Variational principles scattering theory

Manolopoulos D E, Dmello M and Wyatt R E 1989 Quantum reactive scattering via the log derivative version of the Kohn variational principle—general theory for bimolecular chemical reactions J. Chem. Phys. 91 6096... 

Miller W H 1994 S-matrix version of the Kohn variational principle for quantum scattering theory of... 

This part introduces variational principles relevant to the quantum mechanics of bound stationary states. Chapter 4 covers well-known variational theory that underlies modern computational methodology for electronic states of atoms and molecules. Extension to condensed matter is deferred until Part III, since continuum theory is part of the formal basis of the multiple scattering theory that has been developed for applications in this subfield. Chapter 5 develops the variational theory that underlies independent-electron models, now widely used to transcend the practical limitations of direct variational methods for large systems. This is extended in Chapter 6 to time-dependent variational theory in the context of independent-electron models, including linear-response theory and its relationship to excitation energies. 

Nesbet, R.K. (1992). Variational principles for full-potential multiple scattering theory, Mat. Res. Symp. Proc. 253, 153-158. 

The KR variational principle determines a wave function with correct boundary conditions at a specified energy, the typical conditions of scattering theory. Energy values are deduced from consistency conditions. 

In bound-state calculations, the Rayleigh-Ritz or Schrodinger variational principle provides both an upper bound to an exact energy and a stationary property that determines free parameters in the wave function. In scattering theory, the energy is specified in advance. Variational principles are used to determine the wave function but do not generally provide variational bounds. A variational functional is made stationary by choice of variational parameters, but the sign of the residual error is not determined. Because there is no well-defined bounded quantity, there is no simple absolute standard of comparison between different variational trial functions. The present discussion will develop a stationary estimate of the multichannel A -matrix. Because this matrix is real and symmetric for open channels, it provides the most... 

McCurdy, C.W., Rescigno, T.N. and Schneider, B.I. (1987). Interrelation between variational principles for scattering amplitudes and generalized R-matrix theory, Phys. Rev. A 36, 2061-2066. 

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. 

Miller W H 1994 S-matrix version of the Kohn variational principle for quantum scattering theory of chemical reactions Adv. Mol. Vibrations and Collision Dynamics vol 2A, ed J M Bowman (Greenwich, CT JAI Press) pp 1-32... 

R. G. Newton, Scattering Theory of Particles and Waves, 2nd ed., Springer-Verlag, New York, 1982 (Section 11.3, variational principles Section 11.2, resonances as poles of the S matrix). 

Theory. Usually we do not solve the fundamental equations directly. We use a theory, for example, Har-tree-Fock theory [3], Moller-Plesset perturbation theory [4], coupled-cluster theory [5], Kohn s [6, 7], Newton s [8], or Schlessinger s [9] variational principle for scattering amplitudes, the quasiclassical trajectory method [10], the trajectory surface hopping method [11], classical S-matrix theory [12], the close-coupling approximation... 

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. 

Summary. Rate constants of chemical reactions can be calculated directly from dynamical simulations. Employing flux correlation functions, no scattering calculations are required. These calculations provide a rigorous quantum description of the reaction process based on first principles. In addition, flux correlation functions are the conceptual basis of important approximate theories. Changing from quantum to classical mechanics and employing a short time approximation, one can derive transition state theory and variational transition state theory. This article reviews the theory of flux correlation functions and discusses their relation to transition state theory. Basic concepts which facilitate the calculation and interpretation of accurate rate constants are introduced and efficient methods for the description of larger systems are described. Applications are presented for several systems highlighting different aspects of reaction rate calculations. For these examples, different types of approximations are described and discussed. 

These rules were originally suggested so as to impose quantum detailed balance in the results obtained with the quantum-classical model for energy transfer in molecular collisions (for a review see [57]). Later it was found that the initial momentum could be obtained variationally [58], and that the result for the optimal variationally determined momentum was very close to the above simple arithmetic mean velocity. The symmetrization principle is, therefore, theoretically well founded and is well known in gas-phase collisions and has been used also in surface scattering [59, 101]. Thus for the forced oscillator problem this approach is very accurate [58] (see also Table 8.1). By introducing the normal mode oscillators in the theory of gas-surface collisions we are converting the problem to that of many forced independent oscillators. Hence the approach, which is accurate for a single forced oscillator, will also be accurate when used in this context. 

---