Dipole approximations, Schrodinger

Here Hq is the molecular Hamiltonian, and fi e(t) is the interaction between the molecule and the laser field in the dipole approximation, where (i is the transition dipole moment of the molecule. Time evolution of the system is determined by the time-dependent Schrodinger equation,... 

However, from a theoretical and also a practical point of view, several teachings may be deduced. In particular, with the retardation, the Pauli approximation of the Dirac theory is no longer in strict agreement with the Schrodinger theory, as it is the case with the dipole approximation, when for example two states pl/2 and p3/2 are considered as unified in a single state p. Such a feature has a nonnegligible incidence. 

Abstract. This chapter concerns the transitions from the state 1S1/2 to the states Pl/2, P3/2 in the dipole approximation (i.e., the fact that the retardation is not taken into account) and the transitions 1S-P with retardation in the Schrodinger theory. 

In the nonrelativistic calculation the Dirac equation is replaced by the Schrodinger one. The formula that is obtained (see [2]), which is convergent, is, if the dipole approximation is applied (i.e. Tj-(k) are replaced by Tj-(O)), the formula used in [4] for the first calculation proposed for the explanation of the Lamb shift. But this last formula is divergent and its use implies that the integration upon k is cut off for a k = kmax. In [23] the value of kmax = ante2 has been proposed and was used in the following calculations of the Lamb shift. 

To deal with photoinduced chemical processes involving several potential surfaces, we need to incorporate the interaction between the molecule and the incident light pulses into the quantum mechanical description. In the semiclassical dipole approximation, the time-dependent Schrodinger equation for two potential energy surfaces Vi and V2 coupled by the laser field reads as ... 

In the following, we explicitly write out the coupled Schrodinger equation for the case of one neutral and one ionized state extension to any number of states is formally straightforward. Resorting to the dipole approximation, we aim to write the time-dependent Schrodinger equation for the nuclear wavefunctions in the form. 

In the previous chapter we considered a rather simple solvent model, treating each solvent molecule as a Langevin-type dipole. Although this model represents the key solvent effects, it is important to examine more realistic models that include explicitly all the solvent atoms. In principle, we should adopt a model where both the solvent and the solute atoms are treated quantum mechanically. Such a model, however, is entirely impractical for studying large molecules in solution. Furthermore, we are interested here in the effect of the solvent on the solute potential surface and not in quantum mechanical effects of the pure solvent. Fortunately, the contributions to the Born-Oppenheimer potential surface that describe the solvent-solvent and solute-solvent interactions can be approximated by some type of analytical potential functions (rather than by the actual solution of the Schrodinger equation for the entire solute-solvent system). For example, the simplest way to describe the potential surface of a collection of water molecules is to represent it as a sum of two-body interactions (the interac-... 

The Born-Oppenheimer approximation allows us to decouple the electronic and nuclear motions of the free molecule of the Hamiltonian Hq. Solving the Schrodinger equation //o l = with respect to the electron coordinates r = r[, O, gives rise to the electronic states (r, R) = (r n(R)), n = 0,..., Ne, of respective energies En (R) as functions of the nuclear coordinates R, with the electronic scalar product defined as (n(R) n (R))r = j dr rj( r. R) T,-(r, R). We assume Ne bound electronic states. The Floquet Hamiltonian of the molecule perturbed by a field (of frequency co, of amplitude 8, and of linear polarization e), in the dipole coupling approximation, and in a coordinate system of origin at the center of mass of the molecule can be written as... 

Here, [in is the transition dipole moment for the transition between V22 and Vn. In the adiabatic approximation (see, e.g.. Ret. [Stenholm 1994]), the coupled bare potentials V22 and V33 can be replaced by uncoupled adiabatic potentials. Using the rotating-wave approximation, the dynamics of the system is then described by the Schrodinger equation... 

There are basically two numerical approaches to obtain approximate solutions to the Schrodinger equation variational and perturbational. In calculations, we usually apply variational methods, while perturbational methods are often applied to estimate some small physical effects. The result is that most concepts (practically all the ones we know of) characterizing the reaction of a molecule to an external field come from the perturbational approach. This leads to such quantities (see Chapter 12) as dipole moment, polarizability, and hyperpolarizability. The computational role of perturbational theories may, in this ctmtext, be seen as being of the second otder. 

Let us first consider spectroscopy. Linear-response theory, in particular the fluctuation dissipation theorem - which relates the absorption of an incident monochromatic field to the correlation function of (e.g. dipole) fluctuations in equilibrium - has changed our perspective on spectroscopy of dense media. It has moved away from a static Schrodinger picture -phrased in terms of transitions between immutable (but usually incomputable) quantum levels - to a dynamic Heisenberg picture, in which the spectral line shape is related by Fourier transform to a correlation function that describes the decay of fluctuations. Of course, any property that cannot be computed in the Schrodinger picture, cannot be computed in the Heisenberg picture either however, correlation functions, unlike wave-functions, have a clear meaning in the classical limit. This makes it much easier to come up with simple (semi) classical interpretations and approximations. 

The above calculation represents an example of the application to an atom of what is called the finite field method. In this method we solve the Schrodinger equation for the system in a given homogeneous (weak) electric field. Say, we are interested in the approximate values of Uqq/ for a molecule. First, we choose a coordinate system, fix the positions of the nuelei in space (the Born-Oppenheimer approximation) and ealeulate the number of electrons in the molecule. These are the data needed for the input into the reliable method we choose to calculate E S). Then, using eqs. (12.38) and (12.24) we calculate the permanent dipole moment, the dipole polarizability, the dipole hyperpolarizabilities, etc. by approximating E(S) by a power series of Sq A. 

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