All-electron methods

Relativistic all-electron (AE) approaches are discussed here in brief for two reasons on one hand relativistic ab initio effective core potentials (ECPs) are derived from (atomic) AE relativistic calculations, on the other hand they are often calibrated in atomic and molecular calculations against the results from AE relativistic calculations. 

Starting point of the following considerations is a general configuration space Hamiltonian for n electrons and N nuclei, where we assume the Bom-Oppenheimer 

The indices i and j denote electrons, X and ju nuclei. is the charge of the nucleus X. For the one- and two-particle operators h and g various expressions can be inserted (e.g., relativistic, quasirelativistic or nonrelativistic all-electron or valence-only). The basic goal of quantum chemical methods is usually the approximate solution of the time-independent Schrodinger equation for a specific Hamiltonian, the system being in the state 7, i.e.. 

The most accurate electronic structure calculations nowadays applicable for atoms, molecules and also solids are based on the Dirac (D) one-particle Hamiltonian 

In some cases a finite nucleus is used, e.g., a Gaussian-type charge distribution 

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The LMTO method is the fastest among the all-electron methods mentioned here due to the small basis size. The accuracy of the general potential teclmique can be high, but LAPW results remain the gold standard . 

The projector augmented-wave (PAW) DFT method was invented by Blochl to generalize both the pseudopotential and the LAPW DFT teclmiques [M]- PAW, however, provides all-electron one-particle wavefiinctions not accessible with the pseudopotential approach. The central idea of the PAW is to express the all-electron quantities in tenns of a pseudo-wavefiinction (easily expanded in plane waves) tenn that describes mterstitial contributions well, and one-centre corrections expanded in tenns of atom-centred fiinctions, that allow for the recovery of the all-electron quantities. The LAPW method is a special case of the PAW method and the pseudopotential fonnalism is obtained by an approximation. Comparisons of the PAW method to other all-electron methods show an accuracy similar to the FLAPW results and an efficiency comparable to plane wave pseudopotential calculations [, ]. PAW is also fonnulated to carry out DFT dynamics, where the forces on nuclei and wavefiinctions are calculated from the PAW wavefiinctions. (Another all-electron DFT molecular dynamics teclmique using a mixed-basis approach is applied in [84].)... 

As an indication of the types of infonnation gleaned from all-electron methods, we focus on one recent approach, the FLAPW method. It has been used to detennine the band stmcture and optical properties over a wide energy range for a variety of crystal stmctures and chemical compositions ranging from elementary metals [ ] to complex oxides [M], layered dichalcogenides [, and nanoporous semiconductors The k p fonnulation has also enabled calculation of the complex band stmcture of the A1 (100) surface... 

Computational solid-state physics and chemistry are vibrant areas of research. The all-electron methods for high-accuracy electronic stnicture calculations mentioned in section B3.2.3.2 are in active development, and with PAW, an efficient new all-electron method has recently been introduced. Ever more powerfiil computers enable more detailed predictions on systems of increasing size. At the same time, new, more complex materials require methods that are able to describe their large unit cells and diverse atomic make-up. Here, the new orbital-free DFT method may lead the way. More powerful teclmiques are also necessary for the accurate treatment of surfaces and their interaction with atoms and, possibly complex, molecules. Combined with recent progress in embedding theory, these developments make possible increasingly sophisticated predictions of the quantum structural properties of solids and solid surfaces. 

Nemoshkalenko V V and Antonov V N 1998 Computational Methods in Solid State Physics (Amsterdam Gordon and Breach) An explicit introduction to the all-electron methods. 

In all cases the ir net charge of sulfur is positive, whereas its cr net charge is sometimes positive (133,130) and sometimes negative (122,132). The all-electrons methods, like ab initio, give a positive total net charge with the exception of the CNDO/2 method for which it is negative (134). 

We briefly discuss the performance of the relativisitic pseudopotential approximation with respect to all-electron methods, as this is the most widely used relativistic... 

A variety of more advanced, all-electron methods of this type Me available, and are generally referred to as semi-empirical calculations. The acronyms used to name the individual methods are descriptive of the manner in which atomic overlap calculations are performed. Among the more widely used semi-empirical methods are those of complete neglect of differential overlap (CNDO/2) (12), modified intermediate neglect of differential overlap (MINDO/3) (13), and modified neglect of diatomic overlap (MNDO) (14). 

Among all electron methods, that of CNDO in its variants CNDO/2 and CNDO/S has been most used. Particularly worthy of note is the work of Galasso434 where -electron methods are compared with all valence electron methods for the 3a-azapentalene anion 246 and 3a,6a-diazapentalene 262a. The conclusion drawn from this study was that core polarization plays a fundamental role in determining overall charge distribution in the ground state but is relatively less important in interpreting electronic spectra. 

All-valence and all-electron methods (except IEHT) predict that the first ionization potential of cytosine and of several derivatives of a cytosine is of the w-type (Tables XIV and XV). An ionization potential of the n-type is the first one in all tautomers of cytosine. Similarly, the electron affinities in all cytosine tautomers should be of the ir-type. 

Energies op the Lowest Empty (LEMO) and Three Highest (HOMO) Molecular Orbitals of Cytosine Calculated by Different All-Valence or All-Electron Methods... 

Two methods are mainly responsible for the breakthrough in the application of quantum chemical methods to heavy atom molecules. One method consists of pseudopotentials, which are also called effective core potentials (ECPs). Although ECPs have been known for a long time, their application was not widespread in the theoretical community which focused more on all-electron methods. Two reviews which appeared in 1996 showed that well-defined ECPs with standard valence basis sets give results whose accuracy is hardly hampered by the replacement of the core electrons with parameterized mathematical functions" . ECPs not only significantly reduce the computer time of the calculations compared with all-electron methods, they also make it possible to treat relativistic effects in an approximate way which turned out to be sufficiently accurate for most chemical studies. Thus, ECPs are a very powerful and effective method to handle both theoretical problems which are posed by heavy atoms, i.e. the large number of electrons and relativistic effects. 

Density Functional Theory (DFT) has become the method of choice for the study of the electronic structure of solids. Advances in computer technology have made possible the development of DFT-based codes providing a detailed ab initio description of the electronic structme of complex materials. Following the two celebrated papers by Hohenberg and Kohn and Kohn and Sham, a wide variety of approaches have been developed and turned into very efficient computational tools. These approaches differ in the way they represent the density, potential, and Kohn-Sham orbitals. Essentially DFT approaches can be classified in two main groups all electron methods and pseudopotential methods. 

The choice of one or the other of these approaches depends not only on the computing resources but more specially on the nature of the problem. It is obvious that properties relying on the density near the nucleus, hyperfine fields for instance, require the use of an all-electron method. In contrast, efficient structural optimizations when the unit cell changes are much more easily performed in pseudopotential approaches. 

The fortunate fact that a number of the biochemical units are conjugated heterocycles has permitted the first theoretical unraveling of problems connected with their electronic structure in the framework of the 7r-electron approximation. However simple the procedures utilized, a careful analysis of the results has allowed the interpretation of a considerable body of experimental facts as well as a number of predictions later confirmed by subsequent experimentation 0. Although these results have survived the test of the successive refinements of the 7r-electron theories and have been complemented by the introduction of a simple representation of the a-framework, the possibilities of treating the a and 7r electrons simultaneously on an equal footing had to be explored in order to establish the theory on a firmer basis and also to gain further insight into some fine features of electronic properties which are otherwise inaccessible. Thus, the first outcome of the penetration of all-valence and all-electrons methods into biochemistry has been to deepen and refine previous studies. 

There is another area in which the all- or quasi all-electrons methods enable us to go much deeper than the previous approximations, namely the field of the electronic distribution. Thus, both it and o populations are now directly attainable as well as hybridization ratios, which had previously to be assumed, both in the simple representation of the o bonds and in the parameters of the it electron theory 12-14>. 

Vext will arise as a superposition of the potentials associated with each of the nuclei making up the solid. In the most direct of pictures, one can imagine treating the solid by superposing the bare nuclear potentials for each atom, and subsequently solving for the disposition of each and every electron. This approach is the basis of the various all-electron methods. 

Two-component all-electron methods for spin-orbit coupling... 

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