Four-Component Hamiltonians

It is convenient to gather space and time coordinates in a vector, 4-position 

4-velocity is obtained as a time derivative, but since absolute time is abolished in the theory of relativity, a Lorentz invariant proper time r is generated from the infinitesimal interval 

With the introduction of proper time and use of the chain rule 4-velocity is obtained as 

The length of this 4-vector is simply the speed of light, thus by construction Lorentz invariant. Proceeding we multiply 4-vector by Lorentz invariant mass m to obtain 4-momentum 

The space part of 4-moment is relativistic momentum p = ymv, which is often re-written p = Mv, where appears relativistic mass M = ym, which increases with the speed of the particle. The content of the time part of 4-momentum is less obvious, but quantification of 4-momentum, p = -id, suggests that it is proportional to energy so that we may write 

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For the computational investigation of molecular systems containing heavy atoms, such as transition metals, lanthanides, and actinides, we could neglect neither relativity nor electron correlation. Relativistic effects, both spin-free and spin-orbit, increase with the nuclear charge of atoms. Therefore, instead of the nonrelativistic Schrodinger equation, we must start with the Dirac equation, which has four-component solutions. For many-electron systems, the four-component Hamiltonian is constructed from the one-electron Dirac operator with an approximated relativistic two-electron operator, such as the Coulomb, Breit, or Gaunt operator, within the nopair approximation. The four-component method is relativistically rigorous, which includes both spin-free and spin-orbit effects in a balanced way. However it requires much computational time since it contains more variational parameters than the approximated, one or two-component method. 

By inserting equation (10) into the energy expression for the Dirac equation (1) one obtains the effective four-component Hamiltonian HDirac)... 

Since we treat the hyperfine energy contribution by first-order perturbation theory we have a four-component Hamiltonian describing the hyperfine interaction according to... 

Because of the li (i) operators in this expression, this Hamiltonian is (less rigorously) called a four-component Hamiltonian in order to distinguish it from more approximate Hamiltonians that contain one-electron operators refering to 2-spinor representations, which are therefore called two-component Hamiltonians. In Eq. (8.66) the nucleus-nucleus repulsion operators are incorporated in the last term on the right-hand side of that equation abbreviated as... 

Once the basis functions have been converted to the orthonormal-basis representation, a standard hermitean eigenvalue solver can be employed to diagonalize the four-component Hamiltonian matrix. 

In the BSS approach, the free-particle Foldy-Wouthuysen transformation in addition to the orthonormal transformation K is applied to obtain the four-component Hamiltonian matrix to be diagonalized. The free-particle Foldy-Wouthuysen transformation Uq is composed of four diagonal block matrices. 

It is then applied to yield a transformed four-component Hamiltonian matrix... 

For the different parametrization schemes of the HJfc [611], the exponential parametrization can be chosen as it requires the lowest number of matrix multiplications [646]. The number of matrix multiplications can be further reduced by two additional considerations elaborated in Ref. [647]. First, the intermediate operator products which do not contribute to the final DKH Hamiltonian can be neglected. For example, in the fcth step the matrix is multiplied to an intermediate M of order I in the potential. If fc - - 21 > n >k + l and M is even, the multiplication with Wfc can be skipped, because the intermediate term, which is the product of and M/, is odd and then does not contribute to the nth order DKH Hamiltonian. The further multiplication to yields an even matrix but goes beyond nth order. Second, the DKH Hamiltonian matrix is taken from the upper part of the transformed four-component Hamiltonian matrix while the lower part is not required. For instance, if fc + Z = n and M/ is odd, the product with the odd operator W/t contributes to the final DKH Hamiltonian, but the matrix multiplications to obtain the lower part result can be neglected. The symmetry of the matrices can also be exploited [647]. Noting that the odd matrices O are hermitean, = O, and the matrices Wjt are antihermitean, = —W/t- The algorithm requires the evaluation of their commutator. 

The number of two-component complex matrix multiplications is different for the different approaches. The BSS approach differs from the X2C approach only in this term. It requires three more than the X2C approach since the off-diagonal parts of its four-component Hamiltonian matrix are not diagonal. For the DKH approach, the number of two-component complex matrix... 

If we now substitute the approximate unitary transformation of Eq. (14.71) into the block-diagonalization of the four-component Hamiltonian,... 

On matrix form the non-unitary transformations (27) and (30) of the previous section are easily extended to the complete Hamiltonian and have therefore allowed relativistic and non-relativistic spin-free calculations of spectroscopic constants and first-order properties at the four-component level (see, for instance. Refs. [45 7]). In this section, we consider the elimination of spin-orbit interaction in four-component calculations of second-order electric and magnetic properties. Formulas are restricted to the Hartree-Fock [48] or Kohn-Sham [49] level of theory, but are straightforwardly generalized. 

However, it is more appropriate to provide theoretical justifications for such use. In this respect, first, we introduce the third category of decoupling of positive and negative states commonly known as the direct perturbation theory . This approach does not suffer from the singularity problems described previously. However, the four-component form of the Dirac equation remains intact. The new Hamiltonian requires identical computational effort as for the Dirac equation itself, hence it is not an attractive alternative to the Dirac equation. However, it is useful to assess the accuracy of approximate two-component forms derived from the Dirac equation such as Pauli Hamiltonian. Consider the transformation... 

We have focused on the prohlems associated with extending Dirac s one-fermion theory smoothly to many-fermion systems. A brief discussion of QED many-fermion Hamiltonians also was given. A comprehensive account of the problem of decoupling Dirac s four-component equation into two-component form and the serious drawbacks of the Pauli expansion were presented. The origins of the DSO and FC operators have been addressed. The working Hamiltonian which describe NMR spectra is derived. 

The no-pair DCB Hamiltonian (6) is used as a starting point for variational or many-body relativistic calculations [9], The procedure is similar to the nonrelativistic case, with the Hartree-Fock orbitals replaced by the four-component Dirac-Fock-Breit (DFB) functions. The spherical symmetry of atoms leads to the separation of the one-electron equation into radial and spin-angular parts [10], The radial four-spinor has the so-called large component the upper two places and the small component Q, in the lower two. The quantum number k (with k =j+ 1/2) comes from the spin-angular equation, and n is the principal quantum number, which counts the solutions of the radial equation with the same k. Defining... 

Energy levels of heavy and super-heavy (Z>100) elements are calculated by the relativistic coupled cluster method. The method starts from the four-component solutions of the Dirac-Fock or Dirac-Fock-Breit equations, and correlates them by the coupled-cluster approach. Simultaneous inclusion of relativistic terms in the Hamiltonian (to order o , where a is the fine-structure constant) and correlation effects (all products smd powers of single and double virtual excitations) is achieved. The Fock-space coupled-cluster method yields directly transition energies (ionization potentials, excitation energies, electron affinities). Results are in good agreement (usually better than 0.1 eV) with known experimental values. Properties of superheavy atoms which are not known experimentally can be predicted. Examples include the nature of the ground states of elements 104 md 111. Molecular applications are also presented. 

The no-pair DCB Hamiltonian (6) is used as a starting point for variational or many-body relativistic calculations [10]. The procedure is similar to the nonrelativistic case, with the Hartree-Fock orbitals replaced by the four-component... 

From a formal point of view, four-component correlation calculations [5, 6] based on the Dirac-Coulomb-Breit (DCB) Hamiltonian (see [7, 8, 9, 10, 11] and references therein) can provide with very high accuracy the physical and chemical properties of molecules containing heavy atoms. However, such calculations were not widely used for such systems during last decade because of the following theoretical and technical complications [12] ... 

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 most straightforward method for electronic structure calculation of heavy-atom molecules is solution of the eigenvalue problem using the Dirac-Coulomb (DC) or Dirac-Coulomb-Breit (DCB) Hamiltonians [4f, 42, 43] when some approximation for the four-component wave function is chosen. 

When innermost core shells must be treated explicitly, the four-component versions of the GREGP operator can be used, in principle, together with the all-electron relativistic Hamiltonians. The GRECP can describe here some quantum electrodynamics effects (self-energy, vacuum polarization etc.) thus avoiding their direct treatment. One more remark is that the... 

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