Off-Diagonal Part

So while our four-atom carbon system can be described approximately by Eq. (4.6), taking all of the off-diagonal elements to be zero, a better description is as given in Eq. (4.11), where the various interaction elements are included. 

Note that this matrix (and similarly other F matrices) is symmetric about the diagonal. That is, the matrix element in the second column and first row is a stretch-stretch interaction between atoms 1-2 and 2-3. The matrix element in the first column of the second row (not shown) is exactly that same quantity. Hence, the upper triangle (above the diagonal) of the mabix is identical numerically and conceptually to the lower triangle (below the diagonal, which is not shown). This is the normal situation in mechanics, and hence we will normally write out and discuss only the upper triangle of the F matrix. 

Let us think again about the dehydrobutane molecule and its internal degrees of freedom. Evidently, there are three bond lengths that can stretch, two bond angles that can bend, and one torsion angle which that rotate. Thus, there are six internal degrees of freedom in the molecule, and, hence, we will need to solve six simultaneous 

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The situation is more complicated in the adiabatic limit when this inequality is reversed. According to Eq. (4.36) and Eq. (4.4) the off-diagonal parts of T and y are different. To elucidate this difference and explore its consequences we shall examine the spectra of the four-level system passing from non-adiabatic to adiabatic broadening. 

In the slow modulation limit the splittings of spectral components are much larger than the rate of line interference and the off-diagonal part of the impact operator may be simply omitted to a first approximation ... 

To recognize what has to be done we must start from the exact definition of the impact operator [192]. Its off-diagonal part responsible for Q-branch interference and rotational transitions is given by the following expression [61]... 

The projection operator method splits the density matrix into diagonal part and off-diagonal part. Discuss why this method is powerful for treating non-equilibrium statistical mechanics ... 

From the off-diagonal parts of Eq. (3), it is clear that (A )i2(p) and (A XjCp) are similarly effectively independent of. In connection with this, contributions to the... 

Expressing e determinant in Eq. (18), we find that only the diagonal terms remain, since e upper off-diagonal part of the determinant is zero. The final result is coordinate-independent. This is the most convenient choice and is the motivation behind the error placement at the edge of the interval. Placing the error at the center would introduce second derivatives of the potential to the Jacobian, which would make it coordinate-dependent and more expensive to compute. 

Magneto-optical effects are a consequence of the off-diagonal part of the dielectric tensor e (a). In order to show the influence of the plasma resonance on... 

In the spin off-diagonal part of the V intermediate, the symmetrizer again could not be used, instead the explicit contraction was computed and accumulated on the final ajd component of V. It is worth to note that Eqs. (117)-(119) are common for Ansatze 1 and 3, the difference is only at the level of the back-transformations where different dimensions of the contraction (referring to the full computational basis or full occupied space) are used. 

The derivation of the off-diagonal parts of the cutoff errors in the total energy and torques proceeds in the same way. That the sum over contains each pair contribution twice has also been considered in the error estimate of the total energy. The results are given by... 

As an illustration let us summarize the indirect % bonds between carbon atoms in benzene and butadiene [58] in the Hiickel approximation. For the consecutive numbering of carbons in the ring and chain, the relevant CBO matrix elements in benzene read = 1, Yw+i = 2/3, Y ,i+2 = 0, Yi.i+3 = -1/3, while the relevant off-diagonal part of the CBO matrix in butadiene is fuUy characterized by the elements... 

Since the diagonal and the off-diagonal parts of TP do not commute with each other, the propagator containing it in Eq. (38) is further written as... 

The number of two-component complex matrix multiplications is different for the different approaches. The BSS approach differs from the X2C approach only in this term. It requires three more than the X2C approach since the off-diagonal parts of its four-component Hamiltonian matrix are not diagonal. For the DKH approach, the number of two-component complex matrix... 

The second contribution to the perturbation V originates from the off-diagonal part of QqHQq and models interactions inside and between the decay chaimels. Denoting Vab the measure of these interactions and using (22), we get for the perturbative part of the Hamiltonian... 

For any stress tensor a, there is a coordinate system where only the diagonal components of the tensor are non-vanishing, whereas all off-diagonal parts are zero. In this coordinate system, all stresses are thus normal stresses. These stresses are called principal stresses of the stress tensor (see appendix A.7) the axes of the coordinate system are called the principal axes. Principal stresses are denoted with Roman numerals when they are sorted a > an > am ... 

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