Unitary matrix function

The derivation of the D matrix for a given contour is based on first deriving the adiabatic-to-diabatic transformation matrix, A, as a function of s and then obtaining its value at the end of the arbitrary closed contours (when s becomes io). Since A is a real unitary matrix it can be expressed in terms of cosine and sine functions of given angles. First, we shall consider briefly the two special cases with M = 2 and 3. 

The unitary transform does the same thing as a similarity transform, except that it operates in a complex space rather than a real space. Thinking in terms of an added imaginary dimension for each real dimension, the space of the unitary matrix is a 2m-dimensionaI space. The unitary transform is introduced here because atomic or molecular wave functions may be complex. 

In this form, one sees why the array Dj m, m is viewed as a unitary matrix, with M and M as indices, that describes the effect of rotation on the set of functions Yl,m - This... 

Kinetic energy functional, calculated from a Slater determinant Internal energy Unitary matrix... 

It is useful to give a matrix solution to this problem. We affix a superscript to emphasize that we are discussing a matrix solution for n basis functions. Since is Hermitian, it may be diagonalized by a unitary matrix, T = (T )" ... 

Recently an effective method of calculating the absorption mode of a dynamic NMR spectrum by the point by point approach has been suggested. (47) It is based on the symmetric form of the lineshape equation (147). One can always choose a real unitary matrix U such that only the last element of the vector F = Uf is non-zero. The matrix ought to have its last column equal to the vector (fff ) 1/2f. Upon a transformation of equation (149), with the matrix U, one obtains the /abs ([Pg.261]

Let us set up a 2D unitary matrix representation for the transformation of the spin functions a and (1 in Civ. So far, we have established only a relation between 0(3)+ and SU(2). The matrix representations of reflections or improper rotations do not belong to 0(3)+ because their determinants have a value of -1. To find out how a and p behave under reflections, we notice that any reflection in a plane can be thought of as a rotation through n about an axis perpendicular to that plane followed by the inversion operation. For instance, 6XZ may be constructed as xz = Cz(y) i. Herein, it is not necessarily required... 

The unitary matrix of (17) which transforms our ZXp) () matrix into that with a spherical basis according to (19), has with Rose s standard order of m-functions the following form ... 

D. Siminovitch, T. S. Untidt and N. C. Nielsen, Exact effective Hamiltonian theory. II Expansion of matrix functions and entangled unitary exponential operators. J. Chem. Phys., 2004, 120, 51-66. 

The irreducible representations may be classified according to whether j is an integer or half of an odd integer. We shall consider here the former, which are the potentially real representations 22 b, p. 287). These representations in contrastandard form 4) can be transformed into real form by a constant unitary matrix, i.e. the same matrix for every element in the group. The elements of the constant matrix will be chosen such that the contrastandard, self-conjugate sets which form the bases for the potentially real representations in complex standard form are connected to the sets of the usual real spherical harmonics which form the bases for the real standard representations. From the constant matrix, the vector coupling coefficients pertaining to the real functions will be deduced. 

Here the unitary operator generated by L acts only on the argument of the wave function (as in nonrelativistic quantum mechanics), while the unitary matrix exp(—i(pn S) only affects the spinor components. Hence... 

The only problem from such a transformation is the loss of the normalization of the wave function. Yet we may even preserve the normalization. Let us choose such a matrix A. that det A = 1. This condition will hold if A = 17, where is a unitary matrix (when V is real, we call V an orthogonal transformation). This means that... 

Spatial symmetry operations are linear transformations of a coordinate function space. When choosing the space in orthonormal form, symmetry operations will conserve orthonormality, and hence all transformations will be carried out by unitary matrices. This will be the case for all spatial representation matrices in this book. When all elements of a unitary matrix are real, it is called an orthogonal matrix. As unitary matrices, orthogonal matrices have the same properties except that complex conjugation leaves them unchanged. The determinant of an orthogonal matrix wiU thus be equal to 1. The rotation matrices in Chap. 1 are all orthogonal and have determinant -I-1. 

In order to apply the transformation to the interaction operator, we must also consider the effect of R on the column of bra-functions. This simply requires the inverse of the matrix, which, for a unitary matrix, is nothing but its complex conjugate transposed ... 

Note that the action of time reversal on the spin functions precisely corresponds to the C operator and thus is represented hy (Cj). This result may be generalized, in the sense that time reversal can be represented as the product of complex conjugation, denoted as K, and a unitary operator acting on the components of a function space, which we shall denote by the unitary matrix U. We thus write = U/f. When this operator is applied twice, it must return the same state, except possibly for a phase factor, say exp(fx). Following Wigner, we now show that the two cases = 1 are in fact the only possibilities. Hence, the phase factor can be only either H-1 (time-even state) or -1 (time-odd state) [11, Chap. 26]. Taking time reversal twice, we have... 

The Wannier functions obtained in Eq. 6.25 are not unique, because it is possible to mix the Bloch states of different band numbers by a unitary matrix The resulting Wannier functions are also a complete representation of the electronic structure, although their localisation features are different ... 

The linear combination of the Bloch orbitals of different bands can be performed by using a non-unitary matrix, resulting in nonorthogonal Wannier functions [3] 4>oR-... 

As pointed out by Koehler (1968), even the solution obtained from these modified force constants is not the most meaningful one, because it is obtained from the harmonic part only of the wavefunction (3.22). He shows that the complete dispersion curves are not obtained from the force constant matrix alone, but first it has to be transformed by a unitary matrix diagonalizing the matrix of second derivatives of the correlation function/ . For q O, however, the frequencies coincide and (3.27) is sufficient for the calculation of lattice frequencies. 

The steps of this demonstration depend upon the facts that Ri is unitary (so that R > tR i = E), that the transpose conjugate of a product is the product of the transpose conjugates of the individual factors, taken in reverse order, and that summation over T of some matrix function of RT is equivalent to summation over the same matrix function of T because of the group properties. 

Let Uab ( ) be the unitary matrix that diagonalizes the (Hermite) spectral function matrix Cab ( ) -... 

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