Orbital invariance

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),... 

Tew, D.P., Klopper, W., Hattig, C. A diagonal orbital-invariant explicitly-correlated coupled-cluster method. Chem. Phys. Lett. 2008, 452, 326-32. 

Another perspective is offered by considering the size dependence of the self interaction (ii ii). This integral displays Ksize-dependence according to Eq. (2-25) and hence it vanishes for a completely delocalized orbital of an infinitely periodic insulator. For a spatially localized orbital, the same self interaction is nonzero. This is a consequence of the lack of orbital invariance of this integral. In HF theory, the self interaction is cancelled between Coulomb and exchange terms and does not spoil the orbital invariance of the total energy and wave function. In semiempirical DFT and TDDFT, since the cancellation is not exact, the remnant, positive self interaction (ii ii) is usually minimal for a delocalized orbital. This renders semiempirical DFT and TDDFT a universal tendency to favor delocalized wave functions irrespective of the true nature of chemical systems. 

Since the actions of the symmetry operators on local s-atomic orbitals are to leave the orbitals invariant, it is clear that the resultant matrices in Figure 3.1 are the same as those... 

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

Standard CC methods, which have been termed plain old CC (POCC) in the literature [231], are those in which the orbital optimization and correlation steps of the calculation are performed separately. POCC calculations therefore suffer from instability poles in addition to the appropriately located EOM poles, but the width of the former are quite small because of the approximate orbital invariance of CC methods that include single excitations [243]. These methods offer some advantages in treating PJT effects relative to CC approaches in which orbitals and cluster amplitudes are determined simultaneously, as discussed briefly in the next section. 

It turned out that this ansatz is not invariant with respect to a switch from canonical to localized orbitals, and that localized orbitals are actually the better choice [13]. This lack of unitary invariance is, of course, unsatisfactory, and one of us [14] has proposed a generalization in which orbital invariance is established. We come to this later. Let us first apply a pair transformation to (96). 

Since it is difficult to construct the extremal pairs for the CC-R12 calculations in question, we have rather constructed the extremal pairs approximatively by some simple criterion, and compared the results of several such choices with the full orbital invariant approach in those cases where it is stable. The fact that one has not the exact extremal pairs (which diagonalize the c-matrix) leads to some loss in the correlation energy, but this is in most cases negligibly small. 

The only case documented here, where the orbital-invariant approach diverges is that for the all-electron calculations of N2 at CCSD-R12 and CCSD(T)-R12 levels, for all three basis sets. Even for this example the MP2-R12/A and MP2-R12/B results are numerically stable, and in valence-only calculations even CCSD-R12 and CCSD(T)-R12 converge. Our general experience is that the approaches based on extremal pairs don t suffer from numerical instabilities, at variance with the orbital-invariant method. To avoid divergencies we use standard pairs as default option. 

F. A. Evangelista, J. Gauss, Insights into the orbital invariance problem in state-specific multireference couple cluster theory, J. Chem. Phys. 133 (2010) 044101. 

As a result, we may view the orbital invariant SS-MRCEPA, termed by us as SS-MRCEPA(I) (I, for invariant), as the optimal approximation to the parent SS-MRCC method, which includes all the EPV terms exactly and which utilizes only those counter terms of the equations which eliminate the lack of extensivity of the attendant non-EPV terms in an orbital invariant manner [59]. In this article, we will present a couple of invariant SS-MRCEPA methods, viz. SS-MRCEPA(O) and SS-MRCEPA(I), for general open-shell systems using spin-free unitary generator adapted cluster operators starting from explicitly spin-free full-blown parent SS-MRCC formalism. Eor a detailed discussion of the allied issues pertaining to all the SS-MRCEPA-like methods, we refer to our recent SS-MRCEPA papers [58,59] and an earlier expose by Szalay [66]. 

MRCEPA(O) by us, is the simplest among the CEPA-hke approximants to the SS-MRCC theory which is extensive and also avoids intruders. However, the appearance of the CAS energy Eq, rather than the ground state energy E itself renders it rather approximate. As we already mentioned, the complete linearity of the SS-MRCEPA(O) equations in the cluster amplitudes lends the same invariance property to it as in the MR-CISD. This parallels the situation in the single reference CEPA(O), which also possesses the invariance. In our earlier papers [58,59], we suggested other schemes where E appears which, however, did not have the orbital invariance property as that of the SS-MRCEPA(O). 

Recently we have proposed another variant where the state energy E appears in the approximation, which at the same time retains the orbital invariance [59]. In this formalism, there is a set of terms that are non-linear in the cluster amplitudes, but they have a special structure which again leads to the desired orbital-invariance. We approximate Hij. in by all terms which lead to single and double excitations out... 

In the case of CEPA methods, we have considered the orbital invariant versions. The simplest, the SS-MRCEPA(O), has no EPV terms, while in SS-MRCEPA(I), the full set of EPV terms are retained. The performance of SS-MRCEAP(O) methods in most of the cases is better than the SS-MRCEPA(I) version, though the latter has the advantage of preserving the desired restricted orbital invariance. 

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... 

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. 

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