The Dirac Wave Function

Because of the appearance of 4 x 4 matrices in the Dirac equation, the wave function must be a four-component vector  

We have therefore moved from a scalar function in the nonrelativistic regime to a four-component vector function in the relativistic regime, and we might expect that the work to describe this wave function would increase in proportion. 

What is the significance of the four components We noted above that in nonrelativistic quantum mechanics we can introduce the spin by replacing the mechanical momentum nr by a nr and making the wave function a two-component vector, or 2-spinor, where the upper component corresponds to spin j and the lower component corresponds to spin -i. The same concept applies to the Dirac wave function— components 1 and 3 correspond to spin and components 2 and 4 correspond to spin — i, and the wave function is called a 4-spinor. 

It is useful to classify the upper two and lower two components of the Dirac wave function as two-component spinors. From the discussion of spin in the nonrelativistic equation, it might be guessed that these would be Pauli spinors. We write 

4 and are termed the large and small components of the wave function. The reason for this nomenclature will become clear later. For the time-independent case. 

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This expression nicely illustrates the main qualitative features of the order (Za) nuclear size contribution. First, we observe a logarithmic enhancement connected with the singularity of the Dirac wave function at small distances. Due to the smallness of the nuclear size, the effective logarithm of the ratio of the atomic size and the nuclear size is a rather large number it is equal to about —10 for the IS level in hydrogen and deuterium. The result in (6.35) contains all state-dependent contributions of order (Za) . 

We can deduce from relations established in Sect. 9.2 that a direct passage of the vectors T- -(k) of the transitions sl/2 — pl/2 and sl/2 — pl/2 to a vector T- -(k) of a transition s —p is not possible. In other words, one of the effect of the retardation is to break the possibility to find an equivalence between the Pauli approximation and the Schrodinger theory, and the reason lies on the incidence of the retardation on the spherical parts of the Dirac wave functions, related to the presence of the spin. The incidence is already sensible, in the transitions of the discrete spectrum (see (9.38), (9.39), (9.40)) and this incidence may be amplified in the contribution of the continuum, independently of the incidence of the chosen values for the radial functions. 

Pseudopotentials have been the subject of considerable attention in the last two decades, and they have been developed by a number of different groups. They are also the most widely used effective core potentials in chemical applications either for the study of chemical reactions or spectroscopy. A large variety of pseudopotentials are now available, and all the coupling schemes at the SCF step have been implemented four-component, two-component, and scalar relativistic along with spin-orbit pseudopotentials. However, it is well known that four-component calculations can (in the worst cases) be 64 times more expensive than in the non relativistic case. In addition, the small component of the Dirac wave function has little density in the valence region, and pseudopoten-... 

The solution to the spin problem followed when Dirac generalized the SE to a rela-tivistically invariant equation (1928). In the Dirac equation, the spin-orbit coupling appears as a natural part of the formalism. The Dirac wave function r has four components, but only two of them refer to electrons. The other two coordinates are positron coordinates. It is reasonable to assume that our particle is 100% an electron, with no probability density for being a positron. 

What is of further concern is whether the probability density is time-independent, which we expect for a bound state, and whether it is conserved under a Lorentz transformation, since this has implications for the normalization of the wave function. If ca is the velocity operator, we may write the current density for the Dirac wave function as... 

To obtain further information on the nature of the Dirac wave function, we can solve the equation for a simple model system. The simplest case is the time-independent equation for a free particle. In the nonrelativistic case the Schrodinger equation for a free particle moving along the x axis is... 

Before taking the limit c oo, this equation must be rearranged for two reasons first, because we need to change it to a form where c occurs in some form of denominator—this will provide us with terms that vanish and hopefully other terms that remain finite—and, second, because the nonrelativistic wave function is a scalar function, whereas the Dirac wave function is a four-component vector function. If we use the two-component nonrelativistic Schrodinger equation that we derived in section 4.2, we can write the nonrelativistic wave function in terms of spin-orbitals, which can be transformed to two-component spinors. Then it is only necessary to reduce the Dirac equation from four-component to two-component form. 

This is a valid definition, as it only requires that the small component be integrable, which is certainly the case sinee must be square integrable in order to normalize the Dirac wave function. The pseudo-large component now has the same symmetry properties as the large component. The nonrelativistic limit of the pseudo-large component is the large component, since... 

The development is not complete, because the large component of the Dirac wave function is not normalized it is the four-component wave function that is normalized. Of course, in the limit c oo, the large component goes over to the nonrelativistic wave function and is therefore normalized, but we are interested in finite c and a perturbation expansion correct to 0(c ). We therefore need to make the replacement... 

Section VI shows the power of the modulus-phase formalism and is included in this chapter partly for methodological purposes. In this formalism, the equations of continuity and the Hamilton-Jacobi equations can be naturally derived in both the nonrelativistic and the relativistic (Dirac) theories of the electron. It is shown that in the four-component (spinor) theory of electrons, the two exha components in the spinor wave function will have only a minor effect on the topological phase, provided certain conditions are met (nearly nonrelativistic velocities and external fields that are not excessively large). 

As pointed out in the previous paragraph, the total wave function of a molecule consists of an electronic and a nuclear parts. The electrons have a different intrinsic nature from nuclei, and hence can be treated separately when one considers the issue of permutational symmetry. First, let us consider the case of electrons. These are fermions with spin and hence the subsystem of electrons obeys the Fermi-Dirac statistics the total electronic wave function... 

There are several ways to include relativity in ah initio calculations more efficiently at the expense of a bit of accuracy. One popular technique is the Dirac-Hartree-Fock technique, which includes the one-electron relativistic terms. Another option is computing energy corrections to the nonrelativistic wave function without changing that wave function. 

The scalar product of two Dirac wave functions is defined as... 

The expressions (4.22)-(4.23) found in chap. 4 for the isomer shift 5 in nonrelativ-istic form may be applied to lighter elements up to iron without causing too much of an error. In heavier elements, however, the wave function j/ is subject to considerable modification by relativistic effects, particularly near the nucleus (remember that the spin-orbit coupling coefficient increases with Z ). Therefore, the electron density at the nucleus l /(o)P will be modified as well and the aforementioned equations for the isomer shift require relativistic correction. This has been considered [1] in a somewhat restricted approach by using Dirac wave functions and first-order perturbation theory in this approximation the relativistic correction simply consists of a dimensionless factor S (Z), which is introduced in the above equations for S,... 

How does one extract eigenpairs from Chebyshev vectors One possibility is to use the spectral method. The commonly used version of the spectral method is based on the time-energy conjugacy and extracts energy domain properties from those in the time domain.145,146 In particular, the energy wave function, obtained by applying the spectral density, or Dirac delta filter operator (8(E — H)), onto an arbitrary initial wave function ( (f)(0)))1 ... 

The Pauli form factor also generates a small contribution to the Lamb shift. This form factor does not produce any contribution if one neglects the lower components of the unperturbed wave functions, since the respective matrix element is identically zero between the upper components in the standard representation for the Dirac matrices which we use everywhere. Taking into account lower components in the nonrelativistic approximation we easily obtain an explicit expression for the respective perturbation... 

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