Matrix elements factorization

The part of Eq. (5) within the parentheses gives the energy levels in reduced form the constant A, conventionally expressed in units of cm-1, is a scaling factor. Matrix elements for these calculations are readily available.13... 

Under the assumption that the matrix elements can be treated as constants, they can be factored out of the integral. This is a good approximation for most crystals. By comparison with equation Al.3.84. it is possible to define a fiinction similar to the density of states. In this case, since both valence and conduction band states are included, the fiinction is called the joint density of states ... 

Equation (A 1.6.94) is called the KHD expression for the polarizability, a. Inspection of the denominators indicates that the first temi is the resonant temi and the second temi is tire non-resonant temi. Note the product of Franck-Condon factors in the numerator one corresponding to the amplitude for excitation and the other to the amplitude for emission. The KHD fonnula is sometimes called the siim-over-states fonnula, since fonnally it requires a sum over all intennediate states j, each intennediate state participating according to how far it is from resonance and the size of the matrix elements that coimect it to the states i. and The KHD fonnula is fiilly equivalent to the time domain fonnula, equation (Al.6.92). and can be derived from the latter in a straightforward way. However, the time domain fonnula can be much more convenient, particularly as one detunes from resonance, since one can exploit the fact that the effective dynamic becomes shorter and shorter as the detuning is increased. 

The first and second terras contain phase factors identical to those previously met in Eq. (82). The last term has the new phase factor [Though the power of q in the second term is different from that in Eq. (82), this term enters with a physics-based coefficient that is independent of k in Eq. (82), and can be taken for the present illustration as zero. The full expression is shown in Eq. (86) and the implications of higher powers of q are discussed thereafter.] Then a new off-diagonal matrix element enlarged with the third temi only, multiplied by a (new) coefficient X, is... 

The matrix elements (60) represent effective operators that still have to act on the functions of nuclear coordinates. The factors exp( 2iAx) determine the selection rules for the matrix elements involving the nuclear basis functions. 

Note in passing that the common model in the theory of diffusion of impurities in 3D Debye crystals is the so-called deformational potential approximation with C a>)ccco,p co)ccco and J o ) oc co, which, for a strictly symmetric potential, displays weakly damped oscillations and does not have a well defined rate constant. If the system permits definition of the rate constant at T = 0, the latter is proportional to the square of the tunneling matrix element times the Franck-Condon factor, whereas accurate determination of the prefactor requires specifying the particular spectrum of the bath. 

Owing to the separation of the active and inactive modes, in the Condon approximation the matrix element (2.56) breaks up into the product of overlap integrals for inactive modes and a constant factor V responsible for interaction of the potential energy terms due to the active modes. In this approximation the survival probability of Ai develops in time as... 

This is an occupied-virtual off-diagonal element of the Fock matrix in the MO basis, and is identical to the gradient of the energy with respect to an occupied-virtual mixing parameter (except for a factor of 4), see eq. (3.67). If the determinants are constructed from optimized canonical HF MOs, the gradient is zero, and the matrix element is zero. This may also be realized by noting that the MOs are eigenfunctions of the Fock operator, eq. (3.41). 

In other words, the matrix element can be factored in the form... 

Alternatively, one can drop the E0IV factor in (11-121) and, whenever computing any matrix element, omit from consideration the contribution from disconnected vacuum fluctuation diagrams. We shall do so hereafter. 

Upon inserting the factor U (a,A)U(a,A) = 1 into the matrix element (11-393) we find... 

Mixing of LS-states by spin orbit coupling will be stronger with an increasing number of f-electrons. As a consequence, intermediate values of Lande g factor and reduced crystal field matrix elements must be used. 

Here the relative intensities of the components of each branch are determined by the Boltzmann factor Correlation function K (t, J), corresponding to Gq(a>, J), is obviously the correlation function of a transition matrix element in Heisenberg representation... 

To find the coefficient of a given exchange integral in a matrix element, (I/F/PII), draw the vector-bond diagram for structure II, change it as indicated by the permutation diagram for P, and form the superposition pattern of I and PI I. The coefficient is then given, except for the factor (—l)p, by the above rules for the Coulomb coefficient that is, it is (— 1)F(— V)r2 in t>. 

Without loss of generality y = y can be assumed. If the dipole moment can be assumed to be a linear function of coordinate within the spread of the frozen Gaussian wave packet, the matrix element (gy,q,p, Pjt(r) Y,q, p ) can be evaluated analytically. Since the integrand in Eq. (201) has distinct maxima usually, we can introduce the linearization approximation around these maxima. Namely, the Taylor expansion with respect to bqp = Qq — Qo and 8po = Po — Po is made, where qj, and pj, represent the maximum positions. The classical action >5qj, p , ( is expanded up to the second order, the final phase-space point (q, p,) to the first order, and the Herman-Kluk preexponential factor Cy pj to the zeroth order. This approximation is the same as the ceUularization procedure used in Ref. [18]. Under the above assumptions, various integrations in U/i(y, q, p ) can be carried out analytically and we have... 

Next we discuss the effect of deuteratlon on low frequency modes Involving the protons> Because of the anharmonlc variation of the energy as a function of tilt angle a (Fig. 4b), the hindered rotations of H2O and D2O turn out to be qualitatively different. The first vibrational excited state of H2O Is less localized than that of D2O, because of Its larger effective mass. The oscillation frequency of the mode decreases by a factor 1.19 and the matrix elements by a factor 1.51 upon deuteratlon. Therefore, the harmonic approximation, which yields an Isotopic factor 1.4 for both the frequency and the Intensity, Is quite Inappropriate for this mode. 

In Table II we compare the calculated frequency and matrix element (In parenthesis) for the D2O-Al(100) hindered rotation with experimental results for D20-Cu(100) (14). The experimental Isotopic factor Is 1.16 for the frequency and =1.6 for the matrix elements,... 

Taking into account (12), the matrix elements (6), with their phase factor in (4), may be written in the following way ... 

The state mixing term, the first in the r.h.s., usually dominates, at least in the presence of avoided crossings. Its determination reduces to a simple problem of interpolation of the Hu matrix elements, according to eq.(16). The second term corresponds, for large R, to the electron translation factor (see for instance [38]). This term depends on the choice of the reference frame that is, for baricentric frames, it depends on the isotopic masses. It contains the Gn matrix, which may be determined by numerical differentiation of the quasi-diabatic wavefunctions [16] this calculation is more demanding, especially in the case of many internal coordinates. It is therefore interesting to adopt the approximation ... 

As XRF is not an absolute but a comparative method, sensitivity factors are needed, which differ for each spectrometer geometry. For quantification, matrix-matched standards or matrix-correction calculations are necessary. Quantitative XRF makes ample use of calibration standards (now available with the calibrating power of some 200 international reference materials). Table 8.41 shows the quantitative procedures commonly employed in XRF analysis. Quantitation is more difficult for the determination of a single element in an unknown than in a known matrix, and is most complex for all elements in an unknown matrix. In the latter case, full qualitative analysis is required before any attempt is made to quantitate the matrix elements. 

XRF nowadays provides accurate concentration data at major and low trace levels for nearly all the elements in a wide variety of materials. Hardware and software advances enable on-line application of the fundamental approach in either classical or influence coefficient algorithms for the correction of absorption and enhancement effects. Vendors software packages, such as QuantAS (ARL), SSQ (Siemens), X40, IQ+ and SuperQ (Philips), are precalibrated analytical programs, allowing semiquantitative to quantitative analysis for elements in any type of (unknown) material measured on a specific X-ray spectrometer without standards or specific calibrations. The basis is the fundamental parameter method for calculation of correction coefficients for matrix elements (inter-element influences) from fundamental physical values such as absorption and secondary fluorescence. UniQuant (ODS) calibrates instrumental sensitivity factors (k values) for 79 elements with a set of standards of the pure element. In this approach to inter-element effects, it is not necessary to determine a calibration curve for each element in a matrix. Calibration of k values with pure standards may still lead to systematic errors for unknown polymer samples. UniQuant provides semiquantitative XRF analysis [242]. 

The high value of the electron density at the nucleus leads to the enhancement of the electron EDM in heavy atoms. The other possible source of the enhancement is the presence of small energy denominators in the sum over states in the first term of Eq.(29). In particular, this takes place when (Eo — En) is of the order of the molecular rotational constant. (It is imperative that a nonperturbative treatment be invoked when the Stark matrix element e z(v /0 z v / ) is comparable to the energy denominator (Eq En) [33].) Neglecting the second term of the right-hand side of Eq.(29), which does not contain this enhancement factor [8, 27], we get... 

Where XA is the concentration of element A and dI /dXA is the absolute sensitivity factor of element A in matrix M. A calibration curve is established using known samples for each element in each matrix. Unknown concentrations may then be determined provided all instrumental factors and experimental conditions are held constant, which conditions may be difficult to fulfil. 

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