The van der Waals Energy

The van der Waals energy l = - aJV = - ap. On a path at constant total critical density p = p, which is a  [c.621]

We first note errors in total energy means that are not greater than 0.5% for all LN versions tested. Individual energy components show errors that are generally less than 1%, with the exception of the van der Waals energy that can reach 4% for large k2. Of course, this discussion of relative errors reflects practical rather than mathematical considerations, since constants can be added to individual terms without affecting the dynamics. The relative errors  [c.253]

Figure 2-117. Dependence of the van der Waals energy on the distance between two non-con-nected atom nuclei. With decreasing atoiTiic distance the energy between the two atoms becomes attraction, going through a minimum at the van der Waals distance. Then, upon a further decrease in the distance, a rapid increase in repulsion energy is observed. Figure 2-117. Dependence of the van der Waals energy on the distance between two non-con-nected atom nuclei. With decreasing atoiTiic distance the energy between the two atoms becomes attraction, going through a minimum at the van der Waals distance. Then, upon a further decrease in the distance, a rapid increase in repulsion energy is observed.
The annihilation of a particle in a condensed environment to give a dummy atom can lead to another endpoint singularity, arising from the van der Waals energy term. Indeed, if the van der Waals interaction energy is scaled by a factor X", the free energy derivative goes to infinity as when /i 0 [31]. If n = 1 [linear coupling, as in Eq. (10)], there is a free energy singularity and the free energy derivative must be extrapolated to the endpoint /i = 0 with care (e.g., using the theoretical form If n > 4, the free  [c.179]

For each atom type there are two parameters to be determined, the van der Waals radius and the atom softness, Rq and, It should be noted that since the van der Waals energy is calculated between pairs of atoms, but parameterized against experimental data, the derived parameters represent an effective pair potential, which at least partly includes many-body contributions.  [c.22]

The van der Waals energy is the interaction between the electron clouds surrounding the nuclei. In the above treatment the atoms are assumed to be spherical. There are two instances where this may not be a good approximation. The first is when one (or both) of the atoms is hydrogen. Hydrogen has only one electron, which is always involved in bonding to the neighbouring atom. For this reason the electron distribution around the hydrogen nucleus is not spherical, rather the electron distribution is displaced towards the other atom. One way of modelling this anisotropy is to displace the position which is used in calculating vdw inwards along the bond. MM2 and MM3 use this approach with a scale factor of 0.92, i.e. the distance which enter vdw is calculated between points located 0.92 times the X-H bond distance, as shown in Figure 2.11.  [c.22]

As for the van der Waals energy, the standard electrostatic term only contains two-body contributions. For polar species the three-body contribution is quite significant, perhaps 10-20% of the two-body term.The three-body effect may be considered as the interaction between two atomic charges being modified because a third atom polarizes the charges. Such many-body effects may be modelled by including an atom polarization. " The electrostatic interaction is then given by an intrinsic contribution due to atomic charges plus a dipolar term arising from the electric field created by the other atomic charges times the polarizability tensor. As each of the atoms contributes to the field, the final set of atomic dipoles is solved self-consistently by iterative methods. Addition of such improvements significantly increases the computational time, and has only seen limited use. The neglect of polarization is probably one of the main limitations of modern force fields, at least for polar systems. It should be noted that the average polarization is included implicitly in the parameterization, since atomic charges are often selected to give a dipole moment which is larger than the observed value for an isolated molecule (i.e. the effective dipole moment for H2O in the solid state is 2.5 D, compared to 1.8 D in the gas phase ).  [c.25]

The introduction of a cut-off distance, beyond which vdw is set to zero, is quite reasonable as the neglected contributions are small. This is not true for the other part of the non-bonded energy, the Coulomb interaction. Contrary to the van der Waals energy, which falls of as the charge-charge interaction varies as This is actually tme only for the interaction between molecules carrying a net charge. The charge distribution in neutral molecules or fragments makes the long-range interaction behave as a dipole-dipole interaction. Consider for example the interaction between two carbonyl groups. The carbons carry a positive and the oxygens a negative charge. Seen from a distance, however, this looks like a bond dipole moment, not two net charges. The interaction between two dipoles behaves like not (the van der Waals interaction is between two induced dipoles, making this interaction — R ).  [c.43]

The Self-Consistent Reaction Field (SCRF) model considers the solvent as a uniform polarizable medium with a dielectric constant of s, with the solute M placed in a suitable shaped hole in the medium. Creation of a cavity in the medium costs energy, i.e. this is a destabilization, while dispersion interactions between the solvent and solute add a stabilization (this is roughly the van der Waals energy between solvent and solute). The electric charge distribution of M will furthermore polarize the medium (induce charge moments), which in turn acts back on the molecule, thereby producing an electrostatic stabilization. The solvation (free) energy may thus be written as  [c.393]

The function of thermodynamics is to provide phenomenological relationships whose validity has the authority of the laws of thermodynamics themselves. One may proceed further, however, if specific models or additional assumptions are made. For example, the use of the van der Waals equation of state allows an analysis of how P - p in Eq. III-40 should vary across the interface Tolman [36,37] made an early calculation of this type. There has been a high degree of development of statistical thermodynamics in this field (see Ref. 47 and the General References and also Sections XV-4 and XVI-3). A great advantage of this approach is that one may derive thermodynamic properties from knowledge of the intermolecular forces in the fluid. Many physical systems can be approximated with model interaction potential energies a widely used system comprises attractive hard spheres where rigid spheres of diameter b interact with an attractive potential energy, att( )-  [c.61]

Often the van der Waals attraction is balanced by electric double-layer repulsion. An important example occurs in the flocculation of aqueous colloids. A suspension of charged particles experiences both the double-layer repulsion and dispersion attraction, and the balance between these determines the ease and hence the rate with which particles aggregate. Verwey and Overbeek [44, 45] considered the case of two colloidal spheres and calculated the net potential energy versus distance curves of the type illustrated in Fig. VI-5 for the case of 0 = 25.6 mV (i.e., 0 = k.T/e at 25°C). At low ionic strength, as measured by K (see Section V-2), the double-layer repulsion is overwhelming except at very small separations, but as k is increased, a net attraction at all distances  [c.240]

Note that the van der Waals forces tliat hold a physisorbed molecule to a surface exist for all atoms and molecules interacting with a surface. The physisorption energy is usually insignificant if the particle is attached to the surface by a much stronger chemisorption bond, as discussed below. Often, however, just before a molecule fonus a strong chemical bond to a surface, it exists in a physisorbed precursor state for a short period of time, as discussed below in section AL7.3.3.  [c.294]

Hard-sphere models lack a characteristic energy scale and, hence, only entropic packing effects can be investigated. A more realistic modelling has to take hard-core-like repulsion at small distances and an attractive interaction at intennediate distances into account. In non-polar liquids the attraction is of the van der Waals type and decays with the sixth power of the interparticle distance r. It can be modelled in the fonn of a Leimard-Jones potential Fj j(r) between segments  [c.2365]

Rare-gas clusters can be produced easily using supersonic expansion. They are attractive to study theoretically because the interaction potentials are relatively simple and dominated by the van der Waals interactions. The Lennard-Jones pair potential describes the stmctures of the rare-gas clusters well and predicts magic clusters with icosahedral stmctures [139, 140]. The first five icosahedral clusters occur at 13, 55, 147, 309 and 561 atoms and are observed in experiments of Ar, Kr and Xe clusters [1411. Small helium clusters are difficult to produce because of the extremely weak interactions between helium atoms. Due to the large zero-point energy, bulk helium is a quantum fluid and does not solidify under standard pressure. Large helium clusters, which are liquid-like, have been produced and studied by Toennies and coworkers [142]. Recent experiments have provided evidence of  [c.2400]

The results of the calculations for the three systems were averaged and are displayed in Fig. 2. For unbound cAPK as well for cAPK complexed with ATP or PKI, the total conformational free energies increase monotonically in the sequence closed - intermediate - open . Binding is an interplay of opposing effects of unfavourable Poisson free energy of hydration that favors the solvation of buried protein groups, and the favourable van der Waals energy and coulombic energy that favour association of both domains.  [c.70]

The errors in the variance values (reflecting the fluctuations about the means) are larger for the total energy, variance errors can be as large as 7% for large 2 (the potential energy is the source rather than the kinetic energy) most other entries for energy components are less than 3%, except for two van der Waals values (LN 96 for BPTI and LN 3 for lysozyme) and all electrostatic entries. Note, however, that for the electrostatic energy the variance of the reference trajectory is a very small percentage of the mean value, namely 1% for BPTI. Thus, for example, the LN 96 variance (worst case for BPTI) for the electrostatic energy is still 1% of the reference energy mean although the value in the table is 33% (indicating an absolute energy variance of 16x 1.33 kcal/mol.). Thus, the values shown in Tables 2 and 3 still reflect a satisfactory agreement between LN trajectories and small-timestep analogs of the same Langevin equation. See [88] for many other examples of thermodynamic and geometric agreement.  [c.254]

The energy function V(r( ) passes a minimum at R y, which is the sum of the van der Waals radii of the atoms i and j ey is the potential well depth of the atom pair and ry is the distance between the interacting atoms. Figure 7-12 shows a plot of Eq. (27) for cy = 2.0 kcal mol and = 1.5 A. Using dV r)/ = 0 at the distance r = iCy, it can be shown that the collision diameter a, where the interaction energy is zero, amounts to R yl2.  [c.346]

Figure 7-12. Plot of the van der Waals interaction energy according to the Lennard-Jones potential given in Eq. (27) (Sj, = 2.0 kcal mol , / (, = 1.5 A). The calculated collision diameter tr is 1.34 A. Figure 7-12. Plot of the van der Waals interaction energy according to the Lennard-Jones potential given in Eq. (27) (Sj, = 2.0 kcal mol , / (, = 1.5 A). The calculated collision diameter tr is 1.34 A.
The computer time required for a molecular dynamics simulation grows with the square of the number of atoms in the system, because of the non-bonded interactions defined in the potential energy function (Eq. (32)). They absolutely dominate the time necessary for performing a single energy evaluation and therefore the whole simulation. The easiest way to speed up calculations is to reduce the number of non-bonded interactions by the introduction of so-called cutoffs. They can be ap-phed to the van der Waals and electrostatic interactions by simply defining a maximum distance at which two atoms are allowed to interact through space. If the distance is greater than this, the atom pair is not considered when calculating the non-bonded interactions. Several cutoff schemes have been introduced, from a simple sphere to switched or shifted cutoffs, which all aim to reduce the distortions in the transition region that are possibly destabilizing the simulation.  [c.362]

The potential energy function given in Eq. (32) contains bonded and non-bonded terms to evaluate the energy of a molecule or molecular system. The non-bonded Coulomb and van der Waals contributions need special attention, because every atom interacts with every other atom in the system. Mathematically expressed by a double sum, it is obvious that the major part of the computing time is consumed by these sums. To speed up simulations, these interactions were simply truncated at a fixed distance, typically 8-10 A. From another look at the potential energy function in Eq. (32), it becomes obvious that the truncation after such a short distance is only a minor problem in the case of the van der Waals interactions. The Lennard-Jones potential uses the distance between two interacting atoms i and j in the power of-6 and -12, so the function becomes very small at the cutoff distance. The problem is much more serious for the electrostatic part of the non-bonded interaction, because Coulomb s law uses the reciprocal of the distance r,j, resulting in long-range electrostatic interactions. The unscreened interaction of two full charges on opposite sides of a protein is still significant. At 100 A, the electrostatic interaction is greater than 3 kcal moT [26].  [c.368]

Figure 4-15 A van der Waals Potential Energy Function. The Energy minimum is shallow and the interatomic repulsion energy is steep near the van der Waals radius. Figure 4-15 A van der Waals Potential Energy Function. The Energy minimum is shallow and the interatomic repulsion energy is steep near the van der Waals radius.
For all except the very simplest systems the potential energy is a complicated, multidimensional function of the coordinates. For example, the energy of a conformation of ethane is a function of the 18 internal coordinates or 24 Cartesian coordinates that are required to completely specify the structure. As we discussed in Section 1.3, the way in which the energy varies with the coordinates is usually referred to as the potential energy burface (sometimes called the hypersurface). In the interests of brevity all references to energy should be taken to mean potential energy for the rest of this chapter, except where explicitly stated otherwise. For a system with N atoms the energy is thus a function of 3N — 6 internal or 3N Cartesian coordinates. It is therefore impossible to visualise the entire energy surface except for some simple cases where the energy is a function of just orie or two coordinates. For example, the van der Waals energy of two argon atoms (as might be modelled using the Lennard-Jones potential function) depends upon just one coordinate the interatomic distance. Sometimes we may wish to visualise just a part of the energy surface. For example, suppose we take an extended conformation of pentane and rotate the two central carbon-carbon bonds so that the torsion angles vary from 0 to 360 , calculating the energy of each structure generated. The energy in this case is a function of just two variables and can be plotted as a contour diagram or as an isometric plot, as shown in Figure 5.1.  [c.271]

The three phases of Fig. IV-4 meet at a line, point J in the figure the line is circular in this case. There exists correspondingly a line tension X, expressed as force or as energy per unit length. Line tension can be either positive or negative from a theoretical point of view experimental estimates have ranged from -10 to +10 dyn for various systems (see Ref. 64). A complication is that various authors have used different defining equations (see Ref. 65 and also Section X-5B). Neumann and co-workers have proposed a means to measure the line tension from the shape of the meniscus formed near a wall comprising vertical stripes of different wettability [66]. Kerins and Widom [67] applied three models including the van der Waals theory (Section III-2B) to the density profiles near the three-phase contact line and find both possitive and negative line tensions.  [c.113]

The classic explanation for the presence of an activation energy in the case where dissociation occurs on chemisorption is that of Lennard-Jones [113] and is illustrated in Fig. XVIII-12 for the case of O2 interacting with an Ag(llO) surface. The curve labeled O2 represents the variation of potential energy as the molecule approaches the surface there is a shallow minimum corresponding to the energy of physical adsorption and located at the sum of the van der Waals radii for the surface atom of Ag and the O2 molecule. The curve labeled O + O, on the other hand, shows the potential energy variation for two atoms of oxygen. At the right, it is separated from the first curve by the O2 dissociation energy of some 120 kcal/mol. As the atoms approach the surface, chemical bond formation develops, leading to the deep minimum located at the sum of the covalent radii for Ag and O. The two curves cross, which means that O2 can first become physically adsorbed and then undergo a concerted dissociation and chemisorption process, leading to chemisorbed O atoms (see Ref. 113a for a more general diagram). In this type of sequence, the molecularly adsorbed species is known as a precursor state (see Refs. 115 and 116).  [c.703]

Implementation of the two-region method requires calculation of the interaction between the ions in region 1 and region 2. For short-range potentials (e.g. the van der Waals contribution) it is only the inner part of region 2 that contributes significantly to the energy, E, and the forces on ions in region 1. Thus in current practical implementations of the method the outer region is subdivided into two regions, 2a and 2b (Figure 11.41). In region 1, an atomistic representation is used with full relaxation of the ions. Region 2a also contains expheit ions, whereas in region 2b it is assumed that the only effect of the defect is to change the polarisation of the ions. An iterative approach is used to identify the configuration in which the forces on the ions in region 1 are zero and the ions in region 2a are at equilibrium. The displacements of the ions in region 2a are commonly determined using just the electrostatic force from the defect species alone and equals the force due to any interstitial species less the force due to any vacancies (based on  [c.640]

There are really only two elements in the geo file that we haven t seen already in the input file, the 10.0 in row 1 and a 1 in row 2. Neither is essential the file will run without them. The first addition is a maximum time for the run, which is set at 10.0 minutes. Exeeeding a time maximum usually indieates a fault in the input file that has sent the eomputer into an infinite loop. In praetiee, this safety eheek on the ealeulation should never be eneountered. The 1 in row 2 is a switeh. A switeh is a number that either turns a ealeulation on or turns it off. In this ease, the original input file had a blank in row 2, eolumn 75, indieating that a preeise van der Waals energy need not be ealeulated until near the end of the eomputer run when the geometry is nearly at its final aeeuraey. Beeause the geo file is output with the eoneet geomehy (within the aeeuraey of the model) the van der Waals ealeulation switeh is on (integer 1 in position 2, 75). Although we haven t discussed van der Waals forees yet, the point here is that there are many features of an MM program that can be switched on or ojf by a properly formatted 0 or 1.  [c.103]

If the methyls in /(-butane are eclipsed, the hydrogens are also eclipsed. If the van der Waals repulsion energy in the /(-butane con formation with methyl groups eclipsed is of a higher energy than the conformation with a methyl group eclipsing a hydrogen atom, we get a potential energy curve very like Fig. 4-13 by the mechanism shown in Fig. 4-14.. Addition of this potential energy to the potential energy of the staggered and eclipsed forms of an ethane-like molecule as in Fig. 4-1 I also gives the right qualitative form for the potential energy of /t-biitane shown in Fig. 4-12. Fmpirical selection of the correct torsional parameters V], V4 and the V - term gives quantitative as well as cpialitative agreement between the composite potential energy curve for butane (Fig. 4-12) and the actual butane molecule.  [c.123]

There are several serious problems in modeling the correct van der Waals potential function. First, it is by no means obvious that the methyl group exerts a greater van der Waals repulsion on another methyl group than it does on hydrogen. Neither methyl-methyl nor hydrogen-hydrogen repulsitms can be studied in //-butane in the absence of the other. Despite what one might suppose from the relative sizes of methyl groups and hydrogen atoms, the van der Waals repulsion of a hydrogen atom is not negligible. Precisely because of its small size and consequent high charge density, hydrogen has a rather large effective atomic radius. Fuilhcrrnorc. it is not obvious whether the potential energy increase of the i auche forms of //-butane over the ami ft/rm is best modeled by inclusion of lower-order Fourier series terms or by an entirely independent van der Waals energy sum. Most recently, the latter solution has been preferred, but the form of the van der Waals potential energy function is difficult to extract from the empirical data available for the reasons just given.  [c.123]

See pages that mention the term The van der Waals Energy : [c.122]    [c.18]    [c.18]    [c.267]    [c.618]    [c.627]    [c.628]    [c.1871]    [c.2766]    [c.152]    [c.185]    [c.204]    [c.222]    [c.228]    [c.230]    [c.247]    [c.413]    [c.592]    [c.605]    [c.609]    [c.654]    [c.114]   
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Introduction to computational chemistry  -> The van der Waals Energy