Nonrelativistic quantum chemistry

Nonrelativistic quantum chemistry has been discussed so far. But transition metal (starting already from the first row) and actinide compounds cannot be studied theoretically without a detailed account of relativity. Thus, the multiconfigurational method needs to be extended to the relativistic regime. Can this be done with enough accuracy for chemical applications without using the four-component Dirac theory Much work has also been done in recent years to develop a reliable and computationally efficient four-component quantum chemistry.25,26 Nowadays it can be combined, for example, with the CC approach for electron correlation. The problem is that an extension to multiconfigurational... 

The self-consistent field approach in relativistic quantum chemistry provides one of the most convenient and useful computational tools for the study of the electronic structure and properties of atoms, molecules and solids just as it does in nonrelativistic quantum chemistry. This chapter describes only methods in which the motion of electrons is described by the Dirac operator, namely... 

We have still to examine the apparently innocuous requirement that y/ E The trial functions of nonrelativistic quantum chemistry usually take the choice of for granted, but the more complicated structure of Dirac spinors requires closer scrutiny. It is the failure to realize that the variational procedure will not, of itself, yield precisely the correct relations between the spinor components that is at the root of the pathologies listed in papers like [38-43,45]. 

The know-how of nonrelativistic quantum chemistry can be fully used. 

Eqs. (l)-(3), (13), and (19) define the spin-free CGWB-AIMP relativistic Hamiltonian of a molecule. It can be utilised in any standard wavefunction based or Density Functional Theory based method of nonrelativistic Quantum Chemistry. It would work with all-electron basis sets, but it is expected to be used with valence-only basis sets, which are the last ingredient of practical CGWB-AIMP calculations. The valence basis sets are obtained in atomic CGWB-AIMP calculations, via variational principle, by minimisation of the total valence energy, usually in open-shell restricted Hartree-Fock calculations. In this way, optimisation of valence basis sets is the same problem as optimisation of all-electron basis sets, it faces the same difficulties and all the experience already gathered in the latter is applicable to the former. 

Relativity adds a new dimension to quantum chemistry, which is the choice of the Hamiltonian operator. While the Hamiltonian of a molecule is exactly known in nonrelativistic quantum mechanics (if one focuses on the dominating electrostatic monopole interactions to be considered as being transmitted instantaneously), this is no longer the case for the relativistic formulation. Numerical results obtained by many researchers over the past decades have shown how Hamiltonians which capture most of the (numerical) effect of relativity on physical observables can be derived. Relativistic quantum chemistry therefore comes in various flavors, which are more or less well rooted in fundamental physical theory and whose relation to one another will be described in detail in this book. The new dimension of relativistic Hamiltonians makes the presentation of the relativistic many-electron theory very complicated, and the degree of complexity is far greater than for nonrelativistic quantum chemistry. However, the relativistic theory provides the consistent approach toward the description of nature molecular structures containing heavy atoms can only be treated correctly within a relativistic framework. Prominent examples known to everyone are the color of gold and the liquid state of mercury at room temperature. Moreover, it must be understood that relativistic quantum chemistry provides universal theoretical means that are applicable to any element from the periodic table or to any molecule — not only to heavy-element compounds. 

The Hamiltonian of the two-electron atom already features all pair-interaction operators that are required to describe a system with an arbitrary number of electrons and nuclei. Hence, the step from one to two electrons is much larger than from two to an arbitrary number of electrons. For the latter we are well advised to benefit from the development of nonrelativistic quantum chemistry, where the many-electron Hamiltonian is exactly known, i.e., where it is simply the sum of all kinetic energy operators according to Eq. (4.48) plus all electrostatic Coulombic pair interaction operators. 

Note that this presentation of the approximation of the relativistic many-electron wave function closely follows the systematic construction of electronic wave functions in nonrelativistic quantum chemistry. Consequently, a truncated Cl expansion that takes only all single substitutions into account... 

This book represents an excellent account of the basics of nonrelativistic quantum chemistry. All essential concepts of many-electron theory are introduced. It is extremely useful, as relativistic, first-quantized quantum chemistry heavily exploits the historically older nonrelativistic quantum chemistry. 

Primas and Mtiller-Herold have written one of the most concise and precise introductions to nonrelativistic quantum chemistry. Unfortunately, this inspiring text, which reaches beyond the standard technique-dominated presentations, is only... 

Technically, the four-component quantum chemical methods for molecules will turn out to be very much like those known from nonrelativistic quantum chemistry. Historically, the latter based on Schrodinger s one-electron Hamiltonian were developed first. Four-component molecular quantum chemistry... 

The reason for the introduction of Gaussian-type basis functions is the fact that two-electron integrals can be solved analytically. All relativistic one- and two-electron integrals can be evaluated using standard techniques that have been developed in nonrelativistic quantum chemistry. For a detailed discussion we may therefore refer to the book by Helgaker, Jorgensen and Olsen [284] and include here only a few general comments. 

The book by Szabo and Ostlund rests on the one-electron basis-set expansion of the orbitals but does not utilize the abstract language developed by Helgaker, Jargensen and Olsen in their book mentioned below. It can be recommended as a first step to readers who want to learn more about basis-set methods in nonrelativistic quantum chemistry. 

But the most important feature for practical purposes comes with a certain approximation, which is the scalar-relativistic variant of DKH. This one-component DKH approximation, in which all spin-dependent operators are separated by Dirac s relation and then simply omitted, is particularly easy to implement in widely available standard nonrelativistic quantum chemistry program packages, as Figure 12.4 demonstrates. Only the one-electron operators in matrix representation are modified to account for the kinematic or (synonymously) scalar-relativistic effects. The inclusion of the spin-orbit terms requires a two-component infrastructure of the computer program. The consequences of the neglect of spin-orbit effects have been investigate in pilot studies such as those reported in Refs. [626,655] and, naturally, the accuracy depends on the systems under consideration (see also section 14.1.3.2 and chapter 16 for further discussion). 

The preceding three chapters have already introduced Hamiltonians of reduced dimension. Particularly successful in variational calculations are the DKH and ZORA approaches. In their scalar-relativistic variant, they can easily be implemented in a computer program for nonrelativistic quantum chemistry so that spin remains a good quantum number leading to great computational advantages (if this approximation is justifiable). Already for these methods we have seen that numerous approximations can be made in order to increase their computational efficiency with little or even no loss of accuracy compared with a four-component reference calculation with the same type of total wave function. 

M. Reiher. On the definition of local spin in relativistic and nonrelativistic quantum chemistry. Faraday Discuss., 135 (2007) 97-124. 

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. 

It is well known from nonrelativistic quantum chemistry that mean-field methods, such as the Hartree-Fock (HF) model, provide mainly qualitative insights into the electronic structure and bonding of molecules. To obtain reliable results of chemical accuracy usually requires models that go beyond the mean field and account for electron correlation. There is no reason to expect that the mean-field approach should perform significantly better in this respect for the relativistic case, and so we are led to develop schemes for introducing correlation into our models for relativistic quantum chemistry. 

There is no fundamental change in the concept of correlation between relativistic and nonrelativistic quantum chemistry in both cases, correlation describes the difference between a mean-field description, which forms the reference state for the correlation method, and the exact description. We can also define dynamical and non-dynamical correlation in both cases. There is in fact no formal difference between a nonrelativistic spin-orbital-based formalism and a relativistic spinor-based formalism. Thus we should be able to transfer most of the schemes for post-Hartree-Fock calculations to a relativistic post-Dirac-Hartree-Fock model. Several such schemes have been implemented and applied in a range of calculations. The main technical differences to consider are those arising from having to deal with integrals that are complex, and the need to replace algorithms that exploit the nonrelativistic spin symmetry by schemes that use time-reversal and double-group symmetry. 

Another correlation method commonly used in nonrelativistic quantum chemistry is the coupled-cluster (CC) method. In this method the wave function is developed by applying an exponential wave operator to an A-particle reference function. 

The special importance of multiconfiguration SCF (MCSCF) methods in relativistic calculations was discussed in the introduction to this chapter. But quite apart from this, we know that MCSCF methods are very useful in nonrelativistic quantum chemistry, where various brands of the method, such as complete active space (CAS) SCF,... 

Within these approximations, we now have a cluster of particles described in a mass-centered, nonrotating system. How does molecular structure arise out of this At this point the issues we face are common to both relativistic and nonrelativistic quantum chemistry. To introduce the concept of structure, we have to mathematically divide the particle cluster into a nuclear and an electronic part. In solving the electronic part of the equation, the other, nuclear, part is treated as a classical semi-rigid framework that we can adjust parametrically. The technique most commonly used to achieve this separation is the Born-Oppenheimer approximation, which may be regarded as the lowest order of an expansion about a system of infinite nuclear masses. The problems inherent in the application of this approximation are the same both for relativistic and nonrelativistic models. Whether the same measures should also be taken when the approximation starts to break down has not, to our knowledge, been explored. 

The emergence of relativistic quantum chemistry has been one of the more remarkable developments within computational chemistry over the past decades. Since the early work of Dirac, relativity has always been a part of the overall quantum chemical picture, but it has mostly been neglected on the grounds that the effects were considered small and the methods to treat them were poorly developed and expensive to use. However, as nonrelativistic quantum chemistry became more powerful and accurate, the lower rows of the periodic system came within reach of computational studies, and it became clear that relativistic effects had a significant influence on a number of physical and chemical properties. The start of the modern era of relativistic quantum chemistry may be traced back to a review article by Pyykko (1978) and to articles by Pitzer (1979) and by Pyykko and Desclaux (1979). 

A number of static perturbations arise from internal interactions or fields, which are neglected in the nonrelativistic Born-Oppenheimer electronic Hamiltonian. The relativistic correction terms of the Breit-Pauli Hamiltonian are considered as perturbations in nonrelativistic quantum chemistry, including Darwin corrections, the mass-velocity correction, and spin-orbit and spin-spin interactions. Some properties, such as nuclear magnetic resonance shielding tensors and shielding polarizabilities, are computed from perturbation operators that involve both internal and external fields. 

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