Wave operator open-shell

In molecules, the interaction of surrogate spins localized at the atomic centers is calculated describing a picture of spin-spin interaction of atoms. This picture became prominent for the description of the magnetic behavior of transition-metal clusters, where the coupling type (parallel or antiparallel) of surrogate spins localized at the metal centers is of interest. Once such a description is available it is possible to analyze any wave function with respect to the coupling type between the metal centers. Then, local spin operators can be employed in the Heisenberg Spin Hamiltonian. An overview over wave-function analyses for open-shell molecules with respect to local spins can be found in Ref. (118). 

In many physical problems we come across excited configurations consisting of several open shells or at least one electron above the open shell. Therefore, we have to be able to transform wave functions and matrix elements from one coupling scheme to the other for such complex configurations. If K denotes the configuration, and m, fifi stand for the quantum numbers of two different coupling schemes, then for the corresponding wave functions formulas of the kind (12.1), (12.2) hold, whereas the matrix element of some scalar operator D transforms as ... 

In (22.37) ip is the wave function of an atom with motionless nucleus. The one-electronic submatrix element of the gradient operator (n/ V ni/i) is non-zero only for h = / 1. Therefore, the matrix element of operator (22.38) inside a shell of equivalent electrons vanishes and one has to account for this interaction only between shells. For the configuration, consisting of j closed and two open shells, it is defined by the following formula [156] ... 

In the central field approximation, when radial wave functions not depending on term are usually employed, the line strengths of any transition may be represented as a product of one radial integral and of a number of 3n./-coefficients, one-electron submatrix elements of standard operators (C(fc) and/or L(1 S(1)), CFP (if the number of electrons in open shells changes) and appropriate algebraic multipliers. It is usually assumed that the radial integral does not depend on the quantum numbers of the vec-... 

Here a complete model space (P-space), defined on eigenfunctions of Ho, representing all possible distributions of electrons between open shells, is utilized. In our case the model space consists of the ls22s22p4 and ls22p6 configurations for the even parity states and of the ls22s2p5 configuration for odd parity states. The wave-operator may be written as... 

Calculations show that cross-sections obtained in the Hartree-Fock approximation utilizing length and velocity forms of the appropriate operator, may essentially differ from each other for transitions between neighbouring outer shells, particularly with the same n. However, they are usually close to each other in the case of photoionization or excitation from an inner shell whose wave function is almost orthogonal with the relevant function of the outer open shell. In dipole approximation an electron from a shell lN may be excited to V = l + 1, but the channel /— / + prevails. For configurations ni/f1 n2l 2 an important role is... 

Offermann ejt al /70/ formulated an open—shell CC formalism in Nuclear Physics in which a Bloch wave-operator Cl n for an n-valence problem is obtained recursively from those of the lower valence problems ... 

Mukherjee/91/ initially proved LCT for incomplete model spaces having n-hole n—particle determinants, showing also at the same time the validity of the core—valence separation. The corresponding open-shell perturbation theory of Brandow/20/ for such cases leads to unlinked terms and a breakdown of the core-valence separation, which used IN for O. Mukherjee emphasized that it is essential to have a valence-universal wave operator O within a Fock space formulation/91/ such that it also correlates the subduced valence sectors. Later on,... 

Another popular approach to the correlation problem is the use of perturbation theory. Fq can be taken as an unperturbed wave function associated with a particular partitioning of the Hamiltonian perturbed energies and wave functions can then be obtained formally by repeatedly applying the perturbation operator to Probably the commonest partitioning is the M ller-Plesset scheme, which is used where Fq is the closed-shell or (unrestricted) open-shell Hartree-Fock determinant. Clearly, the perturbation energies have no upper bound properties but, like the CC results, they are size-consistent. 

Electron-electron repulsion can have a profound effect on the electronic structure of a system. For a closed-shell system described by one Slater determinant, in which the up-spin and down-spin electrons of a given MO are restricted to have an identical spatial function, the effective one-electron Hamiltonian employed in Section 26.2 is given by the Fock operator [3]. When one Slater determinant is used to describe the electronic structure of an open-shell system, the up-spin and down-spin electrons are allowed to have different spatial functions. For a certain open-shell system (e.g. diradical), a proper description of its electronic structures even on a qualitative level requires the use of a configuration interaction (Cl) wave function [6], i.e. a linear combination of Slater determinants. In this section, we probe how electron-electron repulsion affects the concepts of orbital interaction, orbital mixing and orbital occupation by considering a dimer that is made up of two identical sites with one electron and one orbital per site (Fig. 26.3). 

Paldus, elsewhere in this book, discusses that there is as yet no generally applicable, open-shell, size-extensive, coupled cluster method, and the same holds for open-shell S APT methods. Therefore, for the computation of potentials of open-shell van der Waals molecules one has the choice between CASSCF followed by a Davidson-corrected MRCl calculation of the interaction energy, or the single reference, high spin, method RCCSD(T). When the ground state of the open-shell monomer is indeed a high spin state, then RCCSD(T) is the method of choice. With regard to the latter method we recall that a major difficulty in open-shell systems is the adaptation of the wave function to the total spin operator S for the CCSD method a partial spin adaptation was published by Knowles et al. [219,220] who refer to their method as partially spin restricted . When non-iterative triple corrections [221] are included, the spin restricted CCSD(T) method, RCCSD(T), is obtained. 

For open-shell systems, multidimensional model spaces are necessary in general in order to account for that the unperturbed solutions [ 4>oi) of the model operator Ho are degenerate. If we act with the wave operator in second quantization (40) to the right upon some determinant d>a) c M, we see of course that each term of this operator leads to an excitation of occupied electron orbitals. These excitations can be classified due to ... 

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