Intermediate Hamiltonian approach

The basic relativistic equations are described in Sec. 2, and the Fock-space coupled cluster method is discussed in Sec. 3. The recently developed intermediate Hamiltonian approach is described and illustrated by several... 

A major advantage of the intermediate Hamiltonian approach is the flexibility in selecting the model space. This has been a major problem in applying the Fock-space scheme, as described at the beginning of this section. While in the Fock-space coupled cluster method one may feel lucky to find any partitioning of the function space into P and Q with convergent CC iterations, the intermediate Hamiltonian method makes it possible for the first time to vary the model space systematically and study the effect upon calculated properties. An example is given in Table 3, which shows the dependence of the calculated electron affinity of Cs on the model spaces Pm and Pi [55]. 

Excitation energies of atomic barium and radium were calculated in 1996 using the Fock-space coupled cluster method [57]. The model space in the 2-electron sector included all states with two electrons in the 5d, 6s and 6p orbitals, except the 6p states inclusion of the latter led to intruder states and divergence, so that incomplete model spaces had to be employed. In the intermediate Hamiltonian approach all these states (including 6p ) were in Pm, Pi was defined by adding states with occupied 7s-10s, 7p-10p, 6d-... 

The methods used, including the relativistic equations, the FSCC scheme, and the intermediate Hamiltonian approach, are described in Section 2.2. The following section lists representative applications of the methodology to actinide atoms and molecules, demonstrating the level of accuracy achieved where comparison with experiment is available, as well as predictions that can be made for the properties still not experimentally available. The final section provides a summary. 

The intermediate Hamiltonian approaches presented here may be applied within any multiroot multireference infinite order method. Recently [43] we implemented the XIH scheme to another all-order relativistic multiroot multireference approach, the Hilbert space or state universal CC, which is the main alternative to and competitor of Fock-space CC. This approach will not be discussed here. We only mention that it allows mixing P-space sectors, which can interact strongly, e.g., 1-particle with 2-particle 1-hole [37]. 

The structure of the model space P in the Fock-space method and of Pm and P in the intermediate Hamiltonian approach is shown in Table 2.4. All determinants constructed from the orbitals listed in the table constitute the relevant space. is a subspace of P in the IH-FSCC approach. Convergence difficulties of the FSCC formalism in sector (2) made it necessary to use an incomplete model space [62,63], moving certain determinants from P to Q. The IH calculations employ much larger P spaces, which are always complete (i.e., include all combinations of relevant orbitals). Orbital selection was determined primarily on the basis of orbital energies. 

The ionization potentials and lower excitation energies of Th and its ions are reported in Tables 2.5 and 2.6. Very good agreement with experiment [61] is obtained the average error of the 51 Fock-space energies at all ionization levels is 0.062 eV. The intermediate Hamiltonian approach reduces the average error to 0.051 eV. This level of accuracy is obtained in spite of the complicated interactions between different electronic configurations, which lead to a rather dense spectrum. 

Let us consider the 5s, 5p, 5d orbitals of lead and Is orbital of oxygen as the outercore and the ai, a2, os, tti, tt2 orbitals of PbO (consisting mainly of 6s, 6p orbitals of Pb and 2s, 2p orbitals of O) as valence. Although in the Cl calculations we take into account only the correlation between valence electrons, the accuracy attained in the Cl calculation of Ay is much better than in the RCC-SD calculation. The main problem with the RCC calculation was that the Fock-space RCC-SD version used there was not optimal in accounting for nondynamic correlations (see [136] for details of RCC-SD and Cl calculations of the Pb atom). Nevertheless, the potential of the RCC approach for electronic structure calculations is very high, especially in the framework of the intermediate Hamiltonian formulation [102, 131]. 

A feature of the intermediate problem approach, is that the construction of comparison operators involves the splitting of the Hamiltonian, H, into two parts and /, as in per-... 

A. Landau, E. Eliav, Y. Ishikawa, U. Kaldor, Mixed-sector intermediate Hamiltonian Fock-space coupled cluster approach, J. Chem. Phys. 121(14) (2004) 6634. 

A particular variant of the coupled cluster method, called Fock-space or valence-universal [49,50], gave remarkable agreement with experiment for many transition energies of heavy atoms [51]. This success makes the scheme a useful tool for reliable prediction of the structure and spectrum of superheavy elements, which are difficult to access experimentally. A brief description of the method is given below. A more flexible scheme with higher accuracy and extended applicability, the intermediate Hamiltonian Fock-space coupled cluster approach, is shown in the next section. 

First one can build up other effective Hamiltonians based on hierarchized orthogonalization procedures. The Gram-Schmidt procedure is recommended if one starts from the best projected wavefunctions of the bottom of the spectrum. Thus one can obtain a quite reliable effective Hamiltonian with well behaved wavefunctions and good transferability properties (see Section III.D.2). The main drawback of this approach is that the Gram-Schmidt method, which involves triangular matrices, does not lead to simple analytical expressions for perturbation expansions. A partial solution to these limitations is brought about by the new concept of intermediate Hamiltonian,... 

The intermediate Hamiltonian Fock-space coupled-cluster calculations of Infante et al. [1146] were performed in an all-electron approach explicitly correlating 26 electrons and using large de[Pg.625]

The FSCC equation (2.7) is solved iteratively, usually by the Jacobi algorithm. As in other CC approaches, denominators of the form Eq —E ) appear, originating in the left-hand side of the equation. The well-known intruder state problem, appearing when some Q states are close to and strongly interacting with P states, may lead to divergence of the CC iterations. The intermediate Hamiltonian method avoids this problem in many cases and allows much larger and more flexible P spaces. 

The intermediates of an MFEP calculation are usually defined using Hamiltonians related to those of system 0 (J%) and 1 p j). This is generally achieved with a parameter-scaling approach (see Sect. 2.6). Linear scaling is the simplest form... 

The EPR spectrum is a reflection of the electronic structure of the paramagnet. The latter may be complicated (especially in low-symmetry biological systems), and the precise relation between the two may be very difficult to establish. As an intermediate level of interpretation, the concept of the spin Hamiltonian was developed, which will be dealt with later in Part 2 on theory. For the time being it suffices to know that in this approach the EPR spectrum is described by means of a small number of parameters, the spin-Hamiltonian parameters, such as g-values, A-values, and )-values. This approach has the advantage that spectral data can be easily tabulated, while a demanding interpretation of the parameters in terms of the electronic structure can be deferred to a later date, for example, by the time we have developed a sufficiently adequate theory to describe electronic structure. In the meantime we can use the spin-Hamiltonian parameters for less demanding, but not necessarily less relevant applications, for example, spin counting. We can also try to establish... 

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