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We live in a world where the speed of light is finite and does not depend upon the direction of observation as was proved by the MichelsonMorley interferometer measurement in 1887. That experimental evidence leads to many theoretical developments that culminate with A. Einstein’s special theory of relativity at the beginning of the century. This first "revolution" of the century in physics was rapidly followed by a second one of equal importance for our understanding of the physical world at the atomic scale. In 1913 Bohr had introduced a first description of the atom that made it possible to predict the spectrum of hydrogen without introducing free parameters, but one had to wait for another decade to see the birth of quantum mechanics. After L. De Broglie suggested in 1924 that a wave should be associated with any particle, W. Heisenberg and E. Schrödinger established the theory of quantum mechanics in 19251926 which is still one the two pillars (the second being the theory of relativity) of modern physics. The next challenge was to unify these two theories at the atomic scale. This was accomplished by P.A.M. Dirac in 1928 who could write in 1929: "This general theory of quantum mechanics is now almost complete, the imperfections that still remain being in connection with the exact fitting of the theory with relativity ideas. These give rise to difficulties only when highspeed particles are involved, and are therefore of no importance in the consideration of atomic and molecular structure and ordinary chemical reactions, in which it is, indeed, usually sufficiently accurate if one neglects relativity variation of mass with velocity and assumes only Coulomb forces between the various electrons and atomic nuclei. The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much complications" (P.A.M. Dirac, Proceedings of the Royal Society, 123A, 1929, P. 714)
As it happens, people keep only a few words in mind and the above quotation has sometimes been taken as a justification of ignoring relativistic effects in atomic and molecular electronic structure. A more careful reading shows, on the contrary, great perspicacity:
 the need for "approximate practical methods" for "equations much too complicated to be soluble" that could not be fulfilled until computers become a reality about twenty years later,
 or, from a more fundamental point of view "the imperfections that still remain being in connection with the exact fitting of the theory with relativity ideas". A still unsolved problem since the general theory of relativity (gravitational force) and quantum mechanics have not yet been unified. This is still considered as one challenge for theoretical physics (see "What remains to be discovered" by J. Maddox, Brockman, Inc. 1998).
The only missing important comment is the need for the electromagnetic field to be also quantized (as suggested in the same year by W. Heisenberg and W. Pauli) leading to the most accurate theory of physics: Quantum Electrodynamics.
The remaining sections of this chapter are devoted to a brief, and more or less historical survey, of what has been done to fulfil the wish of Dirac of developing "approximate practical methods" (the following chapters of this book will give much more exhaustive descriptions of them) and show how far we have gone today from what V.M. Burke and I.P. Grant could write in 1967: "Little attention appears to have been paid to the effect of relativity on atomic wave functions since White studied the matter in 1931".
In this section we briefly review the main properties of the Dirac equation that is the basic equation to start with to build a relativistic effective Hamiltonian for atomic and molecular calculations. This single particle equation, as already stated in the introduction, was established in 1928 by P.A.M Dirac [1] as the Lorentz invariant counterpart of the Schrödinger equation. On a note let us recall that the first attempts to replace the Schrödinger equation by an equation fulfilling the requirements of special relativity started just after quantum mechanics was established. In 1926 the equation, now known as the KleinGordon equation, was derived independently by several authors [2] but this equation was historically discarded since it has solutions for probability densities r that can be either positive or negative. At that time no one realized that any relativistic equation will describe both a particle and its antiparticle and it was not until 1934 that the KleinGordon equation was correctly reinterpreted by Pauli and Weisskopf [3] as the relativistic quantum mechanical equation for spin zero particles. Ironically, the Dirac equation exhibits, and we understand now why, the same pitfall but was rapidly accepted since it explicitly includes the spin and also because of the prediction of the existence of the positron experimentally discovered in 1932 by C. D. Anderson.
As required by special relativity, space and time variables should appear in a symmetric way and this requirement is most obvious in the covariant form of the Dirac equation:
_{} 
(1) 
where m is the mass of the particle, c the speed of light and the _{} are matrices that fulfil the commutation rule
_{}where _{}is the metric tensor
_{} is the fourvector
_{}
defined in terms of the momentum vector _{} and energy E of the particle and of the vector (_{}) and scalar (_{}) potentials of the external electromagnetic field. As for all the remaining sections of this chapter we shall consider only time independent scalar external fields, we rewrite Eq. (1) in the usual form:
_{} 
(2) 
where from now on we use atomic units by putting _{}. In this system the speed of light c » 137 is the inverse of the fine structure constant. For the (4x4) Dirac matrices a and b we adopt the conventional representation that is useful when considering the nonrelativistic limit of the Dirac equation. In this standard representation the components of a are expressed in terms of the (2x2) Pauli matrices and b is taken to be diagonal, explicitly:
_{} 
_{} 
(3) 
where I and 0 are respectively the (2x2) unit and zero matrices and the Pauli matrices s have their usual definition:
_{} 
_{} 
_{} 
The eigenvalue spectrum of the Dirac equation in the Coulomb field of a nuclear charge Z is sketched in Fig. 1. First one notices the presence of two continua (one below mc^{2} and the other above mc^{2}) instead of the single positive continuum of the Schrödinger equation. These two continua are separated by a gap equal to 2mc^{2}. A nonzero value of the nuclear charge Z introduces bound states in this gap (Z>0 is illustrative of hydrogen, i.e. an electron bound to a proton, while Z<0 corresponds to positronium, i.e. a positron bound to an antiproton). The negative continuum for electronic states was of concern at the early stage of relativistic atomic structure theory and led Dirac to postulate that all states of negative energy are occupied and form the physical vacuum. This was an adhoc and empirically efficient way to insure atomic stability by preventing radiative decay into the negative energy states. We shall see later that there is no need to assume an infinite number of electrons in the vacuum. Indeed, as already pointed out, a relativistic theory describes both a particle and its antiparticle so that only the total charge of the system is conserved but not the number of particles. For physical systems in which no particleantiparticle pairs are created we should add a constraint to take this fact into account. We shall elaborate more on this point in the next section.
Z<0
Z=0
Z>0
Fig 1 Schematic
Dirac spectrum for a Coulomb field
If we remember that the eigenvalues of the Schrödinger equation are directly proportional to the mass m of a particle while the expectation mean value <r> scales as 1/m, we expect that the eigenvalues of the Dirac equation will show both an increase in the binding energy of an electron and a contraction of the radial wave functions towards the origin because of the relativistic increase in the mass given by:
^{ }
_{} 

This is indeed the case as illustrated in Fig. 2 for s orbitals. However, we should avoid oversimplified approximations, like assuming that because the "classical velocity" of an electron is a decreasing function of the principal quantum number n (<v^{2}/c^{2}> = (Za)^{2}/n), relativistic corrections should be monotonically decreasing functions of n as well. From the analytical solution of the Dirac equation for a point nucleus of charge Z it is straightforward to show that the binding energy of the s electrons is given to lowest order in (Z/c) by:
_{} 

from which it is obvious that the relativistic increase in the binding energy (the second term in the bracket) is not a monotonic function of n since it is greater for n=2 than for n=1 while it gives the same correction both for n=1 and n=3.


Fig 2 Relativistic corrections for hydrogenic s orbitals (as percentage of the non relativistic value).
Even if the n scaling is different for the expectation value of r we also notice that, percentage wise, the 2s orbital contracts more than the 1s. These relative changes between the results given by the Schrödinger and Dirac equations hold when solving both equations for the same potential. But, for a many electron system, the selfconsistent field effects change this simple picture quite significantly as we shall illustrate later.
To conclude this short introduction to the Dirac equation, let us summarize the main results for oneelectron systems:
 relativistic binding energies are larger than non relativistic ones, the largest increase is for an electron with total angular momentum j=1/2,
 the non relativistic orbital degeneracy of the energy is partially removed because of the explicit inclusion of the spinorbit coupling,
 radial electron density contracts towards the nucleus,
 because the Dirac wave functions have two components (see next chapter) with radial nodes interlaced, the radial electron density vanishes only at the origin and infinity but never at a finite distance.
If, as stated above, orbital degeneracy is partially removed, it remains that for a given value of the principal quantum number n, levels with the same j value but with different orbital angular momenta (like 2s and 2p_{1/2}) are degenerate. This is due to the fact that the original Dirac theory treats the electromagnetic field as a classical field. Quantization of the electromagnetic field lifts this degeneracy as was observed for the first time by Lamb and Retherford [4] who measured a splitting (Lamb shift) of 0.033 cm^{1} in hydrogen. The lowest order corrections introduced by the quantization of the electromagnetic field are given by the two Feynman diagrams of Fig. 3 in which the double lines stand for a bound electron in the Coulomb field of the nucleus.
Fig 3 Lowest order
QED corrections
The first diagram (labelled SE) corresponds to the emission and reabsorption of a virtual photon by an electron and describes the interaction of an electron with its own radiation field. This contribution, known as the selfenergy correction, is the dominant quantum electrodynamical (QED) contribution to the Lamb shift for electrons. The vacuum polarization given by the second diagram (VP) is associated with the interaction of the electron with a virtual electronpositron pair (remember that the Dirac equation conserves the charge of the system but not the number of particles). QED effects in atoms will be discussed in more detail in chapter 8. It is nevertheless worthwhile to give the order of magnitude for QED effects for a high Z hydrogenic 1s electron.
Table 1
Contributions to
1s electron binding energy in U^{+91}
Contribution 
Value (in eV) 
Point nucleus
Dirac equation 
132 279.61 
Finite nucleus
correction 
198.82 
Self Energy 
355.05 
Vacuum
Polarization 
88.60 
Higher order QED
corrections 
0.64 
Nuclear recoil
and nuclear polarization 
0.28 
Reduced mass
correction 
0.30 
Lamb shift (i.e.
sum of all above corrections) 
464.61 
1s binding
energy 
131 815.00 
From the values given in the above table it is obvious that the finite nucleus and QED corrections contribute with the same order of magnitude to the Lamb shift in a heavy atom. The uncertainty in the theoretical value comes not only from yet uncalculated very high order QED contributions (estimated to be less few tenths of an eV) but also from less well known nuclear parameters of uranium that can amount to an uncertainty of about 0.3 eV. Up till now the best experimental value for the 1s Lamb shift in U^{91+} obtained at the GSI [5] is 468±13 eV. This value is still an order of magnitude too imprecise to allow QED to be tested in high Coulomb fields where Za is not a small parameter.
Let us now consider the extension of the relativistic Dirac theory to many electron systems. As pointed out in the previous section, only the total charge is conserved but not the number of particles, we can thus formally write the Hamiltonian as:
_{} 
(4) 
and for atoms and molecules restrict ourselves to a physical system with N electrons without any positron. This is known as the nopair approach [6,7] and formally implies that the Hamiltonian must include some kind of constraint to prevent the creation of pairs of electronpositron and to avoid the socalled Brown Ravenhall [8] "disease" arising from the fact that discrete levels would be embedded in a sea of continuous levels. If this would be the case the discrete levels will autoionize in conflict with the physical reality that atoms and molecules are stable. To illustrate where the problem come from (the reader should not be surprised that the negative energy states found in the one electron case will not disappear for many electron systems) let us consider a twoelectron atom. The "natural" extension of the Dirac theory would be to consider the Hamiltonian:
_{} 
(5) 
where the one electron Hamiltonians h_{D} are given by Eq. (2) and g(1,2) is any kind of electronelectron interaction. To eliminate electronpositron pairs the use of a projection operator has been advocated to ensure that the twobody interaction g(1,2) connects only positive energy states among themselves. This amounts to replace Eq. (5) by
_{} 
(6) 
where the positive energy state projection operator _{} is defined in terms of one electron positive energy eigenfunctions (for both discrete and continuum eigenvalues) of some oneelectron Hamiltonian by:
_{} 
(7) 
The choice of the oneelectron Hamiltonian that defines the _{} is far from being trivial. Indeed negative and positive energy states have no absolute definition and can be assigned only after a given potential has been chosen (no potential for free electron, nuclear potential for hydrogenic solutions or any other). As for any potential all the eigenstates (both positive and negative) of the Dirac equation form a complete basis, the subspace of positive energy eigenstates in a given potential will include part of the negative subspace associated to another potential. Consequently, inconsistent use of u_{n} basis sets can lead to meaningless results. Thus, when performing HartreeFock calculations the u_{n} functions must be the Fock eigenstates and the projection operator of Eq. (7) reduces to the identity operator. This is in agreement with the result obtained by Mittleman [7] who investigated, from a variational point of view, the best possible choice for the u_{n} when the total wave function is restricted to a single determinant. The conclusion is that the use of Eq. (5) without a projection operator is justified provided that we keep only its positive energy solutions. Studying the selfadjointness of the Dirac operator and the properties of the quadratically integrable oneelectron Dirac wave functions I.P. Grant [9] reached the same conclusion.
At the beginning stage of relativistic electronic structure theory the negative energy continuum was sometimes regarded to be responsible for the failure to produce upper bounds in the total energy leading to the socalled "variational collapse" in molecular calculations when basis sets were applied. This attractive explanation was unfortunately too limited to explain also the appearance of spurious solutions. Indeed, both the existence of spurious solutions and the lack of convergence to expected levels can be traced back to originate from poor basis sets and bad finite matrix representations of the operators (in particular for the kinetic energy). For an extensive discussion see Ref. 10.
Dealing with electrons we know that the dominant interaction between them is the Coulomb repulsion corrected, because electrons are fermions, by interactions induced by their spin. The spinorbit interaction is already included in the oneelectron Dirac Hamiltonian but the twoelectron interaction should also include interactions classically known as spinotherorbit, spinspin etc… Furthermore a relativistic theory should incorporate the fact that the speed of light being finite there is no instantaneous interaction between particles. The most common way of deriving an effective Hamiltonian for a many electron system is to start from the Furry [11] bound interaction picture. A more detailed discussion is given in chapter 8 and we just concentrate on some practical considerations.
One important step is the choice of the gauge in which to express the twobody relativistic interaction. For the exact solution we know that the result should be gauge independent but this will not hold for approximate solutions (remember the length and velocity forms for dipole radiative transition probabilities). The final result for g(1,2) of Eq. (5) reads:
_{} 
(8) 
when selecting the Lorentz gauge, while the same quantity in the Coulomb gauge is given by [12]:
_{} 
(9) 
where _{} is the interelectronic distance _{} and _{} is the frequency of the virtual exchanged photon given in terms of the single particle energies e.
At the zero frequency limit Eq. (9) reduces to the Breit interaction [13] that was first derived through second order perturbation theory. Because of this perturbative derivation it was sometimes argued that the Breit operator should not be included in a selfconsistent process. But, as the bound state Furry picture gives the same result, there is no reason to exclude it from the selfconsistent field if we stay at the zero frequency limit. To go beyond would require replacing the frequency w of the virtual photon by operators.
The expectation values of the two operators defined by Eq. (89) are equal for inshell matrix elements when a local potential is used to define the singleparticle states, but they differ for DiracFock orbitals. The offshell contributions are already not the same even for local potential wave functions [14] so that the choice of gauge is of practical importance. It has been shown [15] that the Coulomb gauge is a better choice since the use of standard manybody perturbation theory includes only the reducible part of the multiphoton exchange (i.e. without crossing photons). The underlying justification is, that to restore gauge invariance to leading relativistic order, the irreducible contributions must be taken into account. As these irreducible contributions give a nonzero contribution in the Lorentz gauge while they vanish in the Coulomb gauge it is thus more efficient to express the interaction in the Coulomb gauge. The same conclusion was reached by J. Sucher [16] on the grounds that the nonpertubative use of Eq. (8) leads to incorrect energy levels already at order a^{4}.
Relativistic HartreeFock (also called DiracFock) calculations for atoms started to appear in the late 60's for light atoms using basis set expansions [17], and at the beginning of the 70's by solving the integrodifferential DiracFock equations by numerical methods [1819]. This latter approach was rapidly capable to cover the full periodic table (see Ref. 20), and a picture rather different from the well studied hydrogenic case started to emerge. Contrary to the single electron solution of the Dirac equation showing mainly the mass variation with velocity, a DiracFock calculation includes the changes in the spatial charge distribution of the electrons induced by the selfconsistent field. As the charge distribution of the inner electrons contracts towards the nucleus, the effective charge experienced by outer electrons is reduced and they may become less bound when compared to nonrelativistic predictions. There will obviously be an interplay between the direct relativistic correction (the mass variation) and the indirect one (the charge redistribution) greatly dependant on the localisation of the wave functions and whether or not they have some finite density close to the nucleus where the local speed of the electron is high. Figure 4 below illustrates the relativistic corrections to the binding energies and mean expectation value of <r> for the outermost nl occupied orbitals of atoms with nuclear charge greater than 40. Besides the discontinuities observed when the principal quantum number is changing by one unit and those due to the shell structure (for example, in the lanthanide series, the 5d orbital is occupied only in cerium, Z=58, and in gadolinium, Z=64) some general trends emerge:
 orbitals with a total angular momentum j=1/2 suffer a strong relativistic contraction as can be seen from the increase in their binding energies and the decrease in their mean expectation value of <r>,
 those with j=3/2 remain essentially unchanged until Z becomes greater than 70,
 for high angular momenta (j=5/2 and above) the indirect relativistic effect dominates and results in loosely bound orbitals with radial charge densities delocalized when compared to their nonrelativistic counterparts.




Fig 4 Relativistic corrections to electron binding energies and <r> mean expectation value for the outermost occupied orbitals (expressed as the ratio to the non relativistic value).
As illustrated in the above figures the changes in charge distributions are strongly dependant on the total angular momentum j of the electron. This j dependence may induce drastic changes in the theoretical prediction of some expectation values: a typical (i.e. exceptional) illustration is the dipole oscillator strength of the resonant 6s^{2} to 6s^{1}6p^{1} transition in mercury being decreased by a factor of 3 when relativistic corrections are taken into account (this can be understood from the fact that the 6s and 6p orbitals contributing to this transition probability are highly oscillating functions. A small shift in the respective position of their nodes induces visible variation in the value of the radial integral). As shown in Fig. 4, the 6s orbital is indeed the s orbital that displays the strongest relativistic contraction (percentage wise) of all s valence orbitals. An enlarged view of the relativistic correction for the 6s orbital is given in Fig 5.

Fig 5 Relativistic contraction of the 6s shell in the elements Cs (Z=55) to Fermium (Z=100).
The contraction increases considerably while the 4f shell is being filled and strikingly when the 5d shell is filled. The pronounced local maximum of the contraction at gold makes Au a unique element, even from this point of view. An equally strong relativistic contraction is not found until reaching fermium (Z=100).
Also evident from Fig. 4 is the fact that relativistic f orbitals undergo a rather large expansion compared to their nonrelativistic analogues. This expansion was confirmed by high quality measurements of neutron magnetic scattering in rare earth elements when compared to nonrelativistic and relativistic calculations. These experiments determine the Fourier components of the magnetic density and thus provide a direct comparison with the Fourier transform of the wave functions. For rare earth elements the localized 4f electrons give the dominant contribution, and we illustrate in Fig. 6 the case of terbium [21] for which, for simplicity, we restrict to the spherical contribution only. The left part of the figure shows a comparison between experiment, DiracFock (DF) and HartreeFock (HF) results and demonstrates that the DiracFock and experimental results are almost identical (they cannot be separated on the present scale) while the HartreeFock values lie slightly above. Being in reciprocal space this comparison clearly shows the contracted distribution of the 4f electron wave functions when relativistic corrections are not taken into account. To amplify the difference between relativistic and nonrelativistic wave functions, we display on the right part of Fig. 6 the differences (in percent) to experiment.


Fig 6 Spherical magnetic form factor of Tb (see text for explanations)
We now turn to the behaviour of electron binding energies and consider just two examples that will provide a smooth transition to the incidence of relativistic corrections in the prediction of chemical properties of molecules. The first example is related to the relative binding energies of electrons with different angular momenta. According to relativistic calculations (and in agreement with experiment) the binding energy of a 4f_{5/2} electron becomes lower than the binding energy of a 5s electron for bismuth (Z=83) and heavier atoms while this crossover is predicted to occur at lower atomic number (Z=78) by nonrelativistic calculations. More drastic changes are observed for superheavy atoms as in the case of lawrencium (Z=103). Extrapolation from lighter atoms of the same column in the Periodic Table would suggest that its ground state configuration should be 7s^{2} 6d. But, because of the strong stabilization of the p_{1/2} orbitals, it was rapidly suggested [22] that the 7s^{2}7p^{1} might indeed be lower in energy. This was confirmed by rather extensive calculations [23] including all possible configurations obtained by distributing the three valence electrons among the 6d, 7s and 7p orbitals for the J=1/2 and 3/2 states, complemented by an estimation of the core polarization contribution. These earlier calculations have been more recently extended by using the accurate coupled cluster method [24].
Accurate predictions of electronic structure require going beyond the independent electron picture given by the HartreeFock approximation and it is obvious that correlation and relativistic corrections should be included simultaneously in a coherent scheme. Not unexpectedly, methods that had proven their efficiency in nonrelativistic calculations started to be extended to the relativistic domain. To give some examples:
 the multiconfiguration method of FroeseFischer [25] was extended to the relativistic case [2627],
 the time dependant HartreeFock method [28], also known as the random phase approximation, gives rise to the relativistic random phase approximation [29],
 the many body perturbation theory pioneered by H. Kelly [30] also found its relativistic counterpart [31],
 the coupledcluster pair correlation approach [32], like other methods, did not resist its extension to relativity [33].
To not leave the reader with the impression that these extensions are trivial, let us recall that a relativistic reformulation cannot ignore the virtual creation of electronpositron pairs nor the fact that the Breit interaction involves the exchange of transverse photons.
For many electron systems, QED corrections must also include manybody contributions. For the time being only a limited number of results, besides semiempirical extrapolations, are available for heavy elements where a perturbative Za approach (in terms of the electron nucleus interaction) is irrelevant. The reason is not only that the most precise numerical methods developed for the oneelectron contributions [34] encounter serious numerical accuracy problems for high angular momentum values but also that, even for twoelectron atoms or ions, the standard QED prescription [35] is unable to deal with quasidegenerate levels. Recent developments [3637] open new perspectives for getting accurate estimates in twoelectron systems without any restriction on the nuclear charge.
Considering the outermost atomic orbitals, the effects of relativistic corrections on oneelectron binding energies and the spatial distribution of the radial charge densities are illustrated by the results displayed in Fig. 4. From the strong dependence of these relativistic corrections upon the total angular momentum we expect that they will not be without impact on molecular properties.
Fig 7 transition
from LS to jj coupling along the IVA column of the Periodic Table
Indeed, as the p_{1/2} orbitals differ in behaviour from the p_{3/2} orbitals, it is obvious that the six electrons of a p orbital will not be equivalent when hybridization is involved to form bonding molecular orbitals including heavy atoms. As an illustration, we display in Fig. 7 the transition from LS to jj coupling down the Group 14 column of the Periodic Table as given by the DiracFock method.
The relative splitting between orbitals with different angular momenta is also very sensitive to relativistic corrections. To illustrate this we consider the outermost d and s levels in silver and gold as shown below in Fig. 8. It is obvious that, a nonrelativistic calculation predicts the valence orbitals of the two atoms to be quite similar while a relativistic calculation shows that the gap between the nd and ns orbitals is reduced by a factor close to two in gold compared to silver. We know from everyday observation that silver is white while gold is yellow, and consequently their electronic structures must show some differences (see Ref. 38 and last chapter in this book). Beside the fact that they differ in colour, gold and silver exhibit quite different chemical properties. Some compounds of gold (CsAu and RbAu) are semiconductors instead of metals, and gold is able to form trivalent and pentavalent compounds in contrast to copper or silver (with the exception of CuF_{4}^{} and AgF_{4}^{}).
Fig 8 ns and nd
eigenvalues for silver and gold
If we look again at the results of Fig. 4 we may expect for molecules including heavy atoms that bond lengths obtained from relativistic calculations differ from those predicted for the nonrelativistic case. Indeed, as chemical bonding results in a welldefined distance between the atoms that form the molecule (equilibrium distance), this distance originates from a subtle balance between the overlap of valence atomic orbitals in the molecule and the Coulomb repulsion between the nuclei. Thus, if valence orbitals undergo a relativistic contraction (like the s and p orbitals), we expect shorter bond lengths in a relativistic calculation compared to a nonrelativistic one. The opposite effect is anticipated for high angular momentum bonding orbitals. This rather crude point of view must be taken with some care since orbitals of quite different angular momenta hybridise to form molecular bonds. Indeed, from atomic results it was suggested that compounds of radium (7s^{2}) should have equilibrium distances lower or equal to those of barium (6s^{2}) compounds. Even the crudest molecular model calculations (see the onecentre expansion section below) showed larger bond lengths for radium compounds (due to strong d contribution) as confirmed by experiment.
All these qualitative changes and trends can be inferred from results of atomic calculations obtained at the beginning of the seventies of the last century, but it remained to obtain more quantitative results. At that time all electron calculations for molecules involving heavy atoms using relativistic four component molecular orbitals were practically intractable even at the simple HartreeFock level. Therefore, early relativistic quantum chemistry emerged from the following various approximations:
 stay with all the electrons in a fully relativistic approach as given by the zero frequency limit of the Hamiltonian in Eq. (9) but restrict to model hydride molecules in the onecentre approximation only (see next section),
 restrict to valence electrons only and replace the core by a pseudopotential (see the chapter by M. Dolg in this book),
 go to a local density approximation (LDA) for the exchange and correlation terms following the pioneering works of Slater[39], and Kohn and Sham [40].
The LDA approach originated from solidstate physics where, the HartreeFock approximation being less useful, it is mandatory to take electron correlation from the beginning into account, and this is almost always done in the framework of density functional theory. As the nonrelativistic density functional had to be changed to take relativity into account [41], the practitioners of band calculations rapidly changed to the Dirac equation, and as early as 1981 a rather complete picture of the influence of relativistic effects in band structures emerged [42]. For application to solids and molecules an efficient numerical integration scheme, known as the discrete variational method, was developed [43], which is still very useful [44]. The density functional method is discussed more extensively in the chapter by Engels in this book.
The DiracFock onecentre method was the first approximation used for relativistic molecular structure calculations and is now only of historical importance. In this method the electronelectron interaction is handled exactly and the oneelectron wave functions are four component Dirac spinors. On the other hand both the nuclear potentials and all the oneelectron orbitals are expanded about a single common centre taken to be the position of the nucleus of the heaviest atom of the system under consideration. Because of this expansion, the method is restricted to hydrides XH_{n} and even for them the expansion is only slowly convergent. Nevertheless, experience gained with nonrelativistic calculations has shown surprisingly good results for equilibrium distances of XH bonds and for force constants.
To construct the potential due to the off centre nuclei, for each of them we choose a coordinate system with its origin at the expansion centre and the z axis passing through the nucleus for which we expand the potential. In doing so we obtain an expression for each of the off centre nuclei. The next step is to rotate all these contributions to a common coordinate system which requires the Wigner rotation matrices. The explicit results for the many symmetries applied are published elsewhere [45] and we write the potential created by the protons located at a distance R from the centre of expansion in the generic form:
_{} 
with r_{< }= min (r,R) and r_{> }= max(r,R) while _{} are the usual spherical harmonics. The value of the a coefficients is determined from spatial symmetry of the hydride considered with only a few of them being nonzero, so that the summation over l and m runs only over a limited number of terms. The next step is to construct a set of oneelectron orbitals belonging to a certain irreducible representations of a double point group. Consider the sorbital of a monohydride expanded in terms of only s and p atomic orbitals:
_{} 
= 
_{} 
_{} 
= 
_{} 
with the orbitals being normalized:
_{} 
This simple example is sufficient to point out that, to be able to solve the selfconsistent equations of the onecentre method, an atomic program needs to be modified only in the following form:
 all the atomic orbitals (s and p in the present example) used to build a given molecular orbital must be associated with the same eigenvalue,
 the relative norm of the atomic orbitals has to be optimized during the self consistent process,
 off diagonal Lagrange multipliers must be introduced to enforce orthogonality between core and valence orbitals even for closed shells.
To conclude this short description of the DiracFock onecentre expansion method (a more extensive presentation can be found in Ref. 45) we list in the table below most of the model molecular systems computed with that method and the main conclusions drawn from these calculations (see Table 7.3 in Ref. 2 for a full list of references to the results summarized here).
Table 2
Hydrides studied
by the onecentre DiracFock method
Molecules 
Main impact of relativistic corrections 
CH_{4} to PbH_{4} 
Bond length contraction and increase of force
constants 
CuH, AgH and 
Increase of the dissociation energy. Explanation of
the difference between Ag and Au. 
BH to TlH 
Decrease of the dissociation energy for TlH and
monovalency of TlH partially due to 6p spinorbit splitting. Transition from
LS to jj coupling in bonding orbitals. 
TiH_{4} to (104)H_{4} 
Small bond length expansion for TiH_{4} and
ZrH_{4}. 
CeH_{4} , ThiH_{4} ; UH_{6}
, 
5d orbital of W moves to bonding region and WH
bonds strengthened. Further evidence for 5f participation in UH bonds.
Contraction in actinide series found of the order of 30 pm. 
MH^{+} and MH_{2} with 
Increasingly strong d contributions to the bonds
from Ca to Ra. RaH bonds longer than BaH ones. YbH and NoH bonds are
about the same. Explanation of the linear two coordination of Hg. 
^{ } ^{1}S states of ScH to AcH, TmH, LuH and LrH 
Trends in group 3. LuH and LrH bond lengths
comparable. 
Table 2 includes almost all calculations done before 1980 and illustrates the large number of systems studied under the impulse of P. Pyykkö. Despite its obvious limitations, the onecentre method clearly demonstrated (at least qualitatively) that relativity and chemistry of heavy elements cannot be treated separately in contrast to what Dirac thought 50 years before (see introduction) and as was still believed some decades later by prominent physicists:
"Modern elementaryparticle physics is founded upon the two pillars of quantum mechanics and relativity. I have made little mention of relativity so far because, while the atom is very much a quantum system, it is not very relativistic at all."
(Sheldon L. Glashow in his book "Interactions", Warner, NewYork, 1988)
To illustrate that not only prominent physicists can be wrong let me correct some interpretations given above that follow the chronological order in the literature. From what was outlined before one may conclude that changes in the predicted bond lengths are directly related to the relativistically induced contraction or expansion of atomic charge distributions. This is not quite correct as was shown by Ziegler et al. [46] using nonrelativistic orbitals while relativistic corrections to the total energy of the system were computed by perturbation theory. In doing so, the bond length contractions obtained with the fully relativistic onecentre method is essentially recovered and thus appears to result more from corrections to the energy than from charge redistribution. Nevertheless, if the relativistic contraction of atomic orbitals and the decrease in bond lengths may be viewed as uncorrelated, they both originate from the relativistic decrease in the kinetic energy due to the massvelocity correction.
Onecentre calculations could only provide qualitative trends of relativistic corrections because they are restricted to few model systems. What remained was to obtain a more detailed understanding of the interplay between relativistic, shell structure and electron correlation effects to underline the importance of relativity to chemistry. At the beginning of quantum relativistic calculations, the extension of standard nonrelativistic quantum chemical methods by using oneelectron wave functions expanded in basis sets to the relativistic domain faced many difficulties in handling the unbounded Dirac operator. The RitzRayleigh variational procedure valid for semibound operators lead to what is known as the variational collapse for the Dirac operator, i.e. the fact that the total electronic energy E was diverging towards minus infinity. To avoid this problem, boundary conditions had to be introduced into the variational procedure (the most well known are the kinetic balance and the nonrelativistic limit of the basis spinors). The kinetic balance condition defines the basis set for the small component from that of the large component at the Pauli limit with a careful choice of the contraction scheme for both components:
_{} 
Furthermore large basis sets are needed for an accurate description of the region close to the nucleus where relativistic effects become important. Methods based on the replacement of the Dirac operator by approximate bound operators (square of the Dirac operator, its absolute value etc…) have not been very successful as can been understood from the fact that they break the Lorentz invariance for fermions.
We shall not elaborate on all the methods now used in relativistic quantum chemistry and following chapters in this book will cover most of them rather extensively. They range from all electron “fully” relativistic at the DiracFock level to semiempirical methods, and from relativistic corrections introduced perturbatively, to pseudopotential approximations and local density functionals. Because of the interest in molecules containing heavy elements, ranging from catalysis to bioinorganic chemistry, the last decade brought a large increase in the number of relativistic molecular calculations. For the heaviest elements, say beyond mercury, most of the calculations are still performed by using pseudopotential or density functional approximations as illustrated in Table 7.10 of the third volume of Ref. 2.
This short historical introduction to relativistic electronic structure, and even more so the chapters that follow, illustrates a very alive and active field of research whose vigour is illustrated by the increasing number of publications in this field. Indeed, if in 1986 a single volume published by Pyykkö [2] was sufficient to list all the related publications on relativistic quantum theory (about 3 100) over a period of 70 years, the next 15 years required two more volumes to hold the list of almost 8 000 new articles or reviews devoted to this subject. Although inflation in publishing is a common feature of all fields of research, these figures clearly show the importance to take relativistic and QED contributions into account. The need to include relativistic effects in quantum chemical calculations has stimulated both conceptual and numerical developments to finally fulfil the wish of Dirac for "approximate practical methods"
The fact that we are now able to compute transition energies, lifetimes of excited states, etc… with an accuracy competitive with the uncertainty of the most precise experimental measurements is not only satisfying for the theoreticians' ego but has also a very fundamental impact. For example, the last value for the fine structure constant recommended by the 2000 CODATA could not have been obtained without the measurement of the anomalous magnetic moment of the electron at a few ppb level combined with the most accurate QED contribution [47]. To confirm this new value (the relative change with respect to the previous value is 7´10^{8}) experiments and calculations are currently carried out to determine the 2^{3}P fine structure splitting in atomic helium [48].
For more complex systems, very accurate electronic structure calculations are also useful to test the most fundamental theories of physics. Being able to compute the parity nonconservation contribution (PNC, see chapter 9) to the hyperfine structure [49] allows testing of quantum chromodynamics in a domain of energy not available to highenergy physics experiments. One of the original relativistic atomic structure programs [26], has recently been modified to handle particles other than electrons, which made it possible to study more "exotic" systems (i.e. systems in which one electron is replaced by a muon, a pion or a kaon). If the "exotic" particle is a boson of spin zero this implies replacement (for that particle) of the Dirac equation by the KleinGordon equation. These new calculations, in connection with highly accurate Xray measurements of trapped pionic atoms, should lead to a substantial increase in the precision of the pion mass [50]. A higher accuracy in the value of the pion mass will result in a more reliable upper limit for the mass of the muonic neutrino, which is of prime importance in cosmology (dark matter of the universe). These last two examples demonstrate that we are now moving way beyond atomic and molecular properties.
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