Schwinger variational principle

Variational principles Newton variational principle, Schwinger variational principle, amplitude density method Truhlar, Kouri and coworkers [69], [70] ... 

Kouri D J, Huang Y, Zhu W and Hoffman D K 1994 Variational principles for the time-independent wave-packet-Schrddinger and wave-packet-Lippmann-Schwinger equations J. Chem. Phys. 100... 

Lippmann BA, Schwinger J (1950) Variation principle for scattering processes I. Phys Rev... 

Variational Principles for the Time-Independent Wave-Packet-Schrodinger and Wave-Packet-Lippmann-Schwinger Equations. 

The precise connection with finite dimensional matrix formulas obtains simply from Lowdin s inner and outer projections [21, 22], see more below, or equivalently from the corresponding Hylleraas-Lippmann-Schwinger-type variational principles [24, 25]. For instance, if we restrict our operator representations to an n-dimensional linear manifold (orthonormal for simplicity) defined by... 

In analogy to the Schwinger variational principle, consider the product functional... 

Lucchese, R.R., Takatsuka, K. and McKoy, V. (1986). Applications of the Schwinger variational principle to electron-molecule collisions and molecular photoionization, Phys. Rep. 131, 147-221. 

Takatsuka, K., Lucchese, R.R. and McKoy, V. (1981). Relationship between the Schwinger and Kohn-type variational principles in scattering theory, Phys. Rev. A 24, 1812-1816. 

The use of a separable potential of the form of Equation 8 in Equation 6 to obtain solutions of the form of Equation 9 can be shown to be equivalent to using the functions ct (r) in the Schwinger variational principle for collisions (13). At this stage the functions ai(r) can be chosen to be entirely discrete basis functions such as Cartesian Gaussian (lA) or spherical Gaussian (15) functions. We note that with discrete basis functions alone the approximate solution satisfies the scattering boundary condition. Such basis... 

As an example of the use of this functional, we now use it to obtain the Schwinger variational principle for the static-exchange potential. The total wavefunction is taken as... 

We use an iterative procedure, based on the Schwinger variational principle, to solve the Lippmann-Schwinger equation associated with Eq. (3.63) [250]. The procedure begins by approximating the static-exchange potential of the relaxed ion core by a separable form... 

ABSTRACT. We explore the factors responsible for the rapid convergence of the Schwinger and Newton variational principles in scattering theory. We find that, contrary to conventional wisdom, these variational methods yield high accuracy not because the error associated with the computed quantity is second oide in the error in the wavefunction, but because variational methods find wavefiinctions that are far more accurate in relevent regions of the potential, compared to nonvariational methods. 

Variational methods are at present used extensively in the study of inelastic and reactive scattering involving atoms and diatomic molecules[l-5]. Three of the most commonly used variational methods are due to Kohn (the KVP)[6], Schwinger (the SVP)[7] and Newton (the NVP)[8]. In the applications of these methods, the wavefunction is typically expanded in a set of basis functions, parametrized by the expansion coefficients. These linear variational parameters are then determined so as to render the variational functional stationary. Unlike the variational methods in bound state calculations, the variational principles of scattering theory do not provide an upper or lower bound to the quantity of interest, except in certain special cases.[9] Neverthless, variational methods are useful because, the minimum basis size with which an acceptable level of accuracy can be achieved using a variational method is often much smaller than those required if nonvariational methods are used. The reason for this is generally explained by showing (as... 

The three variational principles in common use in scattering theory are due to Kohn [9], Schwinger [11] and Newton [12]. Two of these variational principles, those due to Kohn and Newton, have been successfully developed and applied to reactive scattering problems in recent years there is the S-matrix Kohn method of Zhang, Chu, and Miller, the related log derivative Kohn method of Manolopoulos, D Mello, and Wyatt and the L - Amplitude Density Generalized Newton Variational Principle (L -AD GNVP) method of Schwenke, Kouri, and Truhlar. 

The Kohn variational principle is perhaps the simplest of the three scattering variational principles mentioned above [9]. In particular, it requires that one calculate matrix elements only over the total Hamiltonian H of the system, and not over the Green s function Gq E) of some reference Hamiltonian Hq. While matrix elements of H between energy-independent basis functions are also energy-independent, all matrix elements of G E) have to be re-evaluated at each new scattering energy E. The Kohn variational principle is therefore somewhat easier to apply than the Schwinger and Newton... 

We are now in a position to perform a generalized variation of the action integral for an open system to demonstrate that Schwinger s principle of stationary action can be extended in such a manner as to provide a quantum definition of an atom in a molecule. We shall be considering the change in the atomic action integral 2] of eqn (8.111) ensuing from variations... 

Properties of a proper open system 12 are defined by Heisenberg s equation of motion obtained from the variation of the state vector within the system and on its boundaries [1,4], in the manner determined by Schwinger s principle of stationary action [5]. For a stationary state, the equation of motion for an observable G is given in Equation (2),... 

We are interested in molecules in stationary states and in this case Schwinger s principle takes on a particularly simple form. In this instance, it yields Schrodinger s equation for a stationary state, equation 11, and equation 14 for the variation in Schrodinger s energy functional G[T ],... 

---