Green wavefunctions

Fig. 8.1 Three different control scenarios for the population transfer from the left to the right minimum via an excited state. Panel a shows a pump-dump sequence, panel b a pump-dump sequence including chirp, and panel c a freely optimized sequence. The top tmv shows snapshots at 0 (black), 44 (blue), 70 (red), and 140 fs (green) from the controlled wavepacket dynamics. The initial wavepacket is shown in black, the excitation process in blue, the freely evolving wavepacket in red, and the final wavepacket in green. In the middle row the temporal laser fields are depicted and below their time versus frequency representation. The population transfer increases from left to right and can be followed in the amplitude of the green wavefunction. In c close to 100 % of the initial (black) wavefunction has been transferred to the target minimum. Nothing remtiins on the reactant side or in the excited state...

Inglesfield used a Green s function technique to write the one-electron wavefunctions as... 

Notice that Gq is the Fourier time transform of the one-electron energy-dependent Green s function Go(r,r ). Expansion of a particular wavefunction, in terms of site atomic orbitals, as... 

All the methods used in this study are response methods. They deserihe the response of an ohservahle sueh as an eleetrie dipole moment /I or quadrupole moment to an external or internal perturhation, e.g., an eleetrie field or field gradient. Response funetions originated in various diseiplines in physies. In statistieal physies, they were used as time-eorrelation functions in the form of Green s functions [44,45]. Linderherg and Ohrn first showed the usefulness of this idea for quantum chemistry [46]. Since then response functions have been derived for many types of electronic wavefunctions. Four of these methods are employed here. 

By writing down the kinetic energy term T = — (ftV2m)V explicitly, and using Green s theorem, the transition matrix element is finally converted into a surface integral similar to Bardeen s, in terms of modified wavefunctions ... 

The wavefunctions in Eq. (2.34) are different from the wavefunctions of the free tip and free sample. The effect of the distortion potential (V = Us — Uso and V = Us - Uso), can be evaluated through time-independent perturbation. In the following, we present an approximate method based on the Green s function of the vacuum (see Appendix B). To first order, the distorted wavefunction i)i is related to the undistorted one, i]jo, by... 

Using the properties of the Green s function (see Appendix B), the evaluation of the effect of distortion to transmission matrix elements can be greatly simplified. First, because of the continuity of the wavefunction and its derivative across the separation surface, only the multiplier of the wavefunctions at the separation surface is relevant. Second, in the first-order approximation, the effect of the distortion potential is additive [see Eq. (2.39)]. Thus, to evaluate the multiplier, a simpler undistorted Hamiltonian might be used instead of the accurate one. For example, the Green s function and the wavefunction of the vacuum can be used to evaluate the distortion multiplier. 

Fig. 2.10. Different approximate methods for the square harrier problem. (Parameters used W= 2 A = 4 eV Utf= 16 eV.) The original Bardeen theory breaks down when the barrier top comes close to the energy level. The modified Bardeen tunneling theory is accurate with separation surfaces either centered L = lV/2) or off-centered L = VV73). By approximating the distortion of wavefunctions using Green s functions, the error in the entire region is only a few percent.

We choose the bisection plane in the barrier as the separation surface. The correction for tunneling current can be obtained from the correction for the wavefunction on the bisection plane. Using the Green s function method, following Eq. (2.42), the correction factor for the wavefunction at z = W/2 is... 

Therefore, the. r-wave tip wavefunction is equal to the Green s function up to a constant, with the center of the apex atom taken as ro,... 

Using Green s theorem, it can be converted into a volume integral over fir, the tip side from the separation surface. Noticing that the sample wavefunction ip satisfies Schrodinger s equation, Eq. (3.2), in fir, and that the Green s function satisfies Eq. (3.8), we obtain immediately... 

The tunneling matrix elements from the rest of the nine tip wavefunctions can be derived using the relation between the tip and Green s functions established in the previous section. For example, for the d), tip state. 

The physical meaning of this Green s function is as follows. If in the entire space, the potential equals the potential of the vacuum, then the solution of Eq. (B.6) is identically zero in the entire space. In order to have a meaningful physical situation, some place in the space must have a certain potential well and nonvanishing wavefunction. The Green s function Eq. (B.IO) then describes the influence of the potential well and the wavefunction in the nonvacuum region on the wavefunction in the vacuum region. 

Tip treatment 281—293, 301 annealing 286 annealing with a field 288 atomic metallic ion emission 289 controlled collision 293 controlled deposition 288 field evaporation 287 for scanning tunneling spectroscopy 301 high-field treatment 291 Tip wavefunctions 76—81 explicit forms 77 Green s functions, and 78 Tip-state characterization 306, 308 ex situ 306 in situ 308... 

From the zero order Green s function the evaluation of expressions is straightforward. The exponential decay of surface and tip wavefunctions allows the conversion of the fourfold volume integration contained in the trace of Eq. (2) into surface integrals. An evaluation of the trace as well as the energy integral then leads to a modification of the standard Bardeen expression ... 

Figure 5 A steering dominated dissociation event. The purple shows a surface of constant wavepacket probability approaching a PES having very attractive regions into which the wavefunction is drawn (lower panel). The green surface shows die zero of potential.

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