Stark Hamiltonian

The perturbation theory is the convenient starting point for the determination of the polarizability from the Schrodinger equation, restricted to its electronic part and the electric dipole interaction regime. The Stark Hamiltonian —p. describes the dipolar interaction between the electric field and the molecule represented by its... 

The variational theorem which has been initially proved in 1907 (24), before the birthday of the Quantum Mechanics, has given rise to a method widely employed in Qnantnm calculations. The finite-field method, developed by Cohen andRoothan (25), is coimected to this method. The Stark Hamiltonian —fi.S explicitly appears in the Fock monoelectronic operator. The polarizability is derived from the second derivative of the energy with respect to the electric field. The finite-field method has been developed at the SCF and Cl levels but the difficulty of such a method is the well known loss in the numerical precision in the limit of small or strong fields. The latter case poses several interconnected problems in the calculation of polarizability at a given order, n ... 

Fig. 6.13 Part of the excitation spectrum oftheNaw = 29 Stark levels from the 3d state in an electrostatic field of 20.5 V/cm corresponding to nj values of n — 3, n — 4, and n — 5. The energy splitting between m = 0 (highest energy fine in the doublets) and m = 1 states is of the order of 180 MHz. The arrows indicate theoretical positions of energy levels obtained by a numerical diagonalization of the Stark Hamiltonian (from ref. 28).

Before we discuss this problem we notice that the Hamiltonian, Eq. (D.3), is bounded from below in contrast to the Stark Hamiltonian treated earlier, which however is a valid potential for Weyl s theory to hold. Without giving the full details, we first define D(H), the domain of H, as... 

I.W. Herbst, J.S. Howland, The Stark Ladder and Other One-Dimensional External Field Problems, Commun. Math. Phys. 80 (1981) 23 J.S. Howland, Complex Scaling of ac Stark Hamiltonians, J. Math. Phys. 24 (1982) 1240. 

Here a- corresponds to a dispersive correction to the Stark quenching line shape, a-2 represents a small vertex correction (of order a3) to Stark Hamiltonian. Numerical values for these quantities are —1.85 x 10-9 and 5.13 x 10-8, respectively. The details of calculations demand a separate examination and will be discussed in our forthcoming work. 

J.S. Howland, Complex scaling of ac Stark Hamiltonians,. Math. Phys. 24 (1983) 1240-1244. 

This term was encountered previously in the Hamiltonian for a single electron (3.101) (E a 7F,) is essentially equivalent to a magnetic field, which interacts with the spin magnetic moment. The term is, however, usually negligible in laboratory experiments. Summarising, the electric field, or Stark, Hamiltonian may be written... 

We dealt with the effects of applied static fields on the electronic Hamiltonian in section 3.7. In this section we first give the relevant terms for the nuclear Zeeman and Stark Hamiltonians and then perform the same coordinate transformations that proved to be convenient for the field-free molecular Hamiltonian. 

We could now choose explicit vector and scalar potential functions as we did in section 3.7. However, when we come to perform the various coordinate transformations outlined in the previous section, it is more convenient to treat the Zeeman and Stark Hamiltonians for the molecule as a whole. Accordingly we combine the expressions from section 3.7 with equations (3.272) and (3.273) to give... 

The Stark Hamiltonian is more straightforward. We use the scalar potentials... 

The remainder of this section is devoted to a simplified two-level treatment of the Zeeman and Stark effects in the presence of zero-field Stark effect and field-dependent interactions between basis functions 1M) and 2M). In the presence of a static field directed along the space Z-axis, Mj remains a good quantum number. The Zeeman and Stark Hamiltonians involve the interaction between a magnetic field or electric dipole, /r, in the molecule-fixed axis system and the space-fixed magnetic or electric field, F, parallel to the laboratory direction K. The interaction can be expressed in terms of direction cosines... 

The ab initio SCF cluster wavefunction has been used to investigate the bonding of CO and CN- on Cu,0 (5,4,1), (5 surface layer, 4 second layer and 1 bottom layer atoms), and to calculate their field dependent vibrational frequency shifts in fields up to 5.2 x 107 V/cm(46). A schematic view of the Cu10 (5,4,l)CO cluster is shown in Figure 8. In order to assess the significance of Lambert s proposal, that the linear Stark effect is the dominant factor in the field dependent frequency shift, the effect of the field was calculated by three methods. One is by a fully variational approach (i.e., the adsorbate is allowed to relax under the influence of the applied field) in which the Hamiltonian for the cluster in a uniform electric field, F, is given by... 

We first consider the AN regime of a two-level system coupled to a thermal bath. We will use off-resonant dynamic modulations, resulting in AC-Stark shifts (Figure 4.5(a)). The Hamiltonians then assume the following form ... 

It should be remarked that Ex. 3 is analogous to the case of the Hamiltonian of the Stark effect. In these cases the diffusion of the eigenvalue into continuous spectrum takes place, and p.m. is not valid in the sense hitherto considered. Nevertheless p.m. is applied to the Stark effect with successful results. In order to justify the application of p.m. to such problems, more profound study is necessary than that given here. We shall discuss the subject in the next chapter. 

Example 7. Stark effect of the hydrogen-like atom. The Hamiltonian is given by... 

Even at high n s one needs to follow the system for many orbital periods if one is to mimic the experimental results. The difficulty is compounded if one measures the time in units of periods of the core motion. This suggests that the time evolution be characterized using the stationary states of the Hamiltonian rather than propagating the initial state. We have done so, but our experience is that in the presence of DC fields of experimental magnitude (which means that Stark manifolds of adjacent n values overlap), and certainly so in the presence of other ions that break the cylindrical symmetry and hence mix the m/ values, the size of the basis required for convergence is near the limit of current computers. In our experience, truncating the quan-... 

Using the zero field n(m states we calculate the matrix elements (/i m Ez n m ) of the Stark perturbation to the zero field Hamiltonian. Writing the matrix element in spherical coordinates and choosing the z axis as the axis of quantization,... 

In this regime, where the levels are discrete, it is possible to calculate the intensities of the transitions by matrix diagonalization, just as the energies are calculated. It is simply a matter of computing the eigenvectors of the Hamiltonian as well as its eigenvalues. For example, to calculate the intensities in the spectra shown in Fig. 8.12 we calculate the rcp amplitude in each of the Stark states and multiply it by the matrix element connecting the 3s state to the n p state,... 

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