Kramers restriction

Active Space (GAS) Concept for the Relativistic Treatment of Electron Correlation. I. Kramers Restricted Two-Component Cl. 

In Table 6.3, the values of De for RfCU are compared with those obtained within various approximations using relativistic effective core potentials (RECP) Kramers-restricted Hartree-Fock (KRHF) (Han et al 1999), averaged RECP including second-order M0ller-Plesset perturbation theory (AREP-MP2) for the correlation part (Han et al. 1999), RECP coupled-cluster single double (triple) [CCSD(T)] excitations (Han et al. 1999), and a Dirac-Fock-Breit (DFB) method (Malli and Styszynski 1998). The AREP-MP2 calculation of De gives 20.4 eV, while the RECP-CCSD(T) method with correlation leads to 18.8 eV. Our value of De of 19.5 eV is just between these calculated values. 

The incorporation of electron correlation effects in a relativistic framework is considered. Three post Hartree-Fock methods are outlined after an introduction that defines the second quantized Dirac-Coulomb-Breit Hamiltonian in the no-pair approximation. Aspects that are considered are the approximations possible within the 4-component framework and the relation of these to other relativistic methods. The possibility of employing Kramers restricted algorithms in the Configuration Interaction and the Coupled Cluster methods are discussed to provide a link to non-relativistic methods and implementations thereof. It is shown how molecular symmetry can be used to make computations more efficient. 

In the following we assume the use of an (effective) one particle Hamiltonian for which all eigenspinors come in degenerate pairs. This means that external magnetic fields are excluded and that a (Kramers-)restricted algorithm is used in the (D)HF step. 

The alternative to the development of new algorithms to handle relativistic Hamiltonians is to search for a way to extend non-relativistic algorithms such that they can handle the additional couplings. Since most implementations are based on a restricted Hartree-Fock scheme the first step is to mimic the spin-restricted excitation operators used in the non-relativistic methods by Kramers restricted excitation operators. This can be done by employing the so-called X-operator formalism [37]. 

The Kramers-restricted form of the Hamiltonian that was used in Cl theory is not suitable for Coupled Cluster theory because it mixes excitation and deexcitation operators. One possibility is to define another set of excitation operators that keep the Kramers pairing and use these in the exponential parametrization of the wavefunction. This would automatically give Kramers-restricted CC equations upon rederivation of the energy and amplitude equations. A more pedestrian but simpler alternative is to start from the spin-orbital formulation and inspect the relations that follow from the Kramers relation of the two-electron integrals. This method does also readily give the relations between the Kramers symmetry-related amplitudes. We will briefly discuss the basic steps in this approach, a detailed description of a possible algorithm is given in reference [47],... 

If spin-orbit effects are considered in ECP calculations, additional complications for the choice of the valence basis sets arise, especially when the radial shape of the / -f-1/2- and / — 1/2-spinors differs significantly. A noticeable influence of spin-orbit interaction on the radial shape may even be present in medium-heavy elements as 53I, as it is seen from Fig. 21. In many computational schemes the orbitals used in correlated calculations are generated in scalar-relativistic calculations, spin-orbit terms being included at the Cl step [244] or even after the Cl step [245,246]. It therefore appears reasonable to determine also the basis set contraction coefficients in scalar-relativistic calculations. Table 9 probes the performance of such basis sets for the fine structure splitting of the 531 P ground state in Kramers-restricted Hartree-Fock [247] and subsequent MRCI calculations [248-250], which allow the largest flexibility of... 

Fine structure splitting of the ground state of 53 from Kramers-restricted Hartree-Fock and... 

Kramers-restricted Hartree-Fock ground state calculations of the neutral atom (basis set C). The most diffuse primitives of the (7s7p) set were left uncontracted to generate the [nsnp] contracted sets. 

Hence, the relativistic analog of the spin-restriction in nonrelativistic closed-shell Hartree-Fock theory is Kramers-restricted Dirac-Hartree-Fock theory. We should emphasize that our derivation of the Roothaan equation above is the pedestrian way chosen in order to produce this matrix-SCF equation step by step. The most sophisticated formulations are the Kramers-restricted quaternion Dirac-Hartree-Fock implementations [286,318,319]. A basis of Kramers pairs, i.e., one adapted to time-reversal s)mimetry, transforms into another basis under quatemionic unitary transformation [589]. This can be exploited not only for the optimization of Dirac-Hartree-Fock spinors, but also for MCSCF spinors. In a Kramers one-electron basis, an operator O invariant under time reversal possesses a specific block structure. 

The work of Lee et al. [1113] compares results obtained with relativistic effective core potentials (ECP). The ECP calculations have been carried out with a Kramers-restricted (KR) two-component formalism as described in Ref. [836]. These results demonstrate the accuracy of the ECP-CCSD(T) approach when compared with experiment. Tsuchiya et al. [1115] investigated Au2 with MP2, CCSD and CCSD(T) employing a scalar-relativistic third-order DKH3 Hamiltonian. Again CCSD(T) yields results that reproduce all spectroscopic parameters remarkably well. 

T. Fleig, J. Olsen, C. M. Marian. The Generalized Active Space Concept for the Relativistic Treatment of Electron Correlation I. Kramers-restricted two-component configuration interaction. /. Chem. Phys., 114(11) (2001) 4775 790. 

L. Visscher, K. G. DyaU, T. J. Lee. Kramers-Restricted Closed-SheU CCSD Theory. Int. J. Quantum Chem. Quantum Chem. Symp., 29 (1995) 411 19. 

Y. S. Kim, Y. S. Lee. The Kramers restricted complete active space self-consistent-field method for two-component molecular spinors and relativistic effective core potentials including spin-orbit interactions. /. Chem. Phys., 119 (2003) 12169. 

Time Reversal and Kramers-Restricted Representation of Operators... 

In the previous discussion of the symmetry of the Dirac equation (chapter 6), it was shown that the Dirac equation was symmetric under time reversal, and that the fermion functions occur in Kramers pairs where the two members are related by time reversal. We will have to deal with a variety of operators, and in most cases the methodologies will be developed in the absence of an external magnetic field, or with the magnetic field considered as a perturbation. Consequently, we can make the developments in terms of a basis of Kramers pairs, which are the natural representation of the wave function in a system that is symmetric under time reversal. The development here is primarily the development of a second-quantized formalism. We will use the term Kramers-restricted to cover techniques and methods based on spinors that in some well-defined way appear as Kramers pairs. The analogous nonrelativistic situation is the spin-restricted formalism, which requires that orbitals appear as pairs with the same spatial part but with a and spins respectively. Spin restriction thus appears as a special case of Kramers restriction, because of the time-reversal connection between a and spin functions. 

With the use of these relations, the Kramers-restricted Dirac-Coulomb-Breit Hamiltonian may be written as... 

To represent the state in Kramers-restricted form we may partition the creation operators into barred and unbarred sets, and anticommute the barred operators to the right. 

The Kramers-restricted relativistic case is a little more complicated. First, because the basis is complex, we must use the conjugate representation for the bra in any integral. In the notation used above, we can express this as... 

For the two-electron integrals, we want to divide the integrals into symmetry classes, as for the nonrelativistic integrals. We also want to divide the integrals into classes by time reversal symmetry, as we did for the one-electron integrals. Because of the structure of the Kramers-restricted Hamiltonian in terms of the one- and two-particle Kramers replacement operators, we hope to obtain a reduction in the expression for the Hamiltonian from time-reversal symmetry. The classification by time-reversal properties is also important for the construction of the many-electron Hamiltonian matrix, whose symmetry properties we consider in the next section. 

These time-reversal reductions affect the expressions for the second-quantized Kramers-restricted Hamiltonian. Both forms given in chapter 9 contain integrals with an odd number of bars, and in both, these terms vanish if the group has no quaternion irreps. If we want to use the Hamiltonian as expressed in terms of the excitation operators Ep and such as in a coupled-cluster calcula-... 

While the expansion of hpq follows readily from the development for the Dirac operator above, the electron-electron interaction integrals must be considered separately. We also want to develop a Kramers-restricted Dirac-Hartree-Fock (KR-DHF) theory, but first we develop expressions for the general, Kramers-unrestricted case. In the developments below we follow the practice of giving only the basis function index in the integrals. 

Kramers-Restricted 2-Spinor Matrix Dirac-Hartree-Fock Equations... 

So far we have not taken time-reversal symmetry into account. From the preceding chapters, we expect that incorporating time-reversal symmetry in a Kramers-restricted Dirac-Hartree-Fock theory will result in a reduction of the work, and possibly also a reduction in the rank of the Fock matrix. The basis set we will use is a basis set of Kramers pairs. We develop the theory for a closed-shell reference, for which all Kramers pairs are doubly occupied. ... 

The symmetry reductions in nonrelativistic methodology come from spin symmetry and from point-group symmetry. In relativistic methodology, time-reversal symmetry is the equivalent of spin symmetry, but it does not provide the same magnitude of reduction as does spin symmetry. This is due to the presence of spin-dependent terms in the Fock operator. Point-group symmetry is intimately connected with time-reversal symmetry in Kramers-restricted relativistic theory, as we saw in chapter 10. 

The answer to this question is a qualified yes , and depends on the nature of the wave function. For a single electron outside a closed shell, the Kramers-restricted wave function is a single determinant, and it is relatively easy to define the Kramers-restricted Fock matrices for the closed-shell and open-shell electrons. Using the expressions for the Fock matrix in the molecular basis, (11.34), with indices i and j for doubly occupied Kramers pairs, t for the singly occupied Kramers pair, and a for the empty or virtual Kramers pairs, we can define the three nonredundant sections of the Fock matrix as follows ... 

The Kramers-restricted integral transformation is therefore 24 times more expensive than the nonrelativistic integral transformation. 

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