Open-shell perturbation theory

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,... 

This problem can be avoided, however, if an appropriate open-shell perturbation theory is defined such that the zeroth-order Hamiltonian is diagonal in the truly spin-restricted molecular orbital basis. The Z-averaged perturbation theory (ZAPT) defined by Lee and Jayatilaka fulfills this requirement. ZAPT takes advantage of the symmetric spin orbital basis. For each doubly occupied spatial orbital and each unoccupied spatial orbital, the usual a and P spin functions are used, but for the singly occupied orbitals, new spin functions. 

P. M. Kozlowski and E. R. Davidson, Chem. Phys. Lett., 226, 440 (1994). Construction of Open-Shell Perturbation Theory Invariant with Respect to Orbital Degeneracy. 

T. D. Crawford, H. F. Schaefer, and T. J. Lee, /. Chem. Phys., 105, 1060 (1996). On the Energy Invariance of Open-Shell Perturbation Theory with Respect to Unitary Transformations of Molecular Orbitals. 

Two-electron operators can be added to Hq. A small number of important terms were suggested by Murray and Davidson for open-shell perturbation theory [16] and later by Kozlowski and Davidson for more general multiconfiguration perturbation theory [17]. Such formulas, which... 

An optimum set of MOs obtained by means of Eq. (22) allows us to construct a well-defined open-shell perturbation theory for excited states which is a natural extension of the popular closed-shell MP2. For example, the zeroth-order Hamiltonian for the first excited state is as follows ... 

C. W. Murray and N. C. Handy,/. Chem. Phys., 97, 6509 (1993). Comparison and Assessment of Different Forms of Open-Shell Perturbation Theory. 

Perturbative approximation methods are usually based on the Mpller-Plesset (MP) perturbation theory for correcting the HF wavefunction. Energetic corrections may be calculated to second (MP2), third (MP3), or higher order. As usual, the open- versus closed-shell character of the wavefunction can be specified by an appropriate prefix, such as ROMP2 or UMP2 for restricted open-shell or unrestricted MP2, respectively. 

We have previously defined the one-electron spin-density matrix in the context of standard HF methodology (Eq. (6.9)), which includes semiempirical methods and both the UHF and ROHF implementations of Hartree-Fock for open-shell systems. In addition, it is well defined at the MP2, CISD, and DFT levels of theory, which permits straightforward computation of h.f.s. values at many levels of theory. Note that if the one-electron density matrix is not readily calculable, the finite-field methodology outlined in the last section allows evaluation of the Fermi contact integral by an appropriate perturbation of the quantum mechanical Hamiltonian. 

Second, where nondynamical correlation is unimportant (that is, mainly closed-shell molecules near their equilibrium geometries), size-extensive (or nearly size-extensive) treatments like CPF, CCSD, and CISD+Q perform well. However, as bonds are stretched, or in species where nondynamical correlation is important because of near-degeneracies or open-shell effects, the performance of these methods deteriorates fairly rapidly. As might be expected, the CCSD method generally deteriorates slowest. In general, low-order perturbation theory methods do not agree well... 

The majority of the above-mentioned problems will be discussed in more or less detail in this book. The book is based on the variational approach, which is the most universal and efficient method of theoretical study of the spectra of any atom or ion of the Periodical Table. However, generalization of the perturbation theory to cover the case of configurations with several open shells is also presented [26, 51, 52],... 

In this section, we briefly discuss some of the electronic structure methods which have been used in the calculations of the PE functions which are discussed in the following sections. There are variety of ab initio electronic structure methods which can be used for the calculation of the PE surface of the electronic ground state. Most widely used are Hartree-Fock (HF) based methods. In this approach, the electronic wavefunction of a closed-shell system is described by a determinant composed of restricted one-electron spin orbitals. The unrestricted HF (UHF) method can handle also open-shell electronic systems. The limitation of HF based methods is that they do not account for electron correlation effects. For the electronic ground state of closed-shell systems, electron correlation effects can be accounted for relatively easily by second-order Mpller-Plesset perturbation theory (MP2). In modern implementations of MP2, linear scaling with the size of the system has been achieved. It is thus possible to treat quite large molecules and clusters at this level of theory. 

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