First-principles method

Quantum chemical first-principles methods are based on the fundamental principles of quantum mechanics. Despite the complicated nature of these methods, we try to give in this section a brief but comprehensive formal description of the most important ones. Technical and practical aspects are left aside as this scientific account is not intended to be a manual for their usage. We may refer the reader interested in these aspects to the excellent general books on computational chemistry by Cramer [46] and by Jensen [47] or to the classic text [48]. 

The central physical quantity of interest for chemistry and supramolecular chemistry is the total electronic energy E i of a molecular system consisting of N electrons and M atomic nuclei. This energy is the solution of the time-independent electronic Schrodinger equation. 

The most simple ansatz d o for the complicated function Tj, = i d(ri, r2. r ) which depends on 3N variables, namely, on the coordinates of all electrons, is an antisymmetrized product of one-electron functions, i.e. of molecular (spin) orbitals V i. 

It was soon realized that the error introduced by the Hartree-Fock model, which is the so-called correlation energy A = E i — E , is small for closed-shell systems (of the order of a few percent) but decisive for chemical reaction energetics. Moreover, weak interactions of the van der Waals type cannot be described with such a single-determinant Hartree-Fock model. Consequently, Hartree-Fock calculations on supramolecular assemblies, whose interaction is governed by weak dispersion forces, cannot provide accurate quantitative results and are likely to yield a wrong qualitative picture. However, in certain cases Hartree-Fock results may be of value. For instance, Houk and coworker have investigated the role of [C-H O] interactions in supramolecular complexes by means of dynamic and static Hartree-Fock calculations [59]. 

In order to improve on the Hartree-Fock model, the use of perturbation theory is common. The first energy correction is obtained at second order and the corresponding method is calles second-order Moller-Plesset perturbation theory (MP2). MP2 calculations provide a first estimate for the correlation energy SE, which turned out to be also useful for estimates of the interaction energy in cases dominated by dispersive interactions (see the next section for an overview on how interaction energies can be calculated from total electronic energy estimates). 

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First-principles Methods for Predicting Protein Structure... 

HyperChem currently supports one first-principle method ab initio theory), one independent-electron method (extended Hiickel theory), and eight semi-empirical SCFmethods (CNDO, INDO, MINDO/3, MNDO, AMI, PM3, ZINDO/1, and ZINDO/S). This section gives sufficient details on each method to serve as an introduction to approximate molecular orbital calculations. For further details, the original papers on each method should be consulted, as well as other research literature. References appear in the following sections. 

At the other end of the spectrum are the ab initio ( from first principles ) methods, such as the calculations already discussed for H2 in Chapter 4. I am not trying to imply that these calculations are correct in any strict sense, although we would hope that the results would bear some relation to reality. An ab initio HF calculation of the potential energy curve for a diatomic Aj will generally give incorrect dissociation products, and so cannot possibly be right in the absolute sense. The phrase ab initio simply means that we have started with a certain Hamiltonian and a set of basis functions, and then done all the intermediate calculations with full rigour and no appeal to experiment. 

At present there have been no attempts to explore the molecular electronic basis for liquid crystal anchoring or to calculate Rapini-Popoular coefficients for real systems using first principles methods. However, there have been a number of theoretical treatments which have suggested that the ordering of... 

First Principles Methods and the Density Functional Principle... 

In addition to all these considerations and possible routes for further work, there is also the salient point that first principles methods are, in principle, applicable to hypothetical and as yet unsynthesised systems. The implication of this is that the combination of such first principles techniques with state of the art molecular synthesis and characterisation technology holds substantial promise for solutions to structure property problems. 

Unlike electrostatic forces, chemical forces between the probing tip and the probed surface have been shown to profoundly affect the tunneling current from a certain onset. Owing to the advent of first-principle methods and powerful computers, it could finally be resolved by a calculation of the combined tip-sample system [ 15 ]. The point of onset for chemical bonding on metals was found to be at a distance of 4—5 A. As the tip approaches the surface, chemical forces rapidly become large enough to... 

Ab-initio (nonempirical, from first principles ) methods also use the HF-SCF model but includes all electrons and uses minimal approximation. Basis sets of functions based on linear combinations of atomic orbitals (LCAO) increase in complexity from the simplest (STO-3G) to more complex (3-21G( )) to extended basis sets (6-311 + G ) for the most accurate (and most time-consuming) results. Treat systems up to 50 atoms. 

The first-principle method used in the present study will be outlined below. Population balance analysis on a perfectly mixed batch crystallizer with negligible crystal breakage and agglomeration yields the familiar nucleation rate equation used by Misra and White (5)... 

At one end of the spectrum are first-principles methods where the only input requirements are the atomic numbers Za, Zb,. .. the relevant mole fractions and a specified crystal structure. This is a simple extension to the methods used to determine the lattice stability of the elements themselves. Having specified the atomic numbers, and some specific approximation for the interaction of the relevant wave functions, there is no need for any further specification of attractive and repulsive terms. Other properties, such as the equilibrium atomic volumes, elastic moduli and charge transfer, result automatically from the global minimisation of... 

Recent advances in heterogeneous catalysis enabled by first-principles methods... 

For CFD modeling a detailed chemical mechanism for the relevant gas phase and surface reaction steps is necessary. Due to the difficulty involved in determining kinetic and thermodynamic parameters for the elementary steps, these are often based on empiricism and even guessing. Here, theoretical first-principles methods can be very helpful. 

In subsection 3.1, we will present GGA and LDA calculations for Au clusters with 6scalar-relativistic pseudo-potential for LDA and GGA (see Fig 1). These calculations show the crucial relevance of the level of density functional theory (DFT), namely the quality of the exchange-correlation functional, to predict the correct structures of Au clusters. Another, even more critical, example is presented in subsection 3.2, where we show that both approaches, LDA and GGA, predict the cage-like tetrahedral structure of Au2o as having lower energy than amorphous-like isomers, whereas for other Au clusters, namely Auig, Au ... 

In this review we shall first establish the theoretical foundations of the semi-classical theory that eventually lead to the formulation of the Breit-Pauli Hamiltonian. The latter is an approximation suited to make the connection to phenomenological model Hamiltonians like the Heisenberg Hamiltonian for the description of electronic spin-spin interactions. The complete derivations have been given in detail in Ref. (21), but turn out to be very involved and are thus scattered over many pages in Ref. (21). For this reason, we aim here at a summary that is as brief and concise as possible so that all relevant connections between different levels of approximation are evident. This allows us to connect present-day quantum chemical methods to phenomenological Hamiltonians and hence to establish and review the current status of these first-principles methods applied to transition-metal clusters. 

When the limiting conditions of the friction approximation are not valid, e.g., there is strong non-adiabatic coupling or rapid temporal variation of the coupling, there is at present no well-defined first principles method to calculate the breakdown in the BOA. The fundamental problem is that DFT cannot calculate excited states of adsorbates and quantum chemistry techniques, that can in principle calculate excited states, are not possible for extended systems. 

The infancy of these first-principles methods as applied to periodic zeolite lattices means that further detailed work is necessary, particularly in the area of verification of the ability of the pseudopotential to reproduce dynamic as well as static structural properties. However, the results found with these methods demonstrate that the debate concerning the modeling of the activation of methanol within a zeolite is far from concluded. The proton transfer to methanol as a reaction in its own right is, however, of relatively little interest. It does not govern the pathway or energetics of reactions such as dehydration to give dimethyl ether (DME). These are governed instead by the individual transition states that lead to the products, as we discuss in the next section. 

Extensive tests have been carried out to establish the reliability of quantum-chemical schemes for metal oxide investigations. This includes schemes at a variety of levels of sophistication suitable for calculations of very large systems. In particular density functional methods offer good possibilities to treat sufficiently large systems to be applicable to many central problems in the field of photoelectrochemistry with reasonable accuracy and at very competitive computational costs. Semiempirical methods still offer a last possibility to perform reasonably accurate calculations on nanostructured systems containing several hundred atoms where first principles methods still cannot be applied routinely. 

Density functional theory (DFT) has been used extensively for the theoretical study of actinide complexes, making it one of the few first-principles methods that can... 

The ever increasing demands from distributed information systems are stimulating research and technology development. Theory has a central role because a microscopic understanding represents a fundamental step towards the innovation, design and fabrication of new materials and devices. The ability to describe structural, electronic and optical properties of new materials with accurate first-principle methods is hence of fundamental importance. 

As in the MD method, PES for KMC can be derived from first-principles methods or using empirical energy functionals described above. However, the KMC method requires the accurate evaluation of the PES not only near the local minima, but also for transition regions between them. The corresponding empirical potentials are called reactive, since they can be used to calculate parameters of chemical reactions. The development of reactive potentials is quite a difficult problem, since chemical reactions usually include the breaking or formation of new bonds and a reconfiguration of the electronic structure. At present, a few types of reactive empirical potentials can semi-quantitatively reproduce the results of first-principles calculations these are EAM and MEAM potentials for metals and bond-order potentials (Tersoff and Brenner) for covalent semiconductors and organics. 

An example of comparison of the results obtained by semiempirical and first-principles methods is given in table 2, which contains energies of the Pr3+ ion (4f2 electron configuration). In the results of the semiempirical calculations, two leading contributions to the intermediately coupled eigenstates of the Hamiltonian (1) are given as well. As can be seen from table 2, there is a systematic overestimation (of about 20-30%) of all energy levels 2 obtained by the first principles calculations. This overestimation, presumably due to underestimation of... 

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