Amplitude matrix

The entire amplitude matrix can be calculated in a straightforward fashion if the complete set of eigenpairs En,bn of the total Hamiltonian (H) is known ... 

Now, we see that if we replace the operator B by the Hamiltonian H of a simple oscillator, these equations are identical to the corresponding equations of the simple quantum oscillator [30, 31]. According to this strong analogy, we are able to determine the amplitude matrix and the matrices of the Bohlin operator B, the displacement operator and the momentum operator. The results are as follows ... 

The components of the mean-square displacement amplitude matrix (MSDA)... 

An important question which can be asked concerns the magnitude of the contributions of certain types of amplitudes which correspond to different classes of excitations (which are represented by /). To consttuct an effective local theory it is necessary to include excitations that give the main connibutions to the total wave-function for the system under study and to neglect those that have smaller influence on the results of the calculation. As the measure of the connibution of a particular / layer of particular amplitudes (excitations which correspond to the cue(Q-CCSD level of theory see Sect. 3.2) it is convenient to use the sum of squares of the amplitude matrix elements ... 

As can be seen from the data, the dominant contribution to the wave-function of cue-CCSD method are made by an extremely small number of amplitudes of the T2 operator matrix (the size of corresponding amplitude matrix, N, mainly determines the computational complexity of the method). Including Z-layers with larger / (starting from the 4th for the polyenes, and 5th for polyacenes) involves larger amount of the matrix elements, but the contribution made by them is insignificant (less than 3 %). This clearly demonstrates the previously discussed fact of the correlation-effects locality. 

The size of amplitude matrix, presented in the last two rows... 

This set of equations can be solved directly, i.e., without proceeding by the pseudo-canonical transformation described in Appendix 3 for the more general case. The only quantities needed are the spherically averaged s and / associated with localized orbitals and the long-range dipole-dipole tensors. The update formula to get the nth approximation to the amplitude matrix element is... 

Our presentation is focused on the analysis of the scattered field in the far-held region. We begin with a basic representation theorem for electromagnetic scattering and then introduce the primary quantities which dehne the single-scattering law the far-held patterns and the amplitude matrix. Because the measurement of the ampUtude matrix is a comphcated experimental problem, we characterize the scattering process by other measurable quantities as for instance the optical cross-sections and the phase and extinction matrices. 

To introduce the concepts of tensor scattering amplitude and amplitude matrix it is necessary to choose an orthonormal unit system for polarization description. In Sect. 1.2 we chose a global coordinate system and used the vertical and horizontal polarization unit vectors Ba and e,g, to describe the polarization state of the incident wave (Fig. 1.9a). For the scattered wave we can proceed analogously by considering the vertical and horizontal polarization unit vectors and eg. Essentially, (e/t, 6/3,60.) are the spherical unit vectors of fee, while er,eg,e ) are the spherical unit vectors of fcs in... 

The above phase matrix is also known as the pure phase matrix, because its elements follow directly from the corresponding amplitude matrix that transforms the two electric field components [100], The phase matrix of a particle in a fixed orientation may contain sixteen nonvanishing elements. Because only phase differences occur in the expressions of Zij i,j = 1,2,3,4, the phase matrix elements are essentially determined by no more than seven real numbers the four moduli S pq and the three differences in phase between the Spq, where p = 6, f and q = f3, a. Consequently, only seven phase matrix elements are independent and there are nine linear relations among the sixteen elements. These linear dependent relations show that a pure phase matrix has a certain internal structure. Several linear and quadratic inequalities for the phase matrix elements have been reported by exploiting the internal structure of the pure phase matrix, and the most important inequalities are Zn > 0 and Zij < Z fori,j = 1,2,3,4 [102-104]. In principle, all scalar and matrix properties of pure phase matrices can be used for theoretical purposes or to test whether an experimentally or numerically determined matrix can be a pure phase matrix. 

Prom the reciprocity relation for the amplitude matrix we easily derive the reciprocity relation for the phase and extinction matrices ... 

To derive the expressions of the tensor scattering amplitude and amplitude matrix, we consider the scattering and incident directions and eu, and express the vector spherical harmonics as... 

Because the elements of the amplitude matrix can be expressed in terms of the elements of the transition matrix, the above relation can be used to express the elements of the amplitude matrix as functions of the particle orientation angles Op, /3p and 7p. The properties of the Wigner D-functions can then be used to compute the integrals over the particle orientation angles. 

To compute the orientation-averaged extinction matrix it is necessary to evaluate the orientation-averaged quantities (S pq(e, e )). Taking into account the expressions of the elements of the amplitude matrix (cf. (1.97)), the equation of the orientation-averaged transition matrix (cf. (1.118) and (1.119)) and the expressions of the vector spherical harmonics in the forward direction (cf. (1.121)), we obtain... 

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