Zeroth order

Note that h is simply the diagonal matrix of zeroth-order eigenvalues In the following, it will be assumed that the zeroth-order eigenfunction a reasonably good approximation to the exact ground-state wavefiinction (meaning that Xfi , and h and v will be written in the compact representations... 

The off-diagonal elements in this representation of h and v are the zero vector of lengtii (for h) and matrix elements which couple the zeroth-order ground-state eigenfunction members of the set q (for v) ... 

However, if the zeroth-order ground-state energy is well separated from low lying excited states, the diagonal... 

Each temi on the right-hand side of the equation involves matrix produets drat eontain v a speeifie number of times, eitlier explieitly or implieitly (for the temis that involve A i,). Reeognizing that is a zeroth-order quantity, it is straightforward to make the assoeiations... 

The long-range interactions between a pair of molecules are detemiined by electric multipole moments and polarizabilities of the individual molecules. MuJtipoJe moments are measures that describe the non-sphericity of the charge distribution of a molecule. The zeroth-order moment is the total charge of the molecule Q = Yfi- where q- is the charge of particle and the sum is over all electrons and nuclei in tlie molecule. The first-order moment is the dipole moment vector with Cartesian components given by... 

When the overlap between the wavefiinctions of the interacting molecules cannot be neglected, the zeroth-order wavefiinction must be anti-symmetrized with respect to all die electrons. The requirement of anti-synmietrization brmgs with it some difficult problems. If electrons have been assigned to individual molecules... 

In all methods, the first-order interaetion energy is just the differenee between the expeetation value of the system Hamiltonian for the antisyimnetrized produet fiinetion and the zeroth-order energy... 

In the limit of low densities, A2.4.15 shows that the zeroth-order approximation forg(r) has the fonn... 

The zeroth-order solution to the above equations is tire Gotiy-Chapman theory dating from the early part of the 20th cenPiry [20], In this solution, the ionic aPnosphere is ignored, as is the mirror image potential for the ion. Equation A2.4.90 can therefore be ignored and equation A2.4.89 reduces to... 

Figure Bl.18.2. Diffraction figure of a grating If only the zeroth-order beam were collected by the lens, only a bright area would be visible without any stnicture indicating the presence of the grating. If the zeroth- and ifirst-order beams are collected, as indicated in the figure, the grating can be observed, albeit with incomplete object fidelity.

At this point it is worth comparing the different techniques of contrast enliancements discussed so far. They represent spatial filtering teclmiques which mostly affect the zeroth order dark field microscopy, which eliminates the zeroth order, the Schlieren method (not discussed here), which suppresses the zerotii order and one side band and, finally, phase contrast microscopy, where the phase of the zeroth order is shifted by nil and its intensity is attenuated. 

In the following, it shall always be assumed that the zeroth-order solution is known, that is, we have a complete set of eigenvalues and wave functions, labeled by the electronic quantum number n, which satisfy... 

Furthennore, we obtain the equation of motion of the zeroth-order nuclear wave function ... 

Thus, the equation we are going to solve for the zeroth-order nuclear motion is... 

For H2, let us write down the zeroth-order electronic Hamiltonian (in atomic unit) ... 

Figure 1- Representatiori of degenerate states from nonrelativistic components, (a) Degenerate zeroth-order states at (b) Spin-orbit interaction splits 11 state, (c) With full spin-orbit...

Employing simplifications arising from the use of asymptotic forms of the electronic basis functions and the zeroth-order kinetic energy operator, we obtain... 

The appearance of the (normally small) linear term in Vis a consequence of the use of reference, instead of equilibrium configuration]. Because the stretching vibrational displacements are of small amplitude, the series in Eqs. (40) should converge quickly. The zeroth-order Hamiltonian is obtained by neglecting all but the leading terms in these expansions, pjjjf and Vo(p) + 1 /2X) rl2r and has the... 

Vo + V2 and = Vo — 2 (actually, effective operators acting onto functions of p and < )), conesponding to the zeroth-order vibronic functions of the form cos(0 —4>) and sin(0 —(()), respectively. PL-H computed the vibronic spectrum of NH2 by carrying out some additional transformations (they found it to be convenient to take the unperturbed situation to be one in which the bending potential coincided with that of the upper electi onic state, which was supposed to be linear) and simplifications (the potential curve for the lower adiabatic electi onic state was assumed to be of quartic order in p, the vibronic wave functions for the upper electronic state were assumed to be represented by sums and differences of pairs of the basis functions with the same quantum number u and / = A) to keep the problem tiactable by means of simple perturbation... 

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