Coupled Schrodinger equations

In his classical paper, Renner [7] first explained the physical background of the vibronic coupling in triatomic molecules. He concluded that the splitting of the bending potential curves at small distortions of linearity has to depend on p2A, being thus mostly pronounced in n electronic state. Renner developed the system of two coupled Schrodinger equations and solved it for n states in the harmonic approximation by means of the perturbation theory. 

In the Time Dependent Density Functional Theory (TDDFT) [16], the correlated many-electron problem is mapped into a set of coupled Schrodinger equations for each single electronic wavefunctions (o7 (r, t),j= 1, ), which yields the so-called Kohn-Sham equations (in atomic units)... 

To capture the essence of the Feshbach resonance phenomenon, we will need to understand what happens to the ground vibrational state 4>o(R) of the ground electronic state, also depicted in Figure 1.13, because of the interaction with the continuum of states excited electronic state. The physical process described above can be formulated as a two coupled channels problem where the solution irg(R) in the closed channel (the ground state) depends on the solution ire(R) in the open channel (the excited state) and vice-versa. The coupled Schrodinger equations read... 

The coupled Schrodinger equations can be projected onto the fa fa subspace by Feshbach partitioning, giving an equation for the coefficient function Xd(q) in the component faxdiq) of the total wave function. The effective Hamiltonian in this equation is tn + Vd(q) + Vopt, which contains an optical potential that is nonlocal in the <7-space. This operator is defined by its kernel in the fa - fa subspace,... 

The LS-coupling case is particularly simple because the target spins s, s are each The potential is spin-independent so the spin coupling is independent of the space coupling. Writing the space-direct and space-exchange amplitudes of the coupled Schrodinger equations (7.24) as D and E respectively we have... 

We may therefore use the arguments of section 7.1, replacing (7.1) by (7.114) to obtain the explicitly-antisymmetrised set of P-projected coupled Schrodinger equations analogous to (7.24,7.35). 

In 51 the author presents a new procedure for constructing efficient embedded modified Runge-Kutta methods for the numerical solution of the Schrodinger equation. The methods of the embedded scheme have algebraic orders five and four. Applications of the new pair to the radial Schrodinger equation and to coupled Schrodinger equations show the efficiency of the approach. 

A sometimes overlooked fact is that the Kohn-Sham equation is exact at this stage. It is much easier to solve than the coupled Schrodinger equations that would have to be solved for the original system, since it decouples into single particle equations. The only problem is that we have introduced the exchange-correlation energy, which is an unknown quantity, and which must be approximated. Fortunately, it will turn out to be relatively easy to find reasonably good local approximations for it. 

Stuckelberg did the most elaborate analysis (15). He applied the approximate complex WKB analysis to the fourth-order differential equation obtained from the original second-order coupled Schrodinger equations. In the complex / -plane he took into account the Stokes phenomenon associated with the asymptotic solutions in an approximate way, and finally derived not only the Landau-Zener transition probability p but also the total inelastic transition probability Pn as... 

The most fundamental quantum mechanical model of curve crossing is the linear potential model (in coordinate R), in which the diabatic crossing potentials V R) and V2(R) are linear functions of R and the diabatic coupling V(R) is constant (=A). The basic coupled Schrodinger equations are (29)... 

It is to be noted that the dependence of the electronic basis wavefunctions S/( rJ R ) on the nuclear coordinates R is a parametric one. Since the field-free electronic Hamiltonian is time independent, the basis set is independent of time. Yet, the expansion coefficients T /( R, f) in this basis are time-dependent and have to obey the reduced coupled Schrodinger equations ... 

For each symmetry, these Xka " st satisfy the set of coupled Schrodinger equations... 

These quantities are the main players in the theory of nonadiabaticity. In addition to these, can contain spin-orbit couplings. We do not, however, discuss the relativistic effects any fmther, since it is beyond the scope of this book. Interested readers are referred to Ref. [103] This multi-state coupled Schrodinger equation can be transformed into a compact matrix form as shown below... 

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. 

Theoretically " the pump and probe process is described by three coupled Schrodinger equations for the three nuclear wave packets, one evolving in the ground state, a second one moving on the first excited PES, and the third wave packet representing the motion in the second excited state (the extension to more states, if required, is straightforward, at least formally). 

Substitution of Eq. (3) into the molecular Schrodinger equation leads to a system of coupled equations in a coupled multistate electronic manifold... 

The important outcome from this transformation is that now the non-adiabatic coupling term t is incorporated in the Schrodinger equation in the same way as a vector potential due to an external magnetic field. In other words, X behaves like a vector potential and therefore is expected to fulfill an equation of the kind [111a]... 

Inserting the separation ansatz, i.e., U , results in two nonlinearly coupled single particle Schrodinger equations, the so-called time dependent self-consistent field (TDSCF) equations ... 

Various kinds of mixed quantum-classical models have been introduced in the literature. We will concentrate on the so-called quantum-classical molecular dynamics (QCMD) model, which consists of a Schrodinger equation coupled to classical Newtonian equations (cf. Sec. 2). 

Beeause there are no terms in this equation that couple motion in the x and y directions (e.g., no terms of the form x yb or 3/3x 3/3y or x3/3y), separation of variables can be used to write / as a product /(x,y)=A(x)B(y). Substitution of this form into the Schrodinger equation, followed by collecting together all x-dependent and all y-dependent terms, gives ... 

RPA, and CPHF. Time-dependent Hartree-Fock (TDFIF) is the Flartree-Fock approximation for the time-dependent Schrodinger equation. CPFIF stands for coupled perturbed Flartree-Fock. The random-phase approximation (RPA) is also an equivalent formulation. There have also been time-dependent MCSCF formulations using the time-dependent gauge invariant approach (TDGI) that is equivalent to multiconfiguration RPA. All of the time-dependent methods go to the static calculation results in the v = 0 limit. 

The Bom-Oppenheimer approximation is usually very good. For the hydrogen molecule the error is of the order of 10 ", and for systems with heavier nuclei, the approximation becomes better. As we shall see later, it is only possible in a few cases to solve the electronic part of the Schrodinger equation to an accuracy of 10 ", i.e. neglect of the nuclear-electron coupling is usually only a minor approximation compared with other errors. 

Application of the time-dependent Schrodinger equation gives equation (26). This 3nelds two coupled equations which, solved for... 

The field- and time-dependent cluster operator is defined as T t, ) = nd HF) is the SCF wavefunction of the unperturbed molecule. By keeping the Hartree-Fock reference fixed in the presence of the external perturbation, a two step approach, which would introduce into the coupled cluster wavefunction an artificial pole structure form the response of the Hartree Fock orbitals, is circumvented. The quasienergy W and the time-dependent coupled cluster equations are determined by projecting the time-dependent Schrodinger equation onto the Hartree-Fock reference and onto the bra states (HF f[[exp(—T) ... 

---