Off-diagonal terms

The method shown affords easy generalization to higher order coupling in the important case where a single mode is engaged, that is, G i = g i = (l/i /2) e . Then the two off-diagonal terms derived above are, after physics-based constant coefficients have been affixed, in the upper right comer... 

The key study for our development of molecular mechanics was that by Scbachtschneider and Snyder (1969), who showed that transferable force constants can be obtained provided that a few off-diagonal terms are not neglected. These authors found that olf-diagonal terms are usually largest when neighbouring atoms are involved. A final point for consideration is that the C atom in OCS is obviously chemically different from a C atom in ethane and from a C atom in ethyne. It is necessary to take account of the chemical environment of a given atom. 

For bonded atoms, the off-diagonal terms (where i j) are taken to depend on tjje type and length of the bond joining the atoms on which the basis functions y- and Xj 0 centred. The entire integral is written as a constant, 0ij, which is not the same as the fixY in Hiickel 7r-electron theory. The are taken to be parameters, fixed by calibration against experiment. It is usual to set Pij to zero when the pair of atoms are not formally bonded. 

The off-diagonal terms (i j) are treated to a similar analysis. Each penetration term involving different basis functions that are on the same atomic centre are given a value of — Vab to maintain invariance. Suppose now that Xi is centred on nucleus A and Xj on nucleus B. We have... 

The qualitative behavior of this system is most conveniently studied by looking at space-time mai)S of the normalized probability values Pj(t) = [Pg.408]

Only the off-diagonal terms in H p or only H need be considered since these are the terms that mix S and T wavefunctions. 

The form of the g tensor depends upon the choice of orthogonal axes. If the axis system is chosen along the molecular symmetry axes, the tensor contains only diagonal terms with the off-diagonal terms being zero. Conveniently, the principal g values derived from the polycrystalline spectrum are just those of a diagonalized matrix, i.e.,... 

The previous discussion refers mainly to diagonal anisotropy terms. 6° 0°, which only contain powers of Jz and are often dominant. Deviations from pure axial symmetry lead to non-zero off-diagonal terms with m 0. These terms mix... 

The adiabatic approximation involves neglect of these off-diagonal terms, on the basis that En(Q) — Em(Q) (xm VQH xn). The diagonal elements (xn Vg x ) are undetermined by this argument, but the gradient of the normalization integral, (x x ) = 1, shows that... 

The analytic evaluation of the density matrix requires the diagonalization of Hi, which can be easily performed for the two extreme cases, a>r( Qg. Indeed, in the case of a low rf-field, only off-diagonal terms related to the CT are retained in the H Hamiltonian (39), which thus behaves like a fictitious spin-1/2 operator, affecting only the CT coherences /1 io- These coherences are thus selectively excited with the nutation frequency ... 

A quantum state loses quantum coherence (decoheres) when Sqd wave functions are peaked along classical trajectories. And it decoheres when each trajectory loses quantum coherence with its neighbors. Quantum decoherence is realized when the diagonal term density matrix dominates over the off-diagonal term (fts-... 

Let us call x the vector of Na and Cl concentrations, x the vector of sample means and S the symmetric, positive-definite covariance matrix, i.e., the 2 x 2 matrix with variances on the diagonal and the covariance between Na and Cl concentrations as off-diagonal terms. The equation of the ellipse to be drawn can be written... 

The solution becomes particularly simple and time-invariant if r is chosen in such a way that the cross-terms vanish. Using basic trigonometric identities, we cancel the off-diagonal term... 

First, as the molecule on which the chromophore sits rotates, this projection will change. Second, the magnitude of the transition dipole may depend on bath coordinates, which in analogy with gas-phase spectroscopy is called a non-Condon effect For water, as we will see, this latter dependence is very important [13, 14]. In principle there are off-diagonal terms in the Hamiltonian in this truncated two-state Hilbert space, which depend on the bath coordinates and which lead to vibrational energy relaxation [4]. In practice it is usually too difficult to treat both the spectral diffusion and vibrational relaxation problems at the same time, and so one usually adds the effects of this relaxation phenomenologically, and the lifetime 7j can either be calculated separately or determined from experiment. Within this approach the line shape can be written as [92 94]... 

As has been described, the correlation of the parameter estimates is also of great interest. This correlation is measured by the off-diagonal terms of p(b), completely ignored by the above equations. If an analysis of variance is conducted, one finds that a joint (1 — a) 100% confidence region is defined by... 

The covariance (off-diagonal) terms of Eq. (53) are also needed to calculate approximate intervals for parameters in the hyperbolic models. For a model such as... 

The 100(1 —a) % point of the F statistic with (vltv2) degrees of freedom Mass feed rate to reactor Partial derivative of nonlinear rate model with respect to the ith parameter and evaluated at the nth experimental point Matrix of unity diagonal terms and zero off-diagonal terms... 

After this step one may already identify the paramagnetic and diamagnetic terms as arising from the pp and pn first order orbital corrections to the zeroth order wave function. The contribution is then further approximated by neglecting all off-diagonal terms in the pn-pn part of the Hessian... 

Rotation of the translated factor axes is an eigenvalue-eigenvector problem, the complete discussion of which is beyond the scope of this presentation. It may be shown that there exists a set of rotated factor axes such that the off-diagonal terms of the resulting S matrix are equal to zero (the indicates rotation) that is, in the translated and rotated coordinate system, there are no interaction terms. The relationship between the rotated coordinate system and the translated coordinate system centered at the stationary point is given by... 

---