Brillouin condition

W. Kutzelnigg and D. Mukherjee, Irreducible Brillouin conditions and contracted Schrodinger equations for n-electron systems. I. The equations satisfied by the density cumulants. J. Chem. Phys. 114, 2047 (2001). 

GENERALIZED NORMAL ORDERING, IRREDUCIBLE BRILLOUIN CONDITIONS, AND CONTRACTED SCHRODINGER EQUATIONS... 

Formulating conditions for the energy to be stationary with respect to variations of the wavefunction P in this generalized normal ordering, one is led to the irreducible Brillouin conditions and irreducible contracted Schrodinger equations, which are conditions on the one-particle density matrix and the fe-particle cumulants k, and which differ from their traditional counterparts (even after reconstruction [4]) in being strictly separable (size consistent) and describable in terms of connected diagrams only. 

Figure 2. Diagrammatic representation of the terms that contribute to the irreducible two-parti-cle Brillouin conditions, Eq. (169).

While the (one-particle) Brillouin condition BCi has been known for a long time, and has played a central role in Hartree-Fock theory and in MC-SCF theory, the generalizations for higher particle rank were only proposed in 1979 [38], although a time-dependent formulation by Thouless [39] from 1961 can be regarded as a precursor. 

This is nothing but the Brillouin condition of MC-SCF theory. Explicitly, in an... 

One can formulate a two-particle analogue of the Brillouin condition, the IBC2 ( I stands for irreducible, which essentially means connected. For details see Refs. [20, 29].)... 

If one formulates the conditions for stationarity of the energy expectation value in terms of generalized normal ordering, one is led to either the irreducible fc-particle Brillouin conditions IBCj or the irreducible A -particle contracted Schrodinger equations (IBC ), which are conditions to be satisfied by y = yj and the k. One gets a hierarchy of k-particle approximations that can be truncated at any desired order, without any need for a reconstruction, as is required for the reducible counterparts. 

In this form, the amplitude equations (21) have been previously studied by Kutzelnigg and named the generalized Brillouin conditions [38]. 

W. Kutzelnigg, Generalized k-particle Brillouin conditions and their use for the construction of correlated electronic wave functions. Chem. Phys. Lett. 64, 383 (1979). 

That this condition is a generalization of the standard WKB (Wentzel-Kramers-Brillouin) condition can be seen by considering a one-degree-of-freedom system, where we have... 

Finally, we require for both the orbital and configurational variation parameters that 0 > satisfies the generalized Brillouin condition... 

Optimization of the APSG wave function requires the fulfillment of both the local and nonlocal Brillouin conditions [Eqs. (51) and (54)]. The former can be achieved by solving the local Scrodinger equations [Sect. 4.2] while the latter requires a laborious orbital optimization. 

We determine the nonrelativistic spin orbitals and hence the model space by the zeroth-order Brillouin condition, i.e. the stationarity condition... 

The second possibility is to use a gradient code, if this is available for the chosen method. The third method is the simplest, namely to evaluate E2 as an expectation value. This method is equivalent to the other two, if Eq satisfies a stationarity condition, like the Brillouin condition of Hartree-Fock theory. For non-stationary approaches, like MP2 or CC, the methods based on differentiations of the expectation value are more reliable. This is related to the validity or non-validity of the Hellmann-Feynman theorem. 

The molecular orbitals satisfy the Brillouin condition which states that the matrix elements of the Fock operator between occupied and unoccupied molecular orbitals vanish... 

These requirements on the matrix elements are precisely the generalized Brillouin conditions for the PHF functions [5]. They are very general and hold also for all the DODS type funtions such as the HPHF function, as well as for the RHF function. 

The part of the Fock potential not defined through the Brillouin condition is often chosen on physical ground [e.g., to have the resultant orbital energies represent ionization potentials and electron afiinities (via Koopmans theorem)] (MeWeeney and Sutcliffe, 1976). For a reference state containing a set of occupied spin-orbitals that we denote by a, / , y, S and a set of unoccupied spin-orbitals denoted ni, i, p, t/, the Fock potential in Eq. (2.89) is defined by the BT only between occupied and unoccupied orbitals. From Eq. (2.89) we get... 

This is the case if the y>i are chosen as the best overlap 5) or Brueck-ner 34,36) spin orbitals. That such a choice is always possible has been diown by Brenig and independently by Nesbet 4). Eq. (54) is usually referred to as the Brueckner condition, in contrast to the Brillouin condition (53). Note that (53) can be regarded as either a theorem, if one defines the Hartree-Fock equation in the conventional way, or as a condition from which the conventional Hartree-Fock equation can be derived. 

Eqs. (130)-(132), derived with the assumption of the generalized Brillouin condition [31], entirely define the CCSD(F12) model. It is worth to mention that this model is an approximation to the full CCSD-F12 method [41], where more explicitly correlated terms... 

The MP2-F12 coupling matrix (7 was obtained without the assumption of the generalized Brillouin condition [49]... 

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