Multireference model spaces

TABLE rV. Correlation energies for the first biexcited state of the H4 MBS model obtained using various multireference correlation techniques within a two-dimensional model space spanned by the ground state configuration 1122) and the first biexcited configuration lll33). For simplicity, we present only differences from corresponding FCI values (A = X - FCI) in mHartree. 

In general, multireference many-body methods are implemented by choosing a model space. This involves dividing orbitals into three types (i) core orbitals, which are doubly occupied in all reference configurations (ii) active orbitals which may or may not be occupied in each of the reference configurations, and (iii) virtual orbitals which are not occupied in all reference configurations. The concept of a complete active space is an important one. It implies that the model space contains all of the determinants which can be obtained by considering all possible occupancies of the active orbitals. 

U. Kaldor, Intruder states and incomplete model spaces in multireference coupled-cluster theory The 2p states of Be, Phys. Rev. A 38 (1988) 6013. 

Recent developments include exact [12-14, 44, 90, 91] and approximate [14, 90, 92-94] iterative schemes to determine Hg, the intermediate Hamiltonian method [21, 24, 95], the use of incomplete model spaces [43, 44] and some multireference open-shell coupled-cluster (CC) formalisms [16-20, 96, 97]. Only some eigenvalues of the intermediate Hamiltonian H, are also eigenvalues of H. The corresponding model eigenvectors of H, are related to their true counterparts as in Bloch s theory. Provided effective operators a are restricted to act solely between these model eigenvectors, the possible a definitions from Bloch s formalism (see Section VI.A) can be used. 

Molecular Applications of Multireference Coupled-Cluster Methods Using an Incomplete Model Space. Direa Calculation of Excitation Energies. (FS-MRCCSD.)... 

The Fock space approach assumes that the reference wave functions for the states of interest can be written in terms of determinants that share a common closed shell core. This closed shell core is then considered as the zero order wave function from which the energy of states of interest can be calculated as an electron affinity or ionization energy. In the calculation of these quantities one may incorporate the necessary multireference character by allowing mixing between different reference wave functions. The first step of the procedure is to define the reference or model space that spans a number of sectors of the Fock... 

The results for this model considered at the minimum basis set (MBS) and double zeta (DZ) levels are given in Tables 2 and 3, respectively. In both cases, the (2,2) model space is used in multireference calculations. Again, both AL-RMR-CCSD-1 and AL-RMR-CCSD-2 represent a good approximation to the full RMR CCSD (both differing by only a few /lihartree at the MBS level and by 10-60 /ihartree at the DZ level). In turn, RMR CCSD represents a significant improvement over the standard CCSD (by more than 7 mhartree for a 0 for a DZ model). 

Comparison of correlation energies (all signs reversed) for the ground state of the H8 MBS model (in mhartree). The multireference methods use a (2,2) model space. The acronym AL-RMR-CCSD-i is abbreviated to AL-RMR-i. 

Pal, S., Rittby, M., Bartlett, R. J., Sinha, D., Mukherjee, D. [1987]. Multireference Coupled-Cluster Methods Using an Incomplete Model Space Application to Ionization Potentials and Excitation Energies of Formaldehyde, Chem. Phys. Lett, 137, 272-27Q. 

In the single reference Cl method, the model space (Fig. 10.8) is formed by a single Slater determinant. In the multireference Cl method, the set of determinants constitute the model space. This time, the Cl expansion is obtained by replacement of the spinorbitals participating in the model space by other virtual orbitals. We proceed further as in Cl. 

Fig. 10.8. Illustration of the model space in the multireference Cl method used mainly in the situation when no single Slater determinant dominates the Cl expansion. The orbital levels of the system are presented here. Part of them are occupied in all Slater determinants considered fttKen spinorbitals"). Above them is a region of closely spaced orbital levels called active space. In the optimal case, a significantly large energy gap occurs between the latter and unoccupied levels lying higher. The model space is spanned by all or some of the Slater determinants obtained by various occupancies of the active space levels.

For a system described in zero order by a multireference function, the Hilbert space, i), is a partition into the model space and the complementary space... 

Unlike the BWCCSD method, the MR-BWCCSD theory, the BWCISD theory and the MR-BWCISD theory do not support energies which scale linearly with the number of electrons in the system. In the multireference cases, that is MR-BWCCSD and MR-BWCISD theories, this lack of ex-tensivity arise both from nonlinear terms in the matrix elements of the effective hamiltonian and from diagonalization of the effective hamiltonian when an incomplete model space is employed. By exploiting the known relation between the Brillouin-Wigner and the Rayleigh-Schrddinger denominators it is possible to devise a posteriori corrections to methods formulated within the framework of Brillouin-Wigner perturbation theory. 

The simplest way to realize an exponential expansion is to employ the exponential ansdtz of Jeziorski and Monkhorst [87] which exploits a complete model space. This is the approach that we followed in Section 4.2.2.2 in developing a multi-root multireference Brillouin-Wigner coupled cluster theory. The Jeziorski and Monkhorst exponential ansatz may be written... 

L. Meissner, S. A. Kucharski, and R. J. Bartlett, A multireference coupled-cluster method for special classes of incomplete model spaces, J. Chem. Phys. 91, 6187 (1989) L. Meissner and R. J. Bartlett, A general model-space coupled-cluster method using a Hilbert-space approach, ... 

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