Unitary decomposition

The unitary decomposition may be applied to any Hermitian, antisymmetric two-particle matrix including the 2-RDM, the two-hole RDM, and the two-particle reduced Hamiltonian. The decomposition is also readily generalized to treat p-particle matrices [80-82]. The trial 2-RDM to be purified may be written... 

The portion of the 2-RDM that may be expressed as wedge products of lower RDMs is said to be unconnected. The unconnected portion of the 2-RDM contains an important portion of the two-particle component from the unitary decomposition D2, and similarly, the trace and one-particle unitary components contain an important portion of the connected 2-RDM A, which corrects the contraction. Both decompositions may be synthesized by examining the unitary decomposition of the connected 2-RDM,... 

The 2-RDM may be made positive semidefinite if each of the negative eigenvalues is set to zero, but this alters not only the positivity but also the contraction of the 2-RDM to the 1-RDM and even the 2-RDM trace. How can we modify the 2-RDM to prevent it from being exposed by the set and yet maintain contraction to the M-representable 1-RDM Again we can employ the unitary decomposition. For a matrix (9, the decomposition is... 

Zeroing the 2-RDM eigenvalue associated with v, is equivalent to adding an appropriate amount of (9, to the 2-RDM. However, this also changes the trace and the underlying 1-RDM because (9, contains the zeroth and the first components of the unitary decomposition. We can modify the two-particle component only by adding just (9, 2 rather than (9,. The adjusted 2-RDM may then be expressed as... 

As in Section IV.A. the eigenvalues of the 1-RDM must lie in the interval [0,1] with the trace of each block equal to N/2. Similarly, with the a/a- and the jS/jS-blocks of the 2-RDM being equal, only one of these blocks requires purification. The purification of either block is the same as in Section IV.B.2 with the normalization being N N/2 — l)/4. The unitary decomposition ensures that the a/a-block of the 2-RDM contracts to the a-component of the 1-RDM. The purification of Section IV.B.2, however, cannot be directly applied to the a/jS-block of the 2-RDM since the spatial orbitals are not antisymmetric for example, the element with upper indices a, i fi, i is not necessarily zero. One possibility is to apply the purification to the entire 2-RDM. While this procedure ensures that the whole 2-RDM contracts correctly to the 1-RDM, it does not generally produce a 2-RDM whose individual spin blocks contract correctly. Usually the overall 1-RDM is correct only because the a/a-spin block has a contraction error that cancels with the contraction error from the a/ S-spin block. 

A better strategy is to introduce a modified unitary decomposition for the a/jS-block. An appropriate decomposition is... 

III. Purification Procedures Based on Unitary Decompositions of Second-Order Reduced Density Matrices... 

Unitary Decomposition of Antisymmetric Second-Order Matrices... 

III. PURIFICATION PROCEDURES BASED ON UNITARY DECOMPOSITIONS OF SECOND-ORDER REDUCED DENSITY MATRICES... 

As has been mentioned, the MZ purification procedure is based on Coleman s unitary decomposition of an antisymmetric Hermitian second-order matrix described earlier. When applied to singlet states of atoms and molecules, the computational cost of this purification procedure is reduced, since the 2-RDM (and thus the 1-RDM obtained by contraction) presents only two different spin-blocks, the aa- and a/i-blocks (and only one spin-block for the 1-RDM). For the remaining part of this section only this type of state will be treated. 

For the a/l-block of the 2-RDM the decomposition was generated ad hoc [70]. This is because this block is not antisymmetric under permutation of the orbital indices within the row or column subsets of indices and thus the unitary decomposition reported by Coleman cannot be applied. Hence the ad hoc decomposition is given here by... 

However, the decomposition of P into C+C is not unique, since, as Pecora wrote, any C = VC (where V is understood to be an (Ax A) unitary matrix) will generate the same P matrix, which" is just a basic fact of Quantum Mechanics or, more generally,... 

This material is a constituent in GB2, which is the binary version of Sarin (C01-A002). It is also commonly found as a decomposition product/impurity in unitary sarin. It may appear as a mixture with Methylphosphonic dichloride (C01-C046), known as Didi, that is also used as a constituent for binary G-series nerve agents. 

Let us briefly mention some formal aspects of the above-introduced formalism, which have been discussed in detail by Blaizot and Marshalek [218]. First, it is noted that the both the Schwinger and the Holstein-Primakoff representations are not unitary transformations in the usual sense. Nevertheless, a transformation may be defined in terms of a formal mapping operator acting in the fermionic-bosonic product Hilbert space. Furthermore, the interrelation of the Schwinger representation and the Holstein-Primakoff representation has been investigated in the context of quantization of time-dependent self-consistent fields. It has been shown that the representations are related to each other by a nonunitary transformation. This lack of unitarity is a consequence of the nonexistence of a unitary polar decomposition of the creation and annihilation operators a and at [221] and the resulting difficulties in the definition of a proper phase operator in quantum optics [222]. 

We notice that it is the analytic continuation which has the effect of breaking the time-reversal symmetry. If we contented ourselves with the continuous spectrum of eigenvalues with Re = 0, we would obtain the unitary group of time evolution valid for positive and negative times. The unitary spectral decomposition is as valid as the spectral decompositions of the forward or... 

The appropriate modification of the 2-RDM may be accomphshed by combining A-representability constraints, known as positivity conditions, with both the unitary and the cumulant decompositions of the 2-RDM. 

If the second-order Hermitian matrix follows the transformation rule for a (2,2) tensor, then this decomposition is the only possible manner of expressing these matrices as a sum of simpler parts so that the decomposition remains invariant under unitary tranformations of the basis [73]. 

To summarize the theory dynamic correlations are described by the unitary operator exp A acting on a suitable reference funchon, where A consists of excitation operators of the form (4). We employ a cumulant decomposition to evaluate all expressions in the energy and amphtude equations. Since we are applying the cumulant decomposition after the first commutator (the term linear in the amplimdes), we call this theory linearized canonical transformation theory, by analogy with the coupled-cluster usage of the term. The key features of the hnearized CT theory are summarized and compared with other theories in Table II. 

Another decomposition of the vectorial field can be found in the literature. This decomposition is formally identical to the previous one, with a notable difference, though, the vector r is now replaced by a constant unitary vector n. Still another decomposition of the field was proposed by Rowe [39]. This decomposition is written as... 

I. Case of Cartesian Coordinates Hansen s decomposition for a constant unitary vector n is written as... 

Something more can be obtained if the perturbation W vanishes also within the Z-th manifold so that the only nonvanishing matrix elements occur between the functions belonging to different manifolds. In this case applying the singular value decomposition allows us to state the following There exist two unitary matrices and U( ) of the sizes gk x gk and gi x gi, respectively, which when respectively applied on the left and on the right to the gk x gt matrix W(kr> formed by the matrix elements of the operator W produce the matrix U fc)W(fc )Ua) which... 

The actual sign ("phase") of the molecular orbital at any given point r of the 3D space has no direct physical significance in fact, any unitary transformation of the MO s of an LCAO (linear combination of atomic orbitals) wavefunction leads to an equivalent description. Consequently, in order to provide a valid basis for comparisons, additonal constraints and conventions are often used when comparing MO s. The orbitals are often selected according to some extremum condition, for example, by taking the most localized [256-260] or the most delocalized [259,260] orbitals. Localized orbitals are often used for the interpretation of local molecular properties and processes [256-260]. The shapes of contour surfaces of localized orbitals are often correlated with local molecular shape properties. On the other hand, the shapes of the contour surfaces of the most delocalized orbitals may provide information on reactivity and on various decomposition reaction channels of molecules [259,260]. 

Moving one step ahead to actual computation, within the quantum world everything is unitary, so the dynamics of computation can be described as U ip) —> I ip). An important result is that any unitary operator U may be simulated by a set of one- and two-qubit operations called gates [Barenco 1995 (b)]. Such a decomposition makes implementation feasible, which is good news. To see why this is so powerful, consider an arbitrary (entangled)... 

It should be mentioned that there are only a few restrictions on the choice of the matrices C/j. Firstly they have to be unitary and analytical (holomorphic) functions on a suitable domain of, and secondly they have to permit a decomposition of Hm in even terms of definite order in the external potential according to Eq. (73). It is thus possible to parametrise them without loss of generality by a power series expansion in an odd and antihermitean operator Wi of ith order in the external potential. In the following, the physical consequences of this freedom in the choice of the unitary transformations will be investigated. Therefore we shall start with a discussion of all possible parametrisations in terms of such power series expansions. Afterwards the most general parametrisation of Ui is applied to the Dirac Hamiltonian in order to derive the fourth-order... 

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