Atomic mean-field approximation

It is better to go one step further and also neglect the multicenter one-electron integrals. The resulting atomic mean-field approximation seems to be good even for molecules composed of light atoms, because it appears that the one- and two-electron multicenter contributions to spin-orbit coupling par-tisdly compensate in a systematic manner. This approxima-... 

Vahtras et al. used the CASSCF(6,6)/DZP and atomic mean-field approximations to calculate the zero-field splitting parameters of benzene, including both spin-spin coupling to the first order and spin-orbit coupling to the second order of... 

Christiansen et al. s tested the coupled-cluster response theory on spin-orbit coupling constants of substituted silylenes (HSiX, X = F, Cl, Br) in the atomic mean-field approximation. Comparison with the full Breit-Pauli showed that the approximation is quite accurate. The calcu-... 

To improve upon die mean-field picture of electronic structure, one must move beyond the singleconfiguration approximation. It is essential to do so to achieve higher accuracy, but it is also important to do so to achieve a conceptually correct view of the chemical electronic structure. Although the picture of configurations in which A electrons occupy A spin orbitals may be familiar and usefiil for systematizing the electronic states of atoms and molecules, these constructs are approximations to the true states of the system. They were introduced when the mean-field approximation was made, and neither orbitals nor configurations can be claimed to describe the proper eigenstates T, . It is thus inconsistent to insist that the carbon atom... 

At the antipodes of the latter description, there is a continuous need for better low-resolution models that involve, for instance, coarse graining of molecules, or implicit solvation. This need is motivated by the expectation that the free energy of a large system can be calculated with sufficient accuracy without requiring that all its components be described at the atomic level. In many cases, this is equivalent to the assumption that a mean-field approximation works, or that many fast degrees of freedom can be removed from the system, yet without any appreciable loss of... 

Fig. 8.13. Concentration dependences of the relative change in the sheer modulus (1, 2) tj(6u) for the equilibrium state of the system and (3, 4) rj (9n) for the non-equilibrium state and (5) a relative change in the volume per palladium atom F(0H) at T = 300 K. Curves 1 and 3 are constructed in the quasi-chemical approximation curves 2 and 4 are in the mean field approximation [213].

Because of the long time scales involved, it is currently not possible to simulate the process of peptide insertion at full atomic level description. As a consequence, in MD simulations a configuration is given, either parallel to the bilayer or inserted into it, and the evolution of the system is followed. Another approach is to perform simulations of orientational preference with the mean field approximation of lipids and water, retaining an atomic level description of the peptide. Several such simulations have been performed to determine the most likely mechanism of action for a given antimicrobial peptide. A recent review on simulations performed with antimicrobial peptides is given in ref. 78. 

Tike all effective one-electron approaches, the mean-field approximation considerably quickens the calculation of spin-orbit coupling matrix elements. Nevertheless, the fact that the construction of the molecular mean-field necessitates the evaluation of two-electron spin-orbit integrals in the complete AO basis represents a serious bottleneck in large applications. An enormous speedup can be achieved if a further approximation is introduced and the molecular mean field is replaced by a sum of atomic mean fields. In this case, only two-electron integrals for basis functions located at the same center have to be evaluated. This idea is based on two observations first, the spin-orbit Hamiltonian exhibits a 1/r3 radial dependence and falls off much faster... 

The one-center approximation allows for an extremely rapid evaluation of spin-orbit mean-field integrals if the atomic symmetry is fully exploited.64 Even more efficiency may be gained, if also the spin-independent core-valence interactions are replaced by atom-centered effective core potentials (ECPs). In this case, the inner shells do not even emerge in the molecular orbital optimization step, and the size of the atomic orbital basis set can be kept small. A prerequisite for the use of the all-electron atomic mean-field Hamiltonian in ECP calculations is to find a prescription for setting up a correspondence between the valence orbitals of the all-electron and ECP treatments.65-67... 

We will skip reactions in which three or more atoms or molecules take place. The rate equations for these reactions can be derived in the same way as those for the bimolecular reactions. We want to address here the question of what to do when the mean-field approximation is too crude. It is possible the simulate the evolution of the adlayer taken into account the exact configuration using dynamic Monte Carlo. The way to do this is described later on. Here we want to show how the rate equations can be improved. Working with these improved equations is often preferable, because they require less (computer) work, and they are easier to interpret than the results of simulations. Even when one does perform simulations, it is useful to have rate equations, or their improved version, to interpret the results of the simulations. 

Computation of the spin-orbit contribution to the electronic g-tensor shift can in principle be carried out using linear density functional response theory, however, one needs to introduce an efficient approximation of the two-electron spin-orbit operator, which formally can not be described in density functional theory. One way to solve this problem is to introduce the atomic mean-field (AMEI) approximation of the spin-orbit operator, which is well known for its accurate description of the spin-orbit interaction in molecules containing heavy atoms. Another two-electron operator appears in the first order diamagnetic two-electron contribution to the g-tensor shift, but in most molecules the contribution of this operator is negligible and can be safely omitted from actual calculations. These approximations have effectively resolved the DET dilemma of dealing with two-electron operators and have so allowed to take a practical approach to evaluate electronic g-tensors in DET. Conventionally, DET calculations of this kind are based on the unrestricted... 

Interstitial impurities (e.g., hydrogen of helium atoms) can easily move over the interstitial cavities in LRC but comparatively large size partial vacancies on the boundaries are traps for small-size atoms. It was established in a series of experiments that amorphous alloys contain traps for hydrogen and helium atoms and that the concentration of these traps is rather high [6.56]. The concentration dependence of interstitial diffusivity in disordered structures was considered phenomenologically [6.56] and by mean-field approximation [6.57],... 

In recent years, there have been many attempts to combine the best of both worlds. Continuum solvent models (reaction field and variations thereof) are very popular now in quantum chemistry but they do not solve all problems, since the environment is treated in a static mean-field approximation. The Car-Parrinello method has found its way into chemistry and it is probably the most rigorous of the methods presently feasible. However, its computational cost allows only the study of systems of a few dozen atoms for periods of a few dozen picoseconds. Semiempirical cluster calculations on chromophores in solvent structures obtained from classical Monte Carlo calculations are discussed in the contribution of Coutinho and Canuto in this volume. In the present article, we describe our attempts with so-called hybrid or quantum-mechanical/molecular-mechanical (QM/MM) methods. These concentrate on the part of the system which is of primary interest (the reactants or the electronically excited solute, say) and treat it by semiempirical quantum chemistry. The rest of the system (solvent, surface, outer part of enzyme) is described by a classical force field. With this, we hope to incorporate the essential influence of the in itself uninteresting environment on the dynamics of the primary system. The approach lacks the rigour of the Car-Parrinello scheme but it allows us to surround a primary system of up to a few dozen atoms by an environment of several ten thousand atoms and run the whole system for several hundred thousand time steps which is equivalent to several hundred picoseconds. 

The unit of length is then ao (the Bohr radius of the ground state foot-noteThe A unit, equal to 10-lom or 0.1 nm, finds favour with spectro-scopists for the simple reason that it is roughly equal to the diameter of a H atom, of H = 5.29177 x 10-11 m), while the unit of time is set by the time taken for an electron in the first Bohr orbit to travel one Bohr radius, which turns out to be 2.41889 x 10-17 s. This last number is particularly interesting even the fastest pulsed lasers (femtoseconds) are still slow on the fundamental atomic timescale. The fact that the atomic unit of time is so short is one reason why the static mean field approximation works so well for many-electron atoms. It gives some idea of the timescale on which the approximation might be expected to break down. 

Theory can now provide much valuable guidance and interpretive assistance to the mechanistic photochemist, and the evaluation of spin-orbit coupling matrix elements has become relatively routine. For the fairly large molecules of common interest, the level of calculation cannot be very high. In molecides composed of light atoms, the use of effective charges is, however, probably best avoided, and a case is pointed out in which its results are incorrect. It seems that the mean-field approximation is a superior way to simplify the computational effort. The use of at least a double zeta basis set with a method of wave function computation that includes electron correlation, such as CASSCF, appears to be imperative even for calculations that are meant to provide only semiquantitative results. The once-prevalent degenerate perturbation theory is now obsolete for quantitative work but will presumably remain in use for qualitative interpretations. 

---