Nonrelativistic electronic structure theory

The two surfaces that cross at the minimum energy crossing point must have a mechanism that couple the two. In the standard nonrelativistic electronic structure theory, there is no mechanism for interaction between the two... 

For the sake of brevity, we proceed in presenting a pragmatic approach to relativistic electronic structure theory, which is justified by its close analogy to the nonrelativistic theory and the fact that most of the finer relativistic aspects must be neglected for calculations on any atom or molecule with more than a few electrons. For a recent comprehensive account on the foundations of relativistic electronic structure theory we refer to Quiney et al. (1998b). 

Needless to say, many-electron atoms and molecules are much more complicated than one-electron atoms, and the realization of the nonrelativistic limit is not easily accomplished in these cases because of the approximations needed for the description of a complicated many-particle system. However, the signature of relativistic effects (see, for example, Chapter 3 in this book) enables us to identify these effects even without calculation from experimental observation. Two mainly experimentally oriented chapters will report astounding examples of relativistic phenomenology, interpreted by means of the methods of relativistic electronic structure theory. These methods for the theoretical treatment of relativistic effects in many-electron atoms and molecules are the subject of most of the chapters in the present volume, and with the help of this theory relativistic effects can be characterized with high precision. 

In this section, we provide a mathematical formulation of the connection between LFT and AI electronic structure theory. Let us start from an elementary and slightly abstract ligand field construction scheme. The exposition will be based on the strong-field coupling scheme which is the one that maps most readily onto AI theory. We give a construction scheme that cleanly connects the two areas. It is certainly not the only possible one but one that we find particularly transparent and illuminating. We stay at the nonrelativistic level in this section as the inclusion of relativistic effects brings in no new aspects. 

The 2 2-component of the EFG operator q is now to be determined by electronic structure theory either from relativistic or nonrelativis-tic wave functions. The expression e Qqzz in eq- (17) is the nuclear quadrupole coupling constant (NQCC) and can be obtained by experiment leading to the sought quadrupole moment eQ. Due to its dependence the EFG operator especially stresses the core region of the electronic wave function and relativity can be expected to play a major role. For the adequate treatment of heavy atoms and molecules containing heavy elements relativity is therefore indispensable and we will mention nonrelativistic results only for comparative purposes. Before we treat the subject of relativistic qzz calculations in detail a few common experimental techniques for accurate NQCC measurements are briefly discussed and the underlying physical principles mentioned. 

The thirty three papers in the proceedings of QSCP-Xni are divided between the present two volumes in the following manner. The first volume, with the subtitle Conceptual and Computational Advances in Quantum Chemistry, contains twenty papers and is divided into six parts. The first part focuses on historical overviews of significance to the QSCP workshop series and quantum chemistry. The remaining five parts, entitled High-Precision Quantum Chemistry, Beyond Nonrelativistic Theory Relativity and QED, Advances in Wave Function Methods, Advances in Density Functional Theory, and Advances in Concepts and Models, address different aspects of quantum mechanics as applied to electronic structure theory and its foundations. The second volume, with the subtitle Dynamics, Spectroscopy, Clusters, and Nanostructures, contains the remaining thirteen papers and is divided into three parts Quantum Dynamics and Spectroscopy, Complexes and Clusters, and Nanostructures and Complex Systems. ... 

Current relativistic electronic structure theory is now in a mature and well-developed state. We are in possession of sufficiently detailed knowledge on relativistic approximations and relativistic Hamiltonian operators which will be demonstrated in the course of this book. Once a relativistic Hamiltonian has been chosen, the electronic wave function can be constructed using methods well known from nonrelativistic quantum chemistry, and the calculation of molecular properties can be performed in close analogy to the standard nonrelativistic framework. In addition, the derivation and efficient implementation of quantum chemical methods based on (quasi-)relativistic Hamiltonians have facilitated a very large amount of computational studies in heavy element chemistry over the last two decades. Relativistic effects are now well understood, and many problems in contemporary relativistic quantum chemistry are technical rather than fundamental in nature. 

The treatment of the many-body system of nuclei and electrons of which solids and metals consist, makes it necessary to introduce approximations. Traditional approximations in electronic structure theory are those due to Hartree and Fock. We shall here only briefly give the main equations relevant for the discussion of the free-electron gas results (Chapter 9) and the density functional theory (Chapter 10). Consider, for instance, a metal of N atoms and each atom having Z electrons, where the number of atoms is of the order Avogadros number. The number of electrons to be considered for each atom can be lowered from the actual number to the number of valence electrons by introducing effective core potentials. The nonrelativistic Schrddinger equation for the electronic part of the problem is then... 

In view of the increasing importance of computational electronic-structure theory in chemical research, it is somewhat surprising that no comprehensive, up-to-date, technical monograph is available on this subject. This book is an attempt to fill this gap. It covers all the important aspects of modem ab initio nonrelativistic wave function-based molecular electronic-structure theory - providing sufficient in-depth background material to enable the reader to appreciate the physical motivation behind the approximations made at the different levels of theory and also to understand the technical machinery needed for efficient implementations on modem computes. 

Some areas of computational electronic-structure theory are not treated in this book. All methods discussed are strictly ab initio. Semi-empirical methods are not treated nor is density-functional theory discussed all techniques discussed involve directly or indirectly the calculation of a wave function. Energy derivatives are not covered, even though these play a prominent role in the evaluation of molecular properties and in the optimization of geometries. Relativistic theory is likewise not treated. In short, the focus is on techniques for solving the nonrelativistic molecular... 

For the case of a purely electrostatic external potential, P = (F , 0), the complete proof of the relativistic HK-theorem can be repeated using just the zeroth component f (x) of the four current (in the following often denoted by the more familiar n x)), i.e. the structure of the external potential determines the minimum set of basic variables for a DFT approach. As a consequence the ground state and all observables, in this case, can be understood as unique functionals of the density n only. This does, however, not imply that the spatial components of the current vanish, but rather that j(jc) = < o[w]liWI oM) has to be interpreted as a functional of n(x). Thus for standard electronic structure problems one can choose between a four current DFT description and a formulation solely in terms of n x), although one might expect the former approach to be more useful in applications to systems with j x) 0 as soon as approximations are involved. This situation is similar to the nonrelativistic case where for a spin-polarised system not subject to an external magnetic field B both the 0 limit of spin-density functional theory as well as the original pure density functional theory can be used. While the former leads in practice to more accurate results for actual spin-polarised systems (as one additional symmetry of the system is take into account explicitly), both approaches coincide for unpolarized systems. 

In principle, all four-component molecular electronic structure codes work like their nonrelativistic relatives. This is, of course, due to the formal similarity of the theories where one-electron Schrbdinger operators are replaced by four-component Dirac operators enforcing a four-component spinor basis. Obviously, the spin symmetry must be treated in a different way, i.e. it is replaced by the time-reversal symmetry being the basis of Kramers theorem. Point group symmetry is replaced by the theory of double groups, since spatial and spin coordinates cannot be treated separately. 

Finally, four-component methods will reach a high degree of applicability such that the relativistic approaches will become the standard tool for electronic structure calculations in the next decades. The four-component theories provide the general framework, in which more approximate methods—such as elimination methods for the small components, reduction methods to one-component wave functions, and also the nonrelativistic approaches—elegantly fit. This function of the four-component theories as the theoretical basis will certainly be reflected in algorithms and computer codes to be developed. 

To get a first idea of what density-functional theory is about, it is useful to take a step back and recall some elementary quantum mechanics. In quantum mechanics we learn that all information we can possibly have about a given system is contained in the system s wave function, T. Here we will exclusively be concerned with the electronic structure of atoms, molecules and solids. The nuclear degrees of freedom (e.g., the crystal lattice in a solid) appear only in the form of a potential u(r) acting on the electrons, so that the wave function depends only on the electronic coordinates.2 Nonrelativistically, this wave function is calculated from Schrodinger s equation, which for a single electron moving in a potential v(r) reads... 

Recently, molecular calculations were performed for simple compounds of element 111 at various levels of theory [95,116]. The electronic structure of the simplest molecule lllH, used as a test system for benchmark calculations similar to AuH, was studied in detail with the use of various methods (HF, DF, DK, PP, PP CCSD(T), DFT, BDF, etc.). Results are compared in Table 9 demonstrating the importance of both relativistic and correlation effects. (A more extended table can be found in [26,95]). A comparison of the relativistic (DF or ARPP) with the nonrelativistic (HF or NRPP) calculations shows that... 

Nowadays, many electronic structure codes include efficient implementations [37—41] of the Ramsey equations [42] for the calciflations of nonrelativistic spin—spin coupling constants. A vast number of publications devoted to the calculation of/-couplings can be found in the Hterature, covering different aspects such as the basis set effects [43-55], the comparison of wave function versus density functional theory (DFT) methods [56-60], or the choice of exchange-correlation functional in DFT approaches [61-68]. Excellent recent reviews of Contreras [69] andHelgaker [70] cover these particular aspects. 

Hamiltonian is not known and, as for the nonrelativistic case, further approximations have to be introduced in the wavefunction, it is tempting to derive approximate computational schemes which are still sufficiently accurate but more efficient. Here we will only summarize those approximate methods that have been used frequently to obtain information about the electronic structure of molecules with lanthanide atoms, i.e. relativistically corrected density-functional approach, pseudopotential method, intermediate neglect of differential overlap method, extended Huckel theory, and ligand field theory. 

However, the short-range behavior of the inhomogeneous part Xf r)/Ri r) (with R being S,P, Q and R, the corresponding radial function) of the electron-electron interaction potentials is different in nonrelativistic and relativistic theory because of the exponents in the series expansions of the radial functions, Eqs. (9.167) and (9.168). This difference has its origin in the structure of the differential equations Eq. (9.120), which are of second order, and Eq. (9.122), which are coupled first-order differential equations. The way in which the structure of the differential equations determines the exponents in the radial function s series expansions has been shown above. In the case of Dirac-Hartree-Fock theory, these exponents additionally depend on the type... 

This monograph presents atomic structure theory from the nonrelativistic perspective with an emphasis on calculations. Relativistic effects are considered by quasi-relativistic HamUtonians, and the Dirac many-electron case is only addressed in the appendix. However, the book provides a good presentation of tire general philosophy and strategy in numerical atomic structure theory. Prior to this monograph, Froese Fischer published a now classic book on numerical nonrelativistic Hartree-Fock theory in the 1970s [475]. Another classic text on this subject was delivered by Hartree in the 1950s [493]. 

The scope of this chapter is to provide a rudimentary imderstanding of response theory as implemented in a number of molecular electronic structure packages based on wave function mo dels or density functional theory. Only the general structure of response theory and its computer implementation are discussed, leaving out the often comphcated details of advanced wave function and density functional models. For these details the reader is referred to the literature mentioned in the last section and references therein. Although the discussion of this chapter is restricted to nonrelativistic theory, it is the same line of reasoning that is applied in relativistic response theory. In conjunction with the chapter on applications of response theory, the reader should become sufficiently familiar with the concepts and practices of response theory to allow educated use of quantum chemistry software packages. 

Within degenerate perturbation theory at the nonrelativistic level, there are in principle two contributing terms arising from expectation values of the spin-Zeeman (O Eq. 11.50) and the magnetic dipole operator (O Eq. 11.44), respectively. The latter can be shown to be zero, and the effect of the spin-Zeeman operator is to recover the free-electron g factor, ge. Thus, within a purely nonrelativistic picture, there would be no effect of the electronic structure of the molecule on the interaction between an external magnetic field and the magnetic moment of the unpaired electron. 

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