Supersystem

Conditions for thermodynamic equilibrium of the lamella can be derived by considering the lamella plus its environment as an isolated supersystem. Assuming the entropy of the supersystem to be fixed, one knows that the... 

FIGURE 1.7. The potential energy surface of the CH4 + C1 supersystem for the collinear hydrogen abstraction reaction CH4 + Cl—> CH3 + HC1. The counter lines are given in spaces of 10 kcal/mol and the coordinates in angstroms. 

Ganti, T. (1979), A Theory of Biochemical Supersystems, University Park Press, Baltimore, MD. 

The two-reactant coupled approach (Nalewajski and Korchowiec, 1997 Nalewajski et al., 1996, 2008 Nalewajski, 1993, 1995, 1997, 2002a, 2003, 2006a,b) can also be envisaged, but the relevant compliant and MEC data would require extra calculations on the reactive system A—B as a whole, with the internal coordinates Q now including those specifying the internal geometries of two subsystems and their mutual orientation in the reactive system. The two-reactant Hessian would then combine the respective blocks of the molecular tensors introduced in Section 30.2. The supersystem relations between perturbations and responses in the canonical geometric representation then read ... 

Elenkov IJ, Wilder RL, Chrousos GP, Vizi SS The sympathetic nerve - an integrative interface between two supersystems the brain and the immune system. Pharmacol Rev 2000 52 595-628. 

The SCF-MI algorithm, recently extended to compute analytic gradients and second derivatives [18,41], furnishes the Hartree Fock wavefunction for the interacting molecules and also provides automatic geometry optimisation and vibrational analysis in the harmonic approximation for the supersystems. The Ml strategy has been implemented into GAMESS-US package [42]. 

The theory is here presented very briefly. The SCF-MI one determinant wavefunction of the supersystem AB is expressed as ... 

Much of the recent literature on RDM reconstruction functionals is couched in terms of cumulant decompositions [13, 27-38]. Insofar as the p-RDM represents a quantum mechanical probability distribution for p-electron subsystems of an M-electron supersystem, the RDM cumulant formalism bears much similarity to the cumulant formalism of classical statistical mechanics, as formalized long ago by by Kubo [39]. (Quantum mechanics introduces important differences, however, as we shall discuss.) Within the cumulant formalism, the p-RDM is decomposed into connected and unconnected contributions, with the latter obtained in a known way from the lower-order -RDMs, q < p. The connected part defines the pth-order RDM cumulant (p-RDMC). In contrast to the p-RDM, the p-RDMC is an extensive quantity, meaning that it is additively separable in the case of a composite system composed of noninteracting subsystems. (The p-RDM is multiphcatively separable in such cases [28, 32]). The implication is that the RDMCs, and the connected equations that they satisfy, behave correctly in the limit of noninteracting subsystems by construction, whereas a 2-RDM obtained by approximate solution of the CSE may fail to preserve extensivity, or in other words may not be size-consistent [40, 42]. 

It is important to emphasize, that the CMOs of the corresponding CP-corrected subsystems can be separated in a similar manner. The SMOs can thus unambigously be attributed to one ore another of the interacting monomers. Furthermore, the different energy quantities in a supersystem can be simply obtained by suming over the corresponding quantities calculated from the contributing subsystems [7-10]. 

The interaction energy of two (weakly) interacting monomers can be calculated as the difference between the total energy of the supersystem and that of the contributing monomers, as follows ... 

When referring to the supersystem it will be convenient to adopt a compact notation as follows ... 

In other words, we get the same result by considering (A+B) as a supersystem as when handling A and B subsystems separately. This is especially important for extended systems, involving n subsystems, in which case the limiting process n —> oo will only make sense if the energy is strictly linear in n in the noninteracting limit. 

It is well known that the CC and MBPT energies are size-extensive, while this is not the case for truncated Cl methods. Indeed, it is not difficult to verify (see, e.g.. Ref. [8]) that for a system consisting of n subsystem [e.g., (He) ], the CISD energy is proportional to rather than to n. This is easy to comprehend when we realize that by restricting the excitations to, say, doubles, we ignore simultaneous double excitations on the subsystems A and B when considering the supersystem (A+B), which now represent quadruples, while such excitations are taken into account when handling the systems A and B individually. 

When one refers to a quantum mechanical solvent model, the word model reverts to its usual sense in die context of QM methods it is the level of electronic structure theory used to describe die solvent. Thus, diere is no real distinction between the solvent and die solute in terms of computational technology - the wave function for the complete supersystem (or the DFT equivalent) is computed without resort to methodological approximations beyond those inherent to the level of electronic structure theory. To avoid problems with basis-set imbalances, one might expect calculations representing the solvent in a fully QM fashion to employ a common level of theory for all particles, but this does not have to be the case. 

Our starting point is the decomposition of the normal modes of a larger system into those of independently computed fragments [12], An exact decomposition is possible if the number of the nuclei of the fragments equals those of the supersystem, and provided all normal modes are considered, which means rotations and translations must be included in the treatment. In order to avoid the otherwise ubiquitous mass factors, it is convenient to use the matrix L which gives the transformation between the mass-weighted excursions of the nuclei a and the normal modes Qp, rather than Lx. The elements of the two matrices are related by Laip = /mJtLxai p [59], A normal mode Lsp of the system S can be written as linear combination of the normal modes Lf, Lf, Lcr of the independent subunits A, B, C - - with the numbers NA, NB, Nc - of nuclei ... 

We have derived a formula for the molecular partition function by considering a system containing many molecules at equilibrium with a heat bath. We can generalize our statistical mechanics by a gedanken experiment of considering a large number of identical systems, each with volume V and number of particles N at equilibrium with the heat bath at temperature T. Such a supersystem is called a canonical ensemble. Our derivation is the same the fraction of systems that are in a state with energy Et is... 

The spin-orbitals 17,) iGA + f form a basis set for the supersystem A —B. One possible procedure of realizing the transformation (218) as well as (219) is the Lowdin symmetric orthonormalization method137. ... 

Thus, we obtain the energy of the supersystem, EAB, from which it is possible to extract exact energies EA and EB for systems A and B and to obtain the expression for the interaction energy up to a particular order of the perturbation expansion. It was not the aim of this chapter to present the details of the MB-RSPT treatment of intermolecular interactions but rather to point out the fundamental ideas involved. Detailed derivation and explicit formulas as well as the physical interpretation of individual terms are presented in Refs.138,139 ... 

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