How To Create A Force Field

Field Theory (Physics)

Computational Chemistry

Matthias Hofmann , Henry F. SchaeferIII, in Encyclopedia of Physical Science and Technology (Third Edition), 2003

II.A.1 Energy Terms

Advanced force fields distinguish several atom types for each element (depending on hybridization and neighboring atoms) and introduce various energy contributions to the total force field energy, E _FF:

${\text{E}}_{\text{FF}}={\text{E}}_{\text{str}}+{\text{E}}_{\text{bend}}+{\text{E}}_{\text{tors}}+{\text{E}}_{\text{vdW}}+{\text{E}}_{\text{elst}}+\dots ,$

where E _str and E _bend are energy terms due to bond stretching and angle bending, respectively; E _tors depends on torsional angles describing rotation about bonds; and E _vdW and E _elst describe (nonbonded) van der Waals and electrostatic interactions, respectively (Fig. 2). In addition to these basic terms common to all empirical force fields there may be extra terms to improve the performance for specific tasks. Each term is a function of the nuclear coordinates and a number of parameters. Once the parameters have been defined, the total energy, E _FF, can be computed and subsequently minimized with respect to the coordinates.

FIGURE 2. Most basic energy terms included in empirical force field (FF) methods.

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Organic van der Waals Epitaxy versus Templated Growth by Organic–Organic Heteroepitaxy

Clemens Simbrunner , Helmut Sitter , in Handbook of Crystal Growth: Thin Films and Epitaxy (Second Edition), 2015

11.1.2.2 The Influence of the Substrate Surface Symmetry on the Molecular Adsorption

The symmetry of each substrate surface can be described by one of 17 possible 2D space- or wallpaper-groups [22–24]. All groups can be constructed by a combination of three symmetry elements, namely (1) glide reflections, (2) mirror planes, and (3) rotation centers. To visualize the influence of these operations, representative substrate surfaces have been chosen and are depicted in Figure 11.4.

FIGURE 11.4. Visualization concerning the influence of mirror planes and rotational centers on the molecular adsorption. Polar plots, indicated in the top panel, reflect the adsorption energy of 6T on muscovite mica (001) (left) and p-6P on KCl (100) (right) as a function of adsorption angle φ. The zero on the energy scale is set to the energy of the least favorable angle. As 6T can adsorb in two different geometries, two curves are depicted, one corresponding to left-handed (red) and the other to right-handed (blue) enantiomer. Experimental observed adsorption geometries are indicated by gray circles [19–21] and corresponding local adsorption sites are depicted in the bottom panel. In general, mirror-reflections lead to point chirality, which is characterized by the generation of two different but energetically equivalent adsorption sites. The influence of rotational centers is demonstrated for p-6P/KCl, where symmetry in general leads to four different, but energetically equivalent, adsorption sites. (For interpretation of the references to color in this figure legend, the reader is referred to the online version of this book.)

Force-field simulations have been performed for the adsorption of a rigid all-trans 6T molecule on a muscovite mica (001) surface and a p-6P molecule on KCl (100). Molecules that are intrinsically achiral but obtain a form of 2D chirality when adsorbed on a substrate surface are also named prochiral [25]. As all-trans 6T represents a prochiral conformation, simulations have been performed for left- and right-handed enantiomers, and corresponding data are indicated by blue and red curves. For each molecular adsorption angle (φ), the preferred adsorption site has been determined, and its adsorption energy is depicted in terms of a polar plot in the top panel of Figure 11.4. As sketched in Figure 11.4, φ characterizes the azimuthal orientation of the long molecular axis (LMA) relative to [110] of muscovite mica (001) and $\left[0\overline{1}\overline{1}\right]$ of KCl (100). The zero on the energy scale is set to the energy of the least favorable angle.

•

Mirror Plane/Glide Plane: For example, the molecular adsorption of 6T on muscovite mica has been chosen and obtained adsorption energies are indicated in the left panel of Figure 11.4 as a polar plot. Experimentally deduced molecular adsorption angles are indicated by gray circles [2]. Energetically preferable adsorption geometries, which are also found from experimental data, are indicated below. Simulated adsorption sites are occupied either by the left- (red, φ  =   60°) or right-handed (blue, φ  =   −60°) enantiomer and can be constructed by mirror operation. Moreover, due to C ₂ point symmetry of both species, energetically equivalent adsorption geometries can be obtained by rotating the molecules by 180°. It can be concluded that mirror- and glide-reflection in general lead to the generation of two energetically equivalent adsorption sites [2,24]. In the case of prochiral molecules, resulting adsorption sites are not energetically equivalent for both enantiomers but are occupied either by a left- or right-handed molecule. Consequently, two energetically equivalent long molecular axis orientations are formed in the general case, which reduces the global anisotropy. Moreover, it should be stated that the obtained molecule–surface combination does not have any mirror elements in general. Consequently, the generated geometry is also called point chirality, and can be induced even by highly symmetric adsorbates, e.g., p-6P [26,27]. The only way to avoid the creation of local chiral motifs by adsorption events is provided if the mirror planes of an achiral-molecule and -substrate surface overlap. Exemplary, such an extraordinary configuration can be achieved if the LMA of p-6P is oriented parallel or normal to the mirror plane of muscovite mica (001) [2]. Importantly, it should be stated that point-, pro-, organizational- and conformational-chirality generated by intrinsic achiral molecules can only be expressed on a local level, e.g., over a restricted surface area. On a global level, e.g., when integrated over the entire surface, chirality disappears due to an equal generation of both enantiomorphs [27].

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Rotational Center: To analyze the influence of rotational symmetry, the adsorption of p-6P molecules on KCl (100) has been simulated. The obtained adsorption energies are presented in term of a polar plot in the top right panel of Figure 11.4. As an adsorbed p-6P molecule is characterized by two mirror symmetry planes and consequently can be described by a C_2v point group, simulations have only to be performed for a single molecular configuration. Again, gray circles represent experimental data [19,20] indicating an energetical preferred LMA orientation along the $〈011〉$ crystallographic directions. An analogous molecular alignment has also been deduced for 6T [28] and can be explained by the presence of strong surface corrugations on the KCl (100) surface [29]. Moreover, simulated adsorption data reflect mirror- (mirror planes along $〈001〉$ and $〈011〉$ are indicated by bold solid lines) and fourfold rotational-symmetry, which can be explained by the p4m space group of the KCl (100) surface. As indicated in the bottom right panel of Figure 11.4, energetically equivalent adsorption sites can be constructed by a 90° rotation at a fourfold symmetry point.

Based on the latter discussion, it can be concluded that the number of energetically equivalent adsorption sites increases with the presence of rotational-, glide- and mirror-symmetry on the substrate surface. An overview for each wallpaper group is listed in Table 11.1 [24].

Table 11.1. Number of Energetically Equivalent Adsorption Sites for Each Wallpaper Group

Equivalent Sites	Wallpaper Group
1	p1
2	p2, pm, pg, cm
3	p3
4	pmm, pmg, pgg, cmm, p4
6	p31m, p3m1, p6
8	p4m, p4g
12	p6m

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Valence Force Field Model and Its Applications

Vyacheslav A. Elyukhin , in Statistical Thermodynamics of Semiconductor Alloys, 2016

Abstract

The valence force field model for semiconductors with the diamond, zinc blende, and wurtzite structures is presented in this chapter. The mixed sublattices in ternary alloys of two binary compounds have the zinc blende and wurtzite structures that are slightly distorted. The deformation energies of the tetrahedral cells in ternary alloys are obtained. These energies demonstrate simultaneous tendencies to the decomposition and the formation of the superstructure in the disordered alloys with several certain compositions. The alloys with such compositions may undergo the discontinuous and continuous order–disorder phase transitions. The possible types of the superstructures in the ternary alloys with the zinc blende and wurtzite structures are described. Carbon and Sn isoelectronic impurities in Ge have the tendency to form 4Sn1C tetrahedral clusters identical in size, shape and composition. The strained energies due to the isoelectronic impurities in binary semiconductor compounds with the zinc blende structure are derived.

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Colloid and Surface Chemistry

Eugene D. Shchukin , ... Andrei S. Zelenev , in Studies in Interface Science, 2001

Publisher Summary

A distinctive force field present at the interface may cause changes in the composition of the near-surface layer: different substances depending on their nature may either concentrate near the surface, or, alternatively, move into the bulk. This phenomenon, referred to as the adsorption, causes changes in the properties of interfaces, including changes in the interfacial (surface) tension. In disperse systems with liquid dispersion medium, adsorption layers present at the surfaces of dispersed particles may significantly influence the interactions between these particles, and hence, affect the properties of disperse system as a whole, including its stability. The laws governing the formation, structure, and properties of the adsorption layers at different interfaces allow to analyze the role such layers play in controlling colloid stability and other properties of disperse systems. Substances that lower the surface tension of the system are referred to as surface active substances, or surfactants. It follows from the Gibbs equation that adsorption of such compounds is positive, that is, their concentration within the surface layer is higher than that in the bulk.

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Simulation and Modelling

J. Gao , in Comprehensive Biophysics, 2012

9.9.2.2.2 The X-Pol potential

The current force fields for biomolecular systems were established by Lifson in the 1960s, ⁶² which have changed very little in the past half a century. Despite its success in biomacromolecular modeling, there are also many shortcomings, including redundancy of empirical parameters and a lack of unified treatment of electronic polarization. A logical step in the future is to go beyond the molecular mechanical representation of biomolecules. To this end, we have introduced the X-Pol method based on quantum mechanics as a framework for the development of next-generation force fields. ^{23,54–56,63–64}

The X-Pol potential is designed with a hierarchy of three approximations. ⁶⁴ The macromolecular system is partitioned into subsystems, which are called fragments or blocks. The first approximation is that the wave function of the full system (Ψ^X-Pol) is a Hartree product of the determinant wave functions of individual fragments:

[14] ${\Psi }^{X\mathrm{-}\mathrm{Pol}}=R_x\prod _{a=1}^M\hat{A}{\Phi }_a$

where M is the number of blocks or residue fragments, R _x is the normalization constant, Â is an antisymmetrization operator, and Φ _a is a product of the occupied spin-orbitals $\left\{{\phi }_i^a\right\}$ in block a (eqn [14]):

[15] ${\Phi }_a={\phi }_1^a{\phi }_2^a\cdot \cdot \cdot {\phi }_{n_a}^a$

The assumption made in eqn [14] neglects the exchange interactions between different fragments. To account for the short-range exchange repulsion as well as the long-range dispersion interactions, the Lennard-Jones potential is used

[16] $E_{ab}^{vdW}=\sum _{\alpha =1}^A\sum _{\beta =1}^{B}4{\epsilon }_{\alpha \beta }\left[{\left(\frac{{\sigma }_{\alpha \beta }}{R_{\alpha \beta }}\right)}^{12}-{\left(\frac{{\sigma }_{\alpha \beta }}{R_{\alpha \beta }}\right)}^6\right]$

where A and B are the number of atoms in fragments a and b, and the parameters ɛ_αβ and σ_αβ are obtained using standard combining rules such that ɛ_αβ=(ɛ_αɛ_β)^1/2 and σ_αβ=(σ_α+σ_β)/2, in which ɛ and σ are atomic empirical parameters.

The use of the R ⁻¹² repulsive terms in eqn [16] to approximate the short-range exchange repulsion significantly reduces the computational cost. Alternatively, the exchange energies can be determined explicitly by antisymmetrizing the X-Pol wave function: ^65–66

[17] ${\Psi }^{X\mathrm{-}\mathrm{Pol}\mathrm{-}X}=R_x^A\hat{A}\left(\prod _{a=1}^M{\Phi }_a\right)$

Here, the notation X-Pol-X is used to indicate that the X-Pol potential includes eXchange explicitly.

The wave functions of the individual fragments are optimized by the self-consistent field method in the presence of the external electrostatic potential of all other blocks until the energy or electron density of the entire system is converged. ^54,63 Thus, for fragment a the external potential, V _a (r), is

[18] $V_a(\mathbf{r})=\sum _{b\ne a}\left\{-\int \frac{d\mathbf{r}\text{'}{\rho }_b(\mathbf{r}\text{'})}{|\mathbf{r}-\mathbf{r}\text{'}|}+\sum _{\beta =1}^B\frac{Z_{\beta }(b)}{|\mathbf{r}-{\mathbf{R}}_{\beta }(b)|}\right\}$

where ρ_b(r′) is the electron density of molecule b, derived from the wave function, ρ_b(r′)=|Ψ _b (r′)|².

The total potential energy of the system is

[19] $E_{tot}=〈\Phi \left|\hat{H}\right|\Phi 〉-\sum _{a=1}^N{E}_a^o$

where Ĥ is the Hamiltonian of the system defined below, $E_a^o$ is the energy of an isolated fragment a whose wave function is ${\Psi }_a^o$ . The individual wave functions of the subsystems can be obtained at any level of theory – ab initio Hartree-Fock, semiempirical molecular orbital theory, correlated wave function theory, or Kohn-Sham DFT. ⁶⁷

Without further approximation, it is necessary to compute the two-electron integrals arising from different molecules, which would be too expensive for a force field designed for condensed phase simulations. Consequently, we introduce the second approximation in the X-Pol method by approximating the "quantum mechanical" electrostatic potential of eqn [18] by a classical multipole representation. We have used the monopole term, that is, partial atomic charges, in early applications, in which the formally two-electron integrals are reduced to one-electron integrals, which are computationally efficient. ^54–55,64

The third approximation is the specific quantum mechanical model to be used for a given problem and a specific purpose. To this end, we adopt semiempirical Hamiltonians based on the neglect diatomic differential overlap (NDDO) approximation. ^68–69

The X-Pol potential has been tested and applied to the simulation of liquid water, ⁵⁵ and has it been extended to molecular dynamics simulations of polypeptide in solution. ²³ Analytic gradient technique has been developed and implemented into the program CHARMM (Chemistry at Harvard Macromolecular Mechanics), ⁷⁰ and the X-Pol method has been illustrated to be feasible for extensive molecular dynamics simulations of fully solvated proteins under periodic boundary conditions. ²³ In that study, 3,200 (3.2 ps) full energy and gradient evaluations can be performed in molecular dynamics simulations of a BPTI protein in water, consisting of nearly 15   000 atoms and 30   000 basis functions, on a single 2.6   GHz processor in 1   day.

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Acoustic radiation force and shear wave elastography techniques

Arsenii V. Telichko , ... Jeremy J. Dahl , in Tissue Elasticity Imaging, 2020

4.1 Shear wave generation by radiation force

The radiation force field that is used in the previous section to generate displacement of tissue creates shear stresses at the boundaries of the region of excitation. These stresses pull the tissue back toward its original location, thereby inducing low-frequency transverse oscillations in the form of a shear wave. The shear wave propagates away from the region of excitation in a direction normal to the direction of applied radiation force. The shear wave oscillations are described by [76]

(5.28) ${\mathbf{F}}_{\mathbf{z}}=\frac{{\partial }^2{s}_z}{\partial t^2}-c_t^2{\mathrm{\Delta }}_{\mathrm{\perp }}{s}_z+\frac{\eta }{\rho }\frac{\partial }{\partial t}{\mathrm{\Delta }}_{\mathrm{\perp }}{s}_z,$

where F _z is the applied radiation force in the z direction, $\eta$ is the shear viscosity, s _z is the $\hat{\mathbf{z}}$ component of the displacement vector s, and Δ_⊥ is the Laplacian operator for the x and y components. c _t is the shear wave velocity and is defined as

(5.29) $c_t=\sqrt{\mu /\rho },$

Where $\mu$ is the shear modulus, and the tissue is assumed to be an isotropic, elastic, and quasi-incompressible medium. In Eq. (5.28) the first two terms describe the linear wave equation with forcing function F _z and the last term on the right is a dissipative term. Eq. (5.28) essentially describes a low-frequency (typically hundreds of hertz) shear wave that propagates with speed c _t away from the axis of the acoustic beam. The shape of the shear wave in the xy-plane is much like an enlarging ring, similar to what one would observe by dropping a pebble into a pool of water. By observation of these shear waves, properties of the medium in which they propagate can be determined. In the following sections, we explore a few of the techniques used for measuring and imaging the viscoelastic properties of tissues, based on shear waves.

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Simulation and Modelling

E. Chow , ... D.E. Shaw , in Comprehensive Biophysics, 2012

9.6.5.1 Recent Improvements to Force Field Accuracy

The functional forms underlying commonly used force field models for proteins and other biomolecules have basically remained unchanged, and recent work has instead focused on improving force field parameters. These improvements are, in part, based on high-level quantum mechanical calculations in combination with condensed phase data for small molecular fragments. However, as the temporal and spatial resolutions accessible to experiment and computation have begun to intersect and overlap, empirical data for complex macromolecular systems can now also be used to assess and improve the accuracy of force fields. This trend is already evident in cases that compare the results of MD simulations to NMR experiments. ⁵⁹

Recent modifications to protein force fields have focused on improvements to the backbone torsion potentials. Torsion potentials have traditionally been the last part of a force field to be parameterized and thus often serve to compensate for inaccuracies present in other parts of the force field, specifically in the nonbonded parameters. Adjustments to these torsion potentials also serve to improve the conformational dynamics of proteins. Recent versions of both the CHARMM and Amber force fields include changes to the backbone torsion potentials based on refitting to high-level quantum mechanical data. In the case of Amber ff99SB (an update to Amber ff99) both the phi and psi torsion terms were refined by fitting them to quantum mechanical data for tetrapeptides. ⁶⁰ CHARMM27, ⁶¹ on the other hand, introduced a new energy term called CMAP. This 2-D spline function was initially fit to high-level quantum mechanical data and subsequently adjusted based on simulations of native protein conformations.

Following the release of Amber ff99SB and CHARMM27, the accuracy of these force fields was studied by reconciling the results of long-timescale MD simulations with NMR and other data. Showalter and Bruschweiler, for example, compared simulations of human ubiquitin to both the crystal structure of this protein as well as to solution state NMR measurements in a study involving the original Amber ff99 and updated Amber ff99SB force fields. ^62,63 The authors found that while both force fields produced similar root-mean-square deviation (RMSD) from the atomic coordinates found by crystallography, predictions of NMR-derived backbone amide order parameters and residual dipolar couplings were much better for the simulation carried out using Amber ff99SB. As long as care is taken to ensure that relevant quantities in MD simulations have converged, ⁶⁴ the number of studies involving the comparison of simulation to NMR experiment for the testing of force fields ^60,65,66 is expected to increase.

The above results suggest that the modifications to Amber and CHARMM give improved structural and dynamical behavior for natively folded proteins. A more rigorous requirement is for force fields to reproduce the correct populations across conformational states. Early versions of Amber, for example, were known to strongly over-stabilize the helical state, and various modifications were made to improve this behavior. In a recent study, Best and Hummer used long equilibrium simulations of small alanine peptides to examine secondary structure propensities across a large set of force fields. ⁶⁷ Although the modified force fields tended to give better agreement with experiment, the overall results were somewhat inconclusive because large discrepancies still existed between these force fields. The same authors, in a more recent study, focused on two variants of the Amber force field, using long MD simulations to refit the backbone torsions to NMR data for a longer helix-forming peptide, in addition to the same small peptides from the earlier study. ⁶⁸ The two resulting force fields produced better agreement for helix propensity and were also validated against other NMR data. Properties such as the temperature dependence of the helix-coil transition, however, were still found to be incorrect. Although such tweaks are useful, more substantial changes to force fields may be necessary before seeing substantial further improvement.

Instead of focusing on backbone NMR data, one can also look at side chain data to evaluate the performance of force field models with respect to side chain conformations. The use of side chain scalar couplings, in particular, is a natural next step in moving beyond backbone conformational preferences, although reliable comparison requires an extensive amount of MD simulation for convergence of the related properties. This is the focus of recent work on further improving the accuracy of the Amber ff99SB force field by modifying side chain torsions. ⁶⁹ This work involved (i) identifying problematic side chains by comparing rotameric preferences from long simulations of helical peptides to those taken from PDB statistics, (ii) fixing the torsions for these problematic side chains by generating new quantum mechanical data and optimizing new torsion parameters to be applied on top of the existing Amber ff99SB force field, and, finally, (iii) using microsecond-long MD simulations to validate the new force field against a large set of NMR data, including side-chain data. These modifications are significant and can be especially important for long MD simulations that exceed the timescales associated with the rotation of buried or partially-buried side chains.

Long MD simulations have been used to assess the accuracy of force fields for other types of biomolecules. One striking example comes from the field of nucleic acids. Until recently, most simulation studies for nucleic acids had been on relatively short timescales, and the force field behavior was deemed adequate based on the observation that nucleic acid structures stayed intact. After simulating these systems on much longer timescales, however, it was found that irreversible transitions destroyed the "good" nucleic acid structures. ⁷⁰ As with proteins, the relevant torsions were refit, ⁷¹ and subsequently validated across a large class of nucleic acids and using long simulations in excess of one microsecond. ^70,72

Lipid molecules represent another class that benefits from the use of long MD simulations to validate the underlying physical models. The relaxation timescales of lipid bilayers are very long and their collective properties are difficult to converge. Using high-level quantum calculations, the CHARMM27 force field for lipids was modified by refitting torsion terms for alkanes, which comprise the aliphatic tails in these molecules. ⁷³ In tests on a fully hydrated DPPC lipid bilayer, the resulting CHARMM27r force field showed improved agreement between simulated and measured NMR order parameters for the aliphatic chains. ⁷³ The same force field also showed improved agreement with experimental data on lipid multilayers and vesicles. ⁷⁴ To complicate matters, force field errors for lipids are sometimes very subtle, and the use of different simulation conditions can reveal errors that are otherwise not apparent. ^75,76 This fact, coupled with the large spatial and long temporal scales needed to study collective motions of lipids, makes the use of long MD simulations to assess force field quality especially important for this class of molecules.

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Vector Analysis

George B. Arfken , ... Frank E. Harris , in Mathematical Methods for Physicists (Seventh Edition), 2013

Divergence, ∇⋅

The divergence of a vector A is defined as the operation

(3.52) $\nabla \cdot A=\frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}.$

The above formula is exactly what one might expect given both the vector and differential-operator character of ∇.

After looking at some examples of the calculation of the divergence, we will discuss its physical significance.

Example 3.5.4

Divergence of Coordinate Vector

Calculate ∇ ⋅ r:

$\begin{array}{llll}\hfill \nabla \cdot r& =\left({\widehat{e}}_x\frac{\partial }{\partial x}+{\widehat{e}}_y\frac{\partial }{\partial y}+{\widehat{e}}_z\frac{\partial }{\partial z}\right)\cdot \left.\right({\widehat{e}}_{x}x+{\widehat{e}}_{y}y+{\widehat{e}}_{z}z\left)\right.& \hfill & \\ \hfill & =\frac{\partial x}{\partial x}+\frac{\partial y}{\partial y}+\frac{\partial z}{\partial z},\end{array}$

which reduces to ∇ ⋅ r = 3.

Example 3.5.5

Divergence of Central Force Field

Consider next $\nabla \cdot f\left(r\right)\widehat{r}$ . Using Eq. (3.48), we write

$\begin{array}{llll}\hfill \nabla \cdot f\left(r\right)\widehat{r}& =\left({\widehat{e}}_x\frac{\partial }{\partial x}+{\widehat{e}}_y\frac{\partial }{\partial y}+{\widehat{e}}_z\frac{\partial }{\partial z}\right)\cdot \left(\frac{xf\left(r\right)}{r}{\widehat{e}}_x+\frac{yf\left(r\right)}{r}{\widehat{e}}_y+\frac{zf\left(r\right)}{r}{\widehat{e}}_z\right).& \hfill & \\ \hfill & =\frac{\partial }{\partial x}\left(\frac{xf\left(r\right)}{r}\right)+\frac{\partial }{\partial y}\left(\frac{yf\left(r\right)}{r}\right)+\frac{\partial }{\partial z}\left(\frac{zf\left(r\right)}{r}\right).& \hfill & \end{array}$

Using

$\frac{\partial }{\partial x}\left(\frac{xf\left(r\right)}{r}\right)=\frac{f\left(r\right)}{r}-\frac{xf\left(r\right)}{r^2}\frac{\partial r}{\partial x}+\frac{x}{r}\frac{df\left(r\right)}{dr}\frac{\partial r}{\partial x}=f\left(r\right)\left[\frac{1}{r}-\frac{x^2}{r^3}\right]+\frac{x^2}{r^2}\frac{df\left(r\right)}{dr}$

and corresponding formulas for the y and z derivatives, we obtain after simplification

(3.53) $\nabla \cdot f\left(r\right)\widehat{r}=2\frac{f\left(r\right)}{r}+\frac{df\left(r\right)}{dr}.$

In the special case f(r) = rⁿ , Eq. (3.53) reduces to

(3.54) $\nabla \cdot r^n\widehat{r}=\left(n+2\right)r^{n-1}.$

For n = 1, this reduces to the result of Example 3.5.4. For n = −2, corresponding to the Coulomb field, the divergence vanishes, except at r = 0, where the differentiations we performed are not defined.

If a vector field represents the flow of some quantity that is distributed in space, its divergence provides information as to the accumulation or depletion of that quantity at the point at which the divergence is evaluated. To gain a clearer picture of the concept, let us suppose that a vector field v(r) represents the velocity of a fluid ⁵ at the spatial points r, and that ρ(r) represents the fluid density at r at a given time t. Then the direction and magnitude of the flow rate at any point will be given by the product ρ(r)v(r).

Our objective is to calculate the net rate of change of the fluid density in a volume element at the point r. To do so, we set up a parallelepiped of dimensions dx, dy, dz centered at r and with sides parallel to the xy, xz, and yz planes. See Fig. 3.9. To first order (infinitesimal d r and dt), the density of fluid exiting the parallelepiped per unit time through the yz face located at x − (dx/2) will be

$\text{Flow\hspace{0.17em}out,\hspace{0.17em}face\hspace{0.17em}at}x-\frac{dx}{2}:-(\rho v_x)|{}_{(x-dx/2,y,z)}dydz.$

Note that only the velocity component v_x is relevant here. The other components of v will not cause motion through a yz face of the parallelepiped. Also, note the following: dy dz is the area of the yz face; the average of ρv_x over the face is to first order its value at (x − dx/2, y, z), as indicated, and the amount of fluid leaving per unit time can be identified as that in a column of area dy dz and height v_x . Finally, keep in mind that outward flow corresponds to that in the −x direction, explaining the presence of the minus sign.

Figure 3.9. Outward flow of ρ v from a volume element in the ± x directions. The quantities ±ρv_x must be multiplied by dy dz to represent the total flux through the bounding surfaces at x ± dx/2.

We next compute the outward flow through the yz planar face at x + dx/2. The result is

$\text{Flow\hspace{0.17em}out,\hspace{0.17em}face\hspace{0.17em}at}x+\frac{dx}{2}:+(\rho v_x)|{}_{(x+dx/2,y,z)}dydz.$

Combining these, we have for both yz faces

$\left(-(\rho v_x)|{}_{x-dx/2}+(\rho v_x)|{}_{x+dx/2}\right)dydz=\left(\frac{\partial (\rho v_x)}{\partial x}\right)dxdydz.$

Note that in combining terms at x − dx/2 and x + dx/2 we used the partial derivative notation, because all the quantities appearing here are also functions of y and z. Finally, adding corresponding contributions from the other four faces of the parallelepiped, we reach

(3.55) $\begin{array}{ll}\begin{array}{l}\text{Net}\text{flow}\text{out}\\ \text{per}\text{unit}\text{time}\end{array}\hfill & =\left[\frac{\partial }{\partial x}(\rho v_x)+\frac{\partial }{\partial y}(\rho v_y)+\frac{\partial }{\partial z}(\rho v_z)\right]dxdydz\hfill \\ \hfill & =\nabla \cdot (\rho v)dxdydz.\hfill \end{array}$

We now see that the name divergence is aptly chosen. As shown in Eq. (3.55), the divergence of the vector ρ v represents the net outflow per unit volume, per unit time. If the physical problem being described is one in which fluid (molecules) are neither created or destroyed, we will also have an equation of continuity, of the form

(3.56) $\frac{\partial \rho }{\partial t}+\nabla \cdot \left(\rho v\right)=0.$

This equation quantifies the obvious statement that a net outflow from a volume element results in a smaller density inside the volume.

When a vector quantity is divergenceless (has zero divergence) in a spatial region, we can interpret it as describing a steady-state "fluid-conserving" flow (flux) within that region (even if the vector field does not represent material that is moving). This is a situation that arises frequently in physics, applying in general to the magnetic field, and, in charge-free regions, also to the electric field. If we draw a diagram with lines that follow the flow paths, the lines (depending on the context) may be called stream lines or lines of force. Within a region of zero divergence, these lines must exit any volume element they enter; they cannot terminate there. However, lines will begin at points of positive divergence (sources) and end at points where the divergence is negative (sinks). Possible patterns for a vector field are shown in Fig. 3.10.

Figure 3.10. Flow diagrams: (a) with source and sink; (b) solenoidal. The divergence vanishes at volume elements A and C, but is negative at B.

If the divergence of a vector field is zero everywhere, its lines of force will consist entirely of closed loops, as in Fig. 3.10(b); such vector fields are termed solenoidal. For emphasis, we write

(3.57) $\nabla \cdot B=0\text{\hspace{0.17em}everywhere}\to B\text{\hspace{0.17em}is\hspace{0.17em}solenoidal.}$

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MOTION OF NUCLEI

In Ideas of Quantum Chemistry, 2007

7.3.1 BONDS THAT CANNOT BREAK

It is worth noting that the force fields correspond to a fixed (and unchangeable during computation) system of chemical bonds. The chemical bonds are treated as springs, most often satisfying Hooke's ²⁶ law (harmonic), and therefore unbreakable. ²⁷ Similarly, the bond angles are forced to satisfy Hooke's law. Such a force field is known as flexible molecular mechanics. To decrease the number of variables, we sometimes use rigid molecular mechanics, ²⁸ in which the bond lengths and the bond angles are fixed at values close to experimental ones, but the torsional angles are free to change. The argument behind such a choice is that the frequencies associated with torsional motion are much lower than those corresponding to the bond angle changes, and much much lower than frequencies of the bond length vibrations. This means that a quantity of energy is able to make only tiny changes in the bond lengths, small changes in the bond angles and large changes in the torsional angles, i.e. the torsional variables determine the overall changes of the molecular geometry. Of course, the second argument is that a smaller number of variables means lower computational costs.

flexible MM

rigid MM

Molecular mechanics represents a method of finding a stable configuration of the nuclei by using a minimization of V ( R ) with respect to the nuclear coordinates (for a molecule or a system of molecules).

The essence of molecular mechanics is that we roll the potential energy hypersurface slowly downhill from a starting point chosen (corresponding to a certain starting geometry of the molecule) to the "nearest" energy minimum corresponding to the final geometry of the molecule. The "rolling down" is carried out by a minimization procedure that traces point by point the trajectory in the configurational space, e.g., in the direction of the negative gradient vector calculated at any consecutive point. The minimization procedure represents a mechanism showing how to obtain the next geometry from the previous one. The procedure ends, when the geometry ceases to change (e.g., the gradient vector has zero length ²⁹ ). The geometry attained is called the equilibrium or stable geometry. The rolling described above is more like a crawling down with large friction, since in molecular mechanics the kinetic energy is always zero and the system is unable to go uphill ³⁰ of V.

A lot of commercial software ³¹ offers force field packets. For example, the Hyperchem package provides the force fields AMBER and MM2, ³² the program Insight offers the CVFF force field. Unfortunately, the results depend to quite a significant degree on the force field chosen. Even using the same starting geometry we may obtain final (equilibrium) results that differ very much one from another. Usually the equilibrium geometries obtained in one force field do not differ much from those from another one, but the corresponding energies may be very different. Therefore, the most stable geometry (corresponding to the lowest energy) obtained in a force field may turn out to be less stable in another one, thus leading to different predictions of the molecular structure.

A big problem in molecular mechanics is that the final geometry is very close to the starting one. We start from a boat (chair) conformation of cyclohexane and obtain a boat (chair) equilibrium geometry. The very essence of molecular mechanics however, is that when started from some, i.e. distorted boat (chair) conformation, we obtain the perfect, beautiful equilibrium boat (chair) conformation, which may be compared with experimental results. Molecular mechanics is extremely useful in conformational studies of systems with a small number of stable conformations, either because the molecule is small, rigid or its overall geometry is fixed. In such cases all or all "reasonable", ³³ conformations can be investigated and those of lowest-energy can be compared with experimental results.

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Characterization of Porous Solids VII

G. Maurin , ... R.G. Bell , in Studies in Surface Science and Catalysis, 2007

5. Conclusions

This work clearly shows that the force field derived from ab initio calculations for representing the interactions between methane and zeolite framework was revealed to be very well transferable. It allowed to reproduce accurately the microcalorimetry data across a wide range of pressure, for two different faujasite forms, DAY and NaX via our Grand Canonical Monte Carlo simulations. Our simulations describe the nature of the CH₄/zeolite interaction depending on the chemical composition of the faujasite. It has been shown that the positions of the extra-framework cations within the supercage significantly modify the adsorption properties of the zeolite material. This work is of high interest for predicting the performance of different types of zeolite materials with respect to CH₄ and thus for defining the main characteristics of the adsorbent materials able to store or separate this gas for environmental or petrochemical applications.

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How To Create A Force Field

Source: https://www.sciencedirect.com/topics/physics-and-astronomy/field-theory-physics

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