Disjoint orbitals

Proof. First suppose is a complex torus T. Taking a basis of V, we may suppose we are in the above situation. (If some coordinates are 0, we replace P by a subspace.) We may also assume that Y is the unique closed orbit in the closure of x. The uniqueness follows from the existence of T -invariant polynomial which separates two disjoint T -invariant closed subsets (Theorem 3.3). 

To illustrate this point, consider a composite system composed of two noninteracting subsystems, one with p electrons (subsystem A) and the other with q = N — p electrons (subsystem B). This would be the case, for example, in the limit that a diatomic molecule A—B is stretched to infinite bond distance. Because subsystems A and B are noninteracting, there must exist disjoint sets Ba and Bb of orthonormal spin orbitals, one set associated with each subsystem, such that the composite system s Hamiltonian matrix can be written as a direct sum. 

Size consistency The DMRG ansatz is size-consistent when using a localized basis (e.g., orthogonalized atomic orbitals) in which the wave function for the separated atoms can be considered to factorize into the wave functions for the individual atoms expressed in disjoint subsets of the localized basis. To see this in an informal way, let us assume that we have two DMRG wave functions Pa) and I Pg) for subsystems A and B separately. Both Pa and VPB have a matrix product structure, that is... 

Here Pa(a = 6, e) is the momentum conjugate to Qa. In the absence of spin-orbit interaction, the e vibration does not mix the orbital components of the 4T2 g and we have vibrational potential energy surfaces consisting of three separate ( disjoint ) paraboloids in the two-dimensional (2D) space of the Qe and Qe coordinates of the e vibration. The Jahn-Teller coupling leads only to a uniform shift (—ZsPJX = — V2/2fia>2 = —Sha>) of all vibronic levels. 

In a different formulation, Borden and Davidson classified alternant 7T-systems as disjoint and nondisjoint.69 Simplistically, disjoint systems can yield spin orbitals having no atoms in common by linear recombination of their SOMOs the spins are separated onto different parts of the molecules, will have limited interelectron exchange interaction, and so have much less preference for HS states. Systems whose SOMOs cannot be so localized are nondisjoint, will have significant interspin interactions, and so will favor HS states where na > np. Open-shell systems having na = np are automatically disjoint because their spin orbitals can be confined to separate a and yS subsets, with low-spin (LS) ground states being more favorable in agreement with Ovchinnikov Klein. 

In systems with two degrees of freedom, two-dimensional tori separate the equi-energy surface into two disjoint parts. Thus, orbits on one part of the equi-energy surface cannot go into the other. Therefore, the existence of two-dimensional tori in systems of two degrees of freedom results in nonergodic behavior of the system. 

Note that commutation of cluster operators holds only when the occupied and virtual orbital spaces are disjoint, as is the case in spin-orbital or spin-restricted closed-shell theories. For spin-restricted open-shell approaches, where singly occupied orbitals contribute terms to both the occupied and virtual orbital subspaces, the commutation relations of cluster operators are significantly more complicated. See Ref. 36 for a discussion of this issue. 

Topologically equivalent vertices constitute disjoint subsets of vertices called orbits. 

Seven vertices give rise to 7 = 5040 mappings onto themselves however, only the four mappings ven in Table 2 are automorphic mappings of This is a consequence of the requirement that the pair relationships most be preserv, i.e. only equivalent vertices can be m ped automorphically onto each other. Equivalent vertices form disjoint subsets of V Q) called orbiis. Obviously, the symmetry of a gra h is reflected by the partitioning of its vertex set into orbits. Fbr Gts the following orbits are obtained ... 

Now, suppose we apply the transformations to the distinct one-electron integrals and add up the results to generate a one-electron matrix. The result is a one-electron matrix with the correct symmetry but the elements are all doubled-, a matrix of 2hij. It is easy to see why this doubling has occurred the result of applying the symmetry operations to the distinct elements does not result in a disjoint set of new elements-, repetitions occur as they do in the list of transformed orbitals. Of course, these repetitions must occur mth the molecular symmetry-, if hn occurs twice, so must 22 and In our case both hn and hi2 occur twice but this need not be so in general there will be a characteristic number of repetitions of each of the symmetry-distinct matrix elements. Thus the method is clear, we must... 

To achieve the first goal, we begin with partitioning a given set of spin orbitals (pp p = 1,..., A into three disjoint sets (1) core (C), (2) active (A) and (3) virtual (V) of dimensions nc, n, and n, respectively. Given the total number of electrons ( e), we construct a model space M =... 

Being orbits, the different conjugacy classes are disjoint and form a set-partition of G. Of course, if G is commutative, then hgh = g, so that Cf (g) = g and this partition of G is trivial. Moreover, the number of fixed points is constant on each conjugacy class ... 

A random generator of zeros and ones is now used in order to associate values 0 or 1 with the cyclic factors of (a) (be) (cd) (/). If it generates the sequence l,0,0,l,say, we obtain the labeled graph that has bonds joining the elements in the pairs a and /, while all the other pairs of vertices remain disjoint. Its orbit is represented by the graph shown below ... 

Thus, a transversal of the G-orbits on X can be obtained as a (disjoint) union of... 

The Fundamental Theorem suggests to break this problem into pieces as follows Decompose the set of orbits of A(P x P) into disjoint subsets of orbits. The union of this is one orbit of a suitable bigger group containing the permutation group induced by A(P x P). Then we have, for each distribution 5 y, according to the Fundamental Theorem the bijection... 

---