Unitary matrix, transformation

In open-shell electronic states, the orbitals are not all doubly occupied, and the preceding procedure is not applicable. However, if the wave function can be written as a single Slater determinant, one can use a modified procedure to obtain energy-localized MOs here also. The procedure is to deal with the a spin-orbitals and the jS spin-orbitals separately, using two different unitary transformation matrices Ba and B in (2.85). 

This defines the fermion contribution to an isovector gauge current density. Although the Euler-Lagrange equation is gauge covariant by construction, this fermion gauge current is not invariant, because the matrix r does not commute with the 5(7(2) unitary transformation matrices. It will be shown below that the... 

If the DLU(nr) approach is employed, the unitary transformation matrices of light atoms are simply set to identity matrices and the explicit form of Hau, Haa, and Hat, is... 

If we consider G as a unitary transformation matrix that diagonalizes the g matrix and 1 is the diagonal matrix with elements toy, j =, N as... 

U(qJ is referred to as an adiabatic-to-diabatic transformation (ADT) matrix. Its mathematical sbucture is discussed in detail in Section in.C. If the electronic wave functions in the adiabatic and diabatic representations are chosen to be real, as is normally the case, U(q ) is orthogonal and therefore has n n — l)/2 independent elements (or degrees of freedom). This transformation mabix U(qO can be chosen so as to yield a diabatic electronic basis set with desired properties, which can then be used to derive the diabatic nuclear motion Schrodinger equation. By using Eqs. (27) and (28) and the orthonormality of the diabatic and adiabatic electronic basis sets, we can relate the adiabatic and diabatic nuclear wave functions through the same n-dimensional unitary transformation matrix U(qx) according to... 

Because IJ is a unitary transformation matrix, equation (216) can be written in the form... 

Step 2. The set of CMOs orthogonal molecular orbitals (LMOs) Xj using, e.g., Ruedenberg s localization criterion205. This is achieved by multiplying up with an appropriate unitary transformation matrix L ... 

The fact that the molecular system can be described within two different frameworks indicates that there must be a relation between them. Such relations are usually formed via a unitary transformation matrix A and since we talk about the adiabatic framework and the diabatic framework it is natural to term it as the ADT. Considering equation (4) we replace by d> where the two are related ... 

Thus the EJR can be viewed as the simplest form of an exponential unitary transformation matrix, expressed in a real space. 

If A C then there exists a unitary transformation matrix Q C such that... 

HnkFn k> with n and k not simultaneously zero, from Hp and corrects HooFQF +ujrN +ijjcN°. The unitary transformation matrix Dp can be defined by a Hermitian operator Sp... 

Natural orbitals [28-39] are defined to be a basis set in terms of which the first-order density matrix is diagonal. Thus if Cb, is a unitary transformation matrix which brings pab into a diagonal form, i.e. if Cbi is a solution to the secular equation... 

In the course of the diagonalization, the corresponding unitary transformation matrix < K Uj ry which diagonalizes the original Hamiltonian matrix according to Eq.1.5), is also determined. 

The unitary transformation matrix A(q) of (71), (73) and (79) is arbitrary, and can be chosen to make (79) have desirable properties which (74) does not display. In many instances the couplings between consecutive PESs are significant only in the small regions of nuclear configuration space where those PESs are close to each other or... 

If we consider G as a unitary transformation matrix that diagonalizes the g matrix and im is the diagonal matrix with elements itOy, j = I,. .., N as the corresponding eigenvalues, it can be shown that, following the unitary transformation performed with G, Eq. (46) becomes... 

The states 0), n> and thus are related through a unitary transformation matrix C. Variations in the expansion coefficients may be achieved either by a direct variation of these linear parameters or alternatively in terms of parameters describing a unitary transformation among the states />. The operator... 

The operator a of the reflection in the plane x = 0 corresponds to the following unitary transformation matrix of... 

This last line defines the Wigner rotation matrices, whose properties are widely tabulated. For each j, T>4co is a unitary transformation matrix note that a rotation can only change m-type quantum numbers, not the total angular momentum j. One of the more useful properties of the D matrices, for us, is... 

Note that is the vector matrix of derivative couplings in the adiabatic electronic basis and the gauge transformation (R) is the unitary transformation matrix connecting the adiabatic and diabatic basis sets. In the above example of two real electronic states, Eq. (32) is identical to Eq. (30a) where x is set to zero ... 

We want the co-ordinates to be perpendicular to each other and achieve that with the orthogonal unitary transformation matrix... 

Directions of generalized modes Qi compose the unitary transformation matrix L, which diagonalizes the covariance matrix K of the mass weighted atomic velocities q, with elements... 

Numerically, the elements in Powell s (5 x 5) matrix form what is called a unitary transformation matrix. The main property of a unitary matrix is that it can scramble the relative amounts of the components but it does not change the overall length (magnimde) of the vector result. This can be shown by evaluating the determinant of the matrix, which is exacdy 1.0, but in this case evaluating a (5 X 5) determinant is a tedious process ... 

Fig. 7 Screenshot of the panel that analyzes the unitary transformation matrix V using a graphicai representation of the matrix. The ROA backward-scattering intensities (in A /amu) associated with the optimum normai modes are aiso given...

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