Self-consistent field calculation with potential well

Assuming that an ab initio or semiempirical technique has been used to obtain p(r), we address the important question of how the calculated electrostatic potential depends on the nature of the wavefunction used for computing p(r). Historically, and today as well, most ab initio calculations of V(r) for reasonably sized molecules have been based on self-consistent-field (SCF) or near Hartree-Fock wavefunctions and therefore do not reflect electron correlation. Whereas the availability of supercomputers has made post-Hartree-Fock calculations of V(r) (which include electron correlation) a realistic possibility even for molecules with 5 to 10 first-row atoms, there is reason to believe that such computational levels are usually not necessary and not warranted. The M0l er-Plesset theorem states that properties computed from Hartree-Fock wavefunctions using one-electron operators, as is V(r), are correct through first order " any errors are no more than second-order effects. Whereas second-order corrections may not always be insignificant, several studies have shown that near-Hartree-Fock electron densities are affected to only a minor extent by the inclusion of correlation.The limited evidence available suggests that the same is true of V(r), ° ° as is indicated also by the following example. 

Analytic gradient methods became widely used as a result of their implementation for closed-shell self-consistent field (SCF) wavefunctions by Pulay, who has reviewed the development of this topic. Since then, these methods have been extended to deal with all types of SCF wavefunctions, - as well as multi-configuration SCF (MC-SCF), - " configuration-interaction (Cl) wavefunctions, and various non-variational methods such as MoUer-Plesset (MP) perturbation theory - - and coupled-cluster (CC) techniques. - In short, it is possible to obtain analytic energy derivatives for virtually all the standard ab initio approaches. The main use of analytic gradient methods is, and will remain, the location of stationary points on a potential energy siuface, to obtain equilibrium and transition-state geometries. However, there is a specialized use in the calculation of quantities such as dipole derivatives. 

The previous result is an important one. It indicates that there can be yet another fruitful route to describe lipid bilayers. The idea is to consider the conformational properties of a probe molecule, and then replace all the other molecules by an external potential field (see Figure 11). This external potential may be called the mean-field or self-consistent potential, as it represents the mean behaviour of all molecules self-consistently. There are mean-field theories in many branches of science, for example (quantum) physics, physical chemistry, etc. Very often mean-field theories simplify the system to such an extent that structural as well as thermodynamic properties can be found analytically. This means that there is no need to use a computer. However, the lipid membrane problem is so complicated that the help of the computer is still needed. The method has been refined over the years to a detailed and complex framework, whose results correspond closely with those of MD simulations. The computer time needed for these calculations is however an order of 105 times less (this estimate is certainly too small when SCF calculations are compared with massive MD simulations in which up to 1000 lipids are considered). Indeed, the calculations can be done on a desktop PC with typical... 

As described in sect. 2, the potential of a free atom from the rare-earth sequence develops two wells. The outer well is the usual atomic well, with an asymptotically coulombic behaviour (long range potential), and contains an infinite number of states. The inner one is a short range well, which is close to the critical binding condition to support one state (4f). This model is backed up by central field self-consistent ab initio Hartree-Fock calculations for free atoms, and involves no adjustable parameters. It correctly describes (a) the order of filling of d and f subshells of the transition elements and R elements (b) the fact that filling occurs deep inside the atom and (c) the behaviour of XAS of free atoms of the Q- and R-element sequences. 

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