Relativistic methods molecular properties

In Science, every concept, question, conclusion, experimental result, method, theory or relationship is always open to reexamination. Molecules do exist Nevertheless, there are serious questions about precise definition. Some of these questions lie at the foundations of modem physics, and some involve states of aggregation or extreme conditions such as intense radiation fields or the region of the continuum. There are some molecular properties that are definable only within limits, for example, the geometrical stmcture of non-rigid molecules, properties consistent with the uncertainty principle, or those limited by the negleet of quantum-field, relativistic or other effects. And there are properties which depend specifically on a state of aggregation, such as superconductivity, ferroelectric (and anti), ferromagnetic (and anti), superfluidity, excitons. polarons, etc. Thus, any molecular definition may need to be extended in a more complex situation. 

Ab initio quantum chemistry has advanced so far in the last 40 years that it now allows the study of molecular systems containing any atom in the Periodic Table. Transition metal and actinide compounds can be treated routinely, provided that electron correlation1 and relativistic effects2 are properly taken into account. Computational quantum chemical methods can be employed in combination with experiment, to predict a priori, to confirm, or eventually, to refine experimental results. These methods can also predict the existence of new species, which may eventually be made by experimentalists. This latter use of computational quantum chemistry is especially important when one considers experiments that are not easy to handle in a laboratory, as, for example, explosive or radioactive species. It is clear that a good understanding of the chemistry of such species can be useful in several areas of scientific and technological exploration. Quantum chemistry can model molecular properties and transformations, and in... 

In this chapter, we therefore consider whether it is possible to eliminate spin-orbit coupling from four-component relativistic calculations. This is a situation quite different from that of more approximate relativistic methods where a considerable effort is required for the inclusion of spin-orbit coupling. We have previously shown that it is indeed possible to eliminate spin-orbit coupling from the calculation of spectroscopic constants [12,13]. In this chapter, we consider the extension of the previous result to the calculation of second-order electric and magnetic properties, i.e., linear response functions. Although the central question of this article may seem somewhat technical, it will be seen that its consideration throws considerable light on the fundamental interactions in molecular systems. We will even claim that four-component relativistic theory is the optimal framework for the understanding of such interactions since they are inherently relativistic. 

Abstract. Investigation of P,T-parity nonconservation (PNC) phenomena is of fundamental importance for physics. Experiments to search for PNC effects have been performed on TIE and YbF molecules and are in progress for PbO and PbF molecules. For interpretation of molecular PNC experiments it is necessary to calculate those needed molecular properties which cannot be measured. In particular, electronic densities in heavy-atom cores are required for interpretation of the measured data in terms of the P,T-odd properties of elementary particles or P,T-odd interactions between them. Reliable calculations of the core properties (PNC effect, hyperfine structure etc., which are described by the operators heavily concentrated in atomic cores or on nuclei) usually require accurate accounting for both relativistic and correlation effects in heavy-atom systems. In this paper, some basic aspects of the experimental search for PNC effects in heavy-atom molecules and the computational methods used in their electronic structure calculations are discussed. The latter include the generalized relativistic effective core potential (GRECP) approach and the methods of nonvariational and variational one-center restoration of correct shapes of four-component spinors in atomic cores after a two-component GRECP calculation of a molecule. Their efficiency is illustrated with calculations of parameters of the effective P,T-odd spin-rotational Hamiltonians in the molecules PbF, HgF, YbF, BaF, TIF, and PbO. 

The electronic structure parameters describing the P,T-odd interactions of electrons (sections 7, 8, and 10) and nucleons (section 9) including the interactions with their EDMs should be reliably calculated for interpretation of the experimental data. Moreover, ab initio calculations of some molecular properties are usually required even for the stage of preparation of the experimental setup. Thus, electronic structure calculations suppose a high level of accounting for both correlations and relativistic effects (see below). Modern methods of relativistic ab initio calculations (including very... 

In summary, conventional relativistic ECP s provide an efficient mean to calculate molecular properties up to and including the third row transition elements in cases where the spin-orbit coupling is weak. ECP s can also be used together with explicit relativistic no-pair operators. Such ECP s are somewhat more precise at at the atomic level, but of essentially the same quality as conventional relativistic ECP s in molecular applications. It should also be possible to combine the ECP formalism with full Fock-Dirac methods, but this has yet not been done. 

Apart from primary structural and energetic data, which can be extracted directly from four-component calculations, molecular properties, which connect measured and calculated quantities, are sought and obtained from response theory. In a pilot study, Visscher et al. (1997) used the four-component random-phase approximation for the calculation of frequency-dependent dipole polarizabilities for water, tin tetrahydride and the mercury atom. They demonstrated that for the mercury atom the frequency-dependent polarizability (in contrast with the static polarizability) cannot be well described by methods which treat relativistic effects as a perturbation. Thus, the varia-tionally stable one-component Douglas-Kroll-Hess method (Hess 1986) works better than perturbation theory, but differences to the four-component approach appear close to spin-forbidden transitions, where spin-orbit coupling, which the four-component approach implicitly takes care of, becomes important. Obviously, the random-phase approximation suffers from the lack of higher-order electron correlation. 

In the particular case of 1- or 2-component approximations to the 4-component relativistic Hamiltonian, finite-field methods is advantageous in that they allow the calculation of molecular properties without taking explicitly picture change effects into account. We will return to this in section 6.1. 

The reader may note that the general theory of molecular properties of section 2 has been developed with very few references to relativity. This is an important point and signals that the methods used in the relativistic domain has essentially the same structure as in the non-relativistic domain. This point can be emphasized further by rewriting the zeroth order Hamiltonian (169) in second quantization [80]... 

During this introduction to molecular properties attention has been on 4-component relativistic methods, but the majority of current calculations of molecular properties that take into relativistic effects do so in a manner that approximate the 4-component level of theory, that is by 1- or 2-component relativistic methods. Before closing this chapter I will therefore give some remarks on inherent difficulties associated with these alternative approaches. These difficulties do not at all invalidate these alternative approaches, but can to some extent limit their range of applicability or enforce special precautions in their application. 

The majority of calculations of molecular properties that take relativistic effects into account are at the time of writing performed with 1-component relativistic methods. The corresponding Hamiltonians are obtained from 2-component relativistic Hamiltonians by the deletion of all spin-dependent terms. 

In the vast field of ionic crystals doped with impurities, many interesting properties are related to large manifolds of excited electronic states well localised in a small, singular portion of the material, made of the impurity and some neighbour atoms, usually called a cluster, which is under the effect of the rest of the host. Relativistic molecular ab initio methods of the Quantum Chemistry like those described in Sections 2.1.1 and 2.1.2 are, in consequence, applicable to the cluster (or pseudomolecule) when the impurities are heavy elements, provided that the embedding effects of the rest of the host are properly... 

J. Autschbach. Analyzing molecular properties calculated with two-component relativistic methods using spin-free Natural Bond Orbitals NMR spin-spin coupling constants. J. Chem. Phys., 127 (2007) 124106. 

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. 

As well as the chemical shift, indirect nuclear spin-spin coupling constant (SSCC) is one of the most important molecular properties measured routinely in NMR experiments. Much effort has been devorted until now to the development of theoretical tools to compute SSCCs from first-principles theory. For SSCCs in heavy element compounds, a relativistic theory is needed. Moreover, SSCCs are sensive to electron correlation. Therefore, in the computation of SSCCs for heavy-atom included systems, density functional theory (DFT) employing two-component relativistic methods plays a major role because of its applicability to relatively large molecules. 

The transformed Hamiltonians that we have derived allow us to calculate intrinsic molecular properties, such as geometries and harmonic frequencies. We would like to be able to calculate response properties as well, with wave functions derived from the transformed Hamiltonian. If we used a method such as the Douglas-Kroll-Hess method, it would be tempting to simply evaluate the property using the nonrelativistic property operators and the transformed wave function. As we saw in section 15.3, the property operators can have a relativistic correction, and for properties sensitive to the environment close to the nuclei where the relativistic effects are strong, these corrections are likely to be significant. To ensure that we do not omit important effects, we must derive a transformed property operator, starting from the Dirac form of the property operator. 

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