Potential density function

Keywords Embedding potential Density functional theory Kinetic energy functional Orbital-free embedding... 

The difference between this Fock operator and the Hartree-Fock counterpart in Eq. (2.51) is only the exchange-correlation potential functional, Exc, which substitutes for the exchange operator in the Hartree-Eock operator. That is, in the electron-electron interaction potential, only the exchange operator is replaced with the approximate potential density functionals of the exchange interactions and electron correlations, while the remaining Coulomb operator, Jj, which is represented as the interaction of electron densities, is used as is. The point is that the electron correlations, which are incorporated as the interactions between electron configurations in wavefunction theories (see Sect. 3.3), are simply included... 

Statistical mechanical theory and computer simulations provide a link between the equation of state and the interatomic potential energy functions. A fluid-solid transition at high density has been inferred from computer simulations of hard spheres. A vapour-liquid phase transition also appears when an attractive component is present hr the interatomic potential (e.g. atoms interacting tlirough a Leimard-Jones potential) provided the temperature lies below T, the critical temperature for this transition. This is illustrated in figure A2.3.2 where the critical point is a point of inflexion of tire critical isothemr in the P - Vplane. 

The first point to remark is that methods that are to be incorporated in MD, and thus require frequent updates, must be both accurate and efficient. It is likely that only semi-empirical and density functional (DFT) methods are suitable for embedding. Semi-empirical methods include MO (molecular orbital) [90] and valence-bond methods [89], both being dependent on suitable parametrizations that can be validated by high-level ab initio QM. The quality of DFT has improved recently by refinements of the exchange density functional to such an extent that its accuracy rivals that of the best ab initio calculations [91]. DFT is quite suitable for embedding into a classical environment [92]. Therefore DFT is expected to have the best potential for future incorporation in embedded QM/MD. 

Stanton, R.V., Heutsough, D.S., Merz, K.M. Jr An Examination of a Density Functional-Molecular Mechanical Coupled Potential. J. Comput. Chem. 16 (1995) 113-128. 

L and W C Mackrodt 1994. Density Functional Theory and Interionic Potentials. Philosophical gazine B69-.871-878. 

To overcome these limitations, the hybrid QM-MM potential can employ ad initio or density function methods in the quantum region. Both of these methods can ensure a higher quantitative accuracy, and the density function methods offer a computaitonally less expensive procedure for including electron correlation [5]. Several groups have reported the development of QM-MM programs that employ ab initio [8,10,13,16] or density functional methods [10,41-43]. 

J Li, MR Nelson, CY Peng, D Bashford, L Noodleman. Incorporating protein environments in density functional theory A self-consistent reaction field calculation of redox potentials of [2Ee2S] clusters in feiTedoxm and phthalate dioxygenase reductase. J Phys Chem A 102 6311-6324, 1998. 

J-M Mouesca, JL Chen, F Noodleman, D Bashford, DA Case. Density functional/Poisson-Boltzmann calculations of redox potentials for iron-sulfur clusters. J Am Chem Soc 116 11898-11914, 1994. 

The stiffness matrix, Cy, has 36 constants in Equation (2.1). However, less than 36 of the constants can be shown to actually be independent for elastic materials when important characteristics of the strain energy are considered. Elastic materials for which an elastic potential or strain energy density function exists have incremental work per unit volume of... 

Fig. 10(a) presents a comparison of computer simulation data with the predictions of both density functional theories presented above [144]. The computations have been carried out for e /k T = 7 and for a bulk fluid density equal to pi, = 0.2098. One can see that the contact profiles, p(z = 0), obtained by different methods are quite similar and approximately equal to 0.5. We realize that the surface effects extend over a wide region, despite the very simple and purely repulsive character of the particle-wall potential. However, the theory of Segura et al. [38,39] underestimates slightly the range of the surface zone. On the other hand, the modified Meister-Kroll-Groot theory [145] leads to a more correct picture. 

The density functional approach has also been used to study capillary condensation in slit-like pores [148,149]. As in the previous section, a simple model of the Lennard-Jones associating fluid with a single associative site is considered. All the parameters of the interparticle potentials are chosen the same as in the previous section. Our attention has been focused on the influence of association on capillary condensation and the evaluation of the phase diagram [42]. 

Let us underline some similarities and differences between a field theory (FT) and a density functional theory (DFT). First, note that for either FT or DFT the standard microscopic-level Hamiltonian is not the relevant quantity. The DFT is based on the existence of a unique functional of ionic densities H[p+(F), p (F)] such that the grand potential Q, of the studied system is the minimum value of the functional Q relative to any variation of the densities, and then the trial density distributions for which the minimum is achieved are the average equihbrium distributions. Only some schemes of approximations exist in order to determine Q. In contrast to FT no functional integrations are involved in the calculations. In FT we construct the effective Hamiltonian p f)] which never reduces to a thermo-... 

Here r is the distance between the centers of two atoms in dimensionless units r = R/a, where R is the actual distance and a defines the effective range of the potential. Uq sets the energy scale of the pair-interaction. A number of crystal growth processes have been investigated by this type of potential, for example [28-31]. An alternative way of calculating solid-liquid interface structures on an atomic level is via classical density-functional methods [32,33]. 

---