Hilbert space partitioning

In order to obtain a bond order formula for open-shell systems that can be applied for both the indep)endent-partide model and correlated wave functions and which simultaneously yields unique bond orders for all spin multiplet components (in the absence of a magnetic field), Alcoba et al. [151, 152] derived a general expression (in the Hilbert space partitioning scheme) from a second-order reduced density matrix. Furthermore, as the first- and second-order reduced density matrices are invariant with respect to the spin projection, they are only a function of the total spin or similarly of the maximum projection S = and the bond order can be evaluated for the highest spin-projected state = S. They arrived at the following expression for the bond order... 

The interpretive tools that satisfy the aforementioned conditions have the advantage of affording properties of atoms, bonds, and molecular fragments that can be regarded as true quantum mechanical observables. Unlike their counterparts produced by Hilbert space partitioning and other arbitrary approaches, these properties are free of bias and enjoy convergence to well-defined limits with improvements in the accuracy of the wavefunctions under analysis. 

Atomic Charges from Hilbert Space Partitioning... 

Although both of these definitions suffer from the obvious limitations imposed by the arbitrariness of the Hilbert space partitioning, equation (17) has been found to produce bond orders that are somewhat less sensitive to the basis set extension effects. In light of the arguments given in the previous sections, neither of these definitions is suitable for rigorous analysis of electronic wavefunctions. The same is true for the shared electron numbers produced by the formalisms developed by Roby and others. ... 

Philosophically and mathematically, the NPA methods are wavefunction-oriented, based on variational and perturbation-theoretic use of Hilbert space partitioning to define model chemistry constructs. For a system described by Schrodinger s equation, Hit = E r, it is advantageous to formulate a physical model in terms of a well-defined model Hamiltonian such that standard perturbation theory can be used to assess the corrections to the model, to determine which of several alternative model is uniquely best (gives most rapid... 

One way of getting rid of distortions and basis set dependence could be that one switches to the formalism developed by Bader [12] according to which the three-dimensional physical space can be partitioned into domains belonging to individual atoms (called atomic basins). In the definition of bond order and valence indices according to this scheme, the summation over atomic orbitals will be replaced by integration over atomic domains [13]. This topological scheme can be called physical space analysis. Table 22.3 shows some examples of bond order indices obtained with this method. Experience shows that the bond order indices obtained via Hilbert space and physical space analysis are reasonably close, and also that the basis set dependence is not removed by the physical space analysis. 

The physical idea behind the PT is the fragmentation of the Hilbert space H of the problem under study into two subspaces, usually termed Q and P, for which solutions of the corresponding SE are obtainable, while the solution of the SE for the full H is computationally formidable. By the partitioning of 7i into two subspaces, one is able to write, in each subspace independently, the SE for the projection of the unknown solution of the SE in H, while the dynamical effect of the complementary subspace is fully incorporated. In this way, one ultimately can construct the solution of the SE in H after solving for its projections in Q and P independently. 

Thus the eigenstates of HSF are labeled by the partitions [ASP] associated with the irreducible representations of S%v. These [Asp] labels are called permutation quantum numbers. The spin-free Hilbert space Fsp of the Hamiltonian may be decomposed into subspaces FSF([ASF]) invariant to 5 p... 

The real Hilbert space is always partitioned into a direct sum of subspaces, each representing a different energy eigenvalue of the spectrum of the hamiltonian operator ... 

The OOA, also known as Kugel-Khomskii approach, is based on the partitioning of a coupled electron-phonon system into an electron spin-orbital system and crystal lattice vibrations. Correspondingly, Hilbert space of vibronic wave functions is partitioned into two subspaces, spin-orbital electron states and crystal-lattice phonon states. A similar partitioning procedure has been applied in many areas of atomic, molecular, and nuclear physics with widespread success. It s most important advantage is the limited (finite) manifold of orbital and spin electron states in which the effective Hamiltonian operates. For the complex problem of cooperative JT effect, this partitioning simplifies its solution a lot. 

The Hilbert space of the wavefunctions is partitioned into a finite n-dimensional inner space, which is generated by the resonances and the unstable states of interest, and its orthogonal complements the outer space. The projectors onto the inner and the outer space are P and Q, respectively P + Q = 1. In the basis of the quasi-bound states = 1, 1,..., n), the projector into the inner space can be written as... 

The Hilbert space of the wavefunctions of quantum mechanics moves to the Hilbert space of the operators describing the observables. The expressions defined within quantum mechanics still holds for the probabilistic approach of mechanics based on the Liouville equation. H has to be replaced by the Liouvillian L and the partition becomes [32]... 

Let us partition the Hilbert space spanned by the y), y>a into — orthogonal subspaces. Each subspace is associated with one of the that are doubly occupied in We can choose an orthogonal basis in any of these subspaces and label these basis orbitals as We also define = < r. 

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