsO, NbsS and NbsSe. The different relative stabilities of two- and three-dimensional structures can be explained on the basis of BCPs in the molecular electrostatic potential. The change in the nature of the bonding between the three compounds cannot be explained by hypervalency or by a qualitative change in the molecular orbitals or orbital energies. 

The electrostatic potential generated by a molecule has a strong influence on molecular interactions and chemical reactions. In particular, the molecular electrostatic potential (MEP) can be calculated, at least approximately, relatively easily even for large molecules, and it is frequently used for the representation of molecular shapes and steric interaaions between polar regions of 

For an approximate calculation of MEP, using multipolar expansions, it is often convenient to use fractional charges on atoms obtained from ah initio or semiempirical quantum chemical MO population analyses.  

A family of molecular surfaces can be defined in terms of the MEP. The MEP is a continuous funaion of the three-dimensional position variable r, hence just as the electronic charge density, it can also be analyzed in terms of level sets F(a) and their contour surfaces G(a), defined analogously to those of charge density  

Here the threshold value a refers to a constant MEP value. Note that in contrast to the case of charge density contours, in MEP analysis both the function V(r) and the threshold parameter a can take both positive and negative values. 

Let us consider an electrical potential at a point in a space where electric charges are distributed as point charges in a uniform dielectric medium. The electrical potential at a point in the space is expressed by  

In an aqueous solution containing positive and negative ions, however, the electroneutrality condition must prevail and the average of the total charge over any large region of such an electrolyte solution is zero  

In a local region, however, taken around a particular ion as center, there is an unequal distribution of the surrounding ions, owing to electrostatic forces. A simple electrostatic example is the situation where the space charge, p, is distributed with spherical symmetry about a center, so that p is a function of r (distance from the center ion) only. 

consider a region where a spherical ion of radius b and charge Ze is at its center, and this charge Ze is considered to be distributed uniformly over the surface of the ion. The electrical potential function of the central ion can be designated iff. It may be expressed as  

the Poisson equation relates the potential function ip to the charge density  

This is a typical function for electrostatic potential (equation 13). 

In this model of electrostatic interactions, two atoms (i and j) have point charges q and qj. The magnitude of the electrostatic energy (Veel) varies inversely with the distance between the atoms, Ry. The effective dielectric constant is 8. For in vacuo simulations or simulations with explicit water molecules, the denominator equals eRij. In some force fields, a distance-dependent dielectric, where the denominator is eRy Rjj, represents solvent implicitly. 

Accurate calculation of molecular electrostatic potentials in an algorithm which scales at most with the cube of molecule size has long remained a challenge. The established method for molecules is to fit the molecular density calculated from the orbital densities with multipolar functions attached to the atoms [39]. This method leads to a term which scales like the cube of the molecule size. It requires the introduction of an auxiliary basis set for the multipolar functions. The choice of such a basis set requires expertise and can easily lead to uncertain results. 

However, the partitioning scheme just described can be used to decompose the electron density into atomic components. Projection onto Y m functions yields multipoles attached to the atoms. The radial resolution is automatically as good as that of the integration. The Poisson equation, relating the static potential to the density, can then be solved with high accuracy for each multipole, since the decomposition has reduced 

The angular decomposition is always a truncated expansion. With a well constructed 

For molecules and molecular solids these requirements seem almost inevitably to lead to an atomic orbital basis concept. Of course this is no new invention in the field. However, it is well known that minimal basis sets, to be used in the context of linear combinations of atomic orbitals (LCAO) can be optimized significantly for each class of bonds. A simple test demonstrates that different types of bonds have different optimum minimal basis sets, even if these are numerically defined. For calculations not relying on expertise or specific optimization, a minimal basis set is not sufficient. A question is whether the wish can be fulfilled with a function set twice as big as the minimal one for the valence orbitals. Investigation convinced the author that this is the case to a surprisingly large extent. 

For benchmark accuracy calculations to be done on polyatomic molecules, without time-consuming explorations and expertise, it is necessary to have a means of systematic improvement not provided by the normal basis sets. It was shown earlier that for the 

Given the complete description of the structure in terms of pseudo atoms as described above, it is possible to derive additional properties that provide insight into the chemical and physical properties of energetic materials. Two of these are described below, the molecular electrostatic potential and the energy density distribution. 

The electrostatic potential at any point, V(r), is the energy required to bring a single positive charge from infinity to that point. As each pseudo atom in the refined model consists of the nucleus and the electron density distribution described by the multipole expansion parameters, the electrostatic potential may be calculated by the evaluation of 

Hartmann, Computer Simulations, World Scientific, Singapore, (2009). 

Cramer, Essentials of Computational Chemistry, Wiley, Chichester, (2002). 

Sutcliffe, The development of computational chemistry in the United Kingdom. Rev. Comput. Chem. 70, 271-316 (1997). 

Mobley, A.P. Graves, J.D. Chodera, A.C. Mcreynolds, B.K. Shoichet, and K.A. Dill, Predicting absolute hgand binding free energies to a simple model site. /. Molecul. Biol. 37 1(4), 1118-1134(2007). 

Massova, and P.A. Kollman, Computational alanine scanning of the 1 1 human growth hormone-receptor complex. J. Comput. Chem. 23(1), 15-27 (2002). 

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The electrostatic potential within a phase, that is, l/e times the electrical work of bringing unit charge from vacuum at infinity into the phase, is called the Galvani, or inner, potential Similarly, the electrostatic potential difference... 

The electrostatic potential generated by a molecule A at a distant point B can be expanded m inverse powers of the distance r between B and the centre of mass (CM) of A. This series is called the multipole expansion because the coefficients can be expressed in temis of the multipole moments of the molecule. With this expansion in hand, it is... 

Momany F A 1978 Determination of partial atomic charges from ab initio molecular electrostatic potentials. Application to formamide, methanol and formic acid J. Phys. Chem. 82 592... 

In these equations the electrostatic potential i might be thought to be the potential at the actual electrodes, the platinum on the left and the silver on the right. However, electrons are not the hypothetical test particles of physics, and the electrostatic potential difference at a junction between two metals is nnmeasurable. Wliat is measurable is the difference in the electrochemical potential p of the electron, which at equilibrium must be the same in any two wires that are in electrical contact. One assumes that the electrochemical potential can be written as the combination of two tenns, a chemical potential minus the electrical potential (- / because of the negative charge on the electron). Wlien two copper wires are connected to the two electrodes, the... 

The model used is the RPM. The average electrostatic potential ifr) at a distance r away from an ion / is related to tire charge density p.(r) by Poisson s equation... 

Experimental investigations of the model system of dye molecules adsorbed onto surfaces of polystyrene spheres have finuly established the sensitivity and surface specificity of the SHG method even for particles of micrometre size [117]. The surface sensitivity of die SHG process has been exploited for probing molecular transport across the bilayer in liposomes [118], for measurement of electrostatic potentials at the surface of small particles [119] and for imaging... 

The atomic scattering factor for electrons is somewhat more complicated. It is again a Fourier transfonn of a density of scattering matter, but, because the electron is a charged particle, it interacts with the nucleus as well as with the electron cloud. Thus p(r) in equation (B1.8.2h) is replaced by (p(r), the electrostatic potential of an electron situated at radius r from the nucleus. Under a range of conditions the electron scattering factor, y (0, can be represented in temis... 

Protems can be physisorbed or covalently attached to mica. Another method is to innnobilise and orient them by specific binding to receptor-fiinctionalized planar lipid bilayers supported on the mica sheets [15]. These surfaces are then brought into contact in an aqueous electrolyte solution, while the pH and the ionic strength are varied. Corresponding variations in the force-versus-distance curve allow conclusions about protein confomiation and interaction to be drawn [99]. The local electrostatic potential of protein-covered surfaces can hence be detemiined with an accuracy of 5 mV. 

Heyes, D.M. Electrostatic potentials arid fields in infinite point charge lattices. J. Chem. Phys. 74 (1981) 1924-1929. 

Task 1 Calculate the first non-vanishing multipole moment of the electrostatic potential of composed objects (i.e., structural units and clusters). 

Gilson, M. K., Sharp, K. A., Honig, B. H. Calculating the electrostatic potential of molecules in solution Method and error assessment. J. Comp. Chem. 9 (1988) 327-335. 

Knowledge of the spatial dimensions of a molecule is insufficient to imderstand the details of complex molecular interactions. In fact, molecular properties such as electrostatic potential, hydrophilic/lipophilic properties, and hydrogen bonding ability should be taken into account. These properties can be classified as scalar isosurfaces), vector field, and volumetric properties. 

Besides molecular orbitals, other molecular properties, such as electrostatic potentials or spin density, can be represented by isovalue surfaces. Normally, these scalar properties are mapped onto different surfaces see above). This type of high-dimensional visualization permits fast and easy identification of the relevant molecular regions. 

To display properties on molecular surfaces, two different approaches are applied. One method assigns color codes to each grid point of the surface. The grid points are connected to lines chicken-wire) or to surfaces (solid sphere) and then the color values are interpolated onto a color gradient [200]. The second method projects colored textures onto the surface [202, 203] and is mostly used to display such properties as electrostatic potentials, polarizability, hydrophobidty, and spin density. 

Bonaccorsi ct al. [204 defined for the first time the molecular electrostatic potential (MEP), wdicli is dearly tfie most important and most used property (Figure 2-125c. The clcctro.static potential helps to identify molecular regions that arc significant for the reactivity of compounds. Furthermore, the MEP is decisive for the formation of protein-ligand complexes. Detailed information is given in Ref [205]. 

These properties arc also relevant if molecular interactions arc considered. In contrast to electrostatic potentials, they only take effect at small distances between interacting molecular regions,... 

Molecular surfaces can express various chemical and physical properties, such as electrostatic potential, atomic charges or hydrophobicity, using colored mapping. 

J.S. Munay, P. Politzer, Electrostatic potential, in Encyclopedia of Gjmpu-tational Chemistry, Vol.2, P. von Rague-Schleyer, N.L. Allinger, T. Clark, J. Gasteiger, P.A. Kollman, H.F. Schaefer III, P.R. Schreiner (Eds.) John Wiley Sons, Chichester. UK, 1998, pp. 912-920. 

If there are ions in the solution, they will try to change their location according to the electrostatic potential in the system. Their distribution can be described according to Boltzmarm. Including these effects and applying some mathematics leads to the final linearized Poisson-Boltzmann equation (Eq. (43)). 

The calculation of autocorrelation vectors of surface properties [25] is similar (Eq. (21), with the distance d XiXj) between two points and Xj on the molecular surface within the interval between d[ and d a certain property p, e.g., the electrostatic potential (ESP) at a point on the molecular surface and the number of distance intervals 1). 

The representation of molecules by molecular surface properties was introduced in Section 2.10. Different properties such as the electrostatic potential, hydrogen bonding potential, or hydrophobicity potential can be mapped to this surface and seiwe for shape analysis [44] or the calculation of surface autocorrelation vectors (refer to Section 8.4.2). 

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