Orbital-invariant formulation

P. Pulay and S. Ssebo, Theor. Chim. Acta, 69,357 (1986). Orbital-Invariant Formulation and Second-Order Gradient Evaluation in Moller-Plesset Perturbation Theory. 

In LMP2 fheory fhe MP2 equations are expressed using an orbital-invariant formulation employing noncanonical orbitals, and a number of approximations are introduced to achieve reduced scaling of the computational cost. 

In the orbital-invariant formulation, the closed-shell MP2 correlation energy can be expressed as follows ... 

Pulay, R, and S. Saebo. Orbital-invariant formulation and second-order gradient evaluation in Moller-Plesset perturbation theory. Theor. Chim. Acta 69 357-368, 1986. 

W. Klopper, Chem. Phys. Lett., 186, 583 (1991). Orbital-Invariant Formulation of the MP2-... 

Contrarily to conventional MP2 theory, the original formulation of MP2-R12 theory (3,4) did not provide the same results when canonical or localized molecular orbitals were used. Indeed, for calculations on extended molecular systems, unphysical results were obtained when the canonical Hartree-Fock orbitals were rather delocalized (5). In order to circumvent this problem, an orbital-invariant MP2-R12 formulation was introduced in 1991, which is the preferred method since then (6),... 

It has the property that/pp = —IPp when the orbital p is doubly occupied and/pp = —EAp when the orbital is empty. The value will be somewhere between these two extremes for active orbitals. Thus, we have for orbitals with occupation number one /pp = — j(IPp + EAp). This formulation is somewhat unbalanced and will favor systems with open shells, leading, for example, to somewhat low binding energies [52]. The problem is that one would like to separate the energy connected with excitation out from an orbital from that of excitation into the orbital. This cannot be done within a one-electron formulation of the zeroth order Hamiltonian. K. Dyall has suggested to use a two-electron operator for the active part [53], but this leads to a too complicated formalism and also breaks important orbital invariance properties (the result is, for... 

Very similar in spirit to CEPA, but formulated as a functional to be made stationary, is the coupled-pair functional (CPF) approach of Ahlrichs and co-workers [28]. CPF can be viewed as modifying the CISD energy functional to obtain size-extensivity for the special case of noninteracting two-electron systems. One disadvantage of some of the CEPA methods is that, unlike CISD or CCSD, the results are not invariant to a unitary transformation that mixes occupied orbitals with one another. CPF... 

LCAO coefficients of the SCF orbitals involved. However, the invariance of the determinantsd wave functions under nonsingular linear transformations of the N basic one-electron functions indicates that the number of really Independent parameters is less than that of the LCAO coefficients. Instead of using them, we may utilize a theorem [45] which in the present case can be formulated as follows the N not necessarily normalized or orthogonalized one-electron functions arbitrary determinant wave function may be presented by the expression ... 

The two fundamental building blocks of Hartree-Fock theory are the molecular orbital and its occupation number. In closed-shell systems each occupied molecular orbital carries two electrons, with opposite spin. The occupied orbitals themselves are only defined as an occupied one-electron subspace of the full space spanned by the eigenfunctions of the Fock operator. Transformations between them leave the total HF wave function invariant. Normally the orbitals are obtained in a delocalized form as the solutions to the HF equations. This formulation is the most relevant one in studies of spectroscopic properties of the molecule, that is, excitation and ionization. The invariance property, however, makes a transformation to locahzed orbitals possible. Such localized orbitals can be valuable for an analysis of the chemical bonds in the system. 

For systems having unpaired electrons, it is usually necessary to include spin-orbit interaction, leading automatically to a two-component treatment. This was actually the original form of the Douglas—Kroll theory—the one-component, spin-free formulation is the result of a further approximation. The mere nature of spin-orbit interaction almost invariably calls for a multistate treatment, and it would thus seem that the level of complexity is dramatically increased as a result of the configuration interaction treatment that is typically required. In all fairness, though, one must keep in mind that the spin-orbit interaction is typically of importance because questions of multistate nature are being asked, and in such cases even a nonrelativistic treatment would often require a Cl treatment. 

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