Kohn variation principle

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

Zhang J Z H and Miller W H 1989 Quantum reactive scattering via the S-matrix version of the Kohn variational principle—differential and integral cross sections for D + Hj —> HD + H J. Chem. Phys. 91 1528... 

On the other hand, employing the Kohn variational principle [67] two lower bound estimates may be obtained [59, 60]... 

Continuum wavefunctions can also be generated by solving the partial differential equation (3.3) directly without first transforming it into a set of ordinary differential equations. One possible scheme is the finite elements method (Askar and Rabitz 1984 Jaquet 1987). Another method, which has been applied for the calculation of multi-dimensional scattering wavefunctions, is the 5-matrix version of the Kohn variational principle (Zhang and Miller 1990). 

In this / -matrix theory, open and closed channels are not distinguished, but the eventual transformation to a A -matrix requires setting the coefficients of exponentially increasing closed-channel functions to zero. Since the channel functions satisfy the unit matrix Wronskian condition, a generalized Kohn variational principle is established [195], as in the complex Kohn theory. In this case the canonical form of the multichannel coefficient matrices is... 

Variational techniques have been used by Mortensen and Gucwa (1969) for H + H2. They based their treatment on the Kohn variational principle, whereby the integral... 

Two other attempts have been made by Crawford. In the first one (1971a) an R-matrix approach was used, following the Wigner-Eisenbud procedure. It was concluded that the R-matrix parameterization did not provide a simple solution. The second one (1971b) employed the Kohn variational principle and a basis of Gaussian functions. Results from a normalized variational S matrix were compared to exact ones and showed some promise. 

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... 

Equation (15) is the key equation of the Kohn variational principle for the -matrix (21). For small problems, when the spectral representation of ft can be obtained, both methods are essentially equivalent. If the linear equations are to be solved iteratively, the present method, Eq. (14), effectively requires to solve half the number of sets of simultaneous linear equations as the basis and xT can chosen real making Eq. (14) real while (15) remains complex. 

Use of Eq. (38) makes it possible to utilize a very convenient expression for the 5-matrix based on the Kohn variational principle (13) ... 

When the basis size is small enough to store the Hamiltonian matrix in the computer core memory, two things can be said with confidence. First, the method presented in Sec. II based on Eq. (1) and Eqs. (2) and (9) (or better to avoid anomalies, (1) and (21)) are very easy to comprehend and implement. This is especially true when the diagonalization of the full Hamiltonian is the key computational step. Second, there are many other approaches, such as the Kohn variational principle (21), the / -matrix theory (35), and the closely related, log-derivative methods (22, 23), that are easy to implement and anomaly free. The methods which use absorbing potentials clearly have a disadvantage relative to the above methods in the sense that they require larger than minimal basis sets and involve non-Hermitian matrices. 

Kohn variational principle for the 5-matrix (21, 8) by a factor of 2. In spite of this saving it is clear from the literature that carrying out the channel-to-channel reactive scattering calculations efficiently for big systems by use of linear equations is still a challenge for the future (23, 8). 

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

The Kohn variational principle (Kohn, 1948) can be formulated in a number of closely related ways (e.g., Nesbet, 1980 Rudge, 1990). Their common feature is that the variational expression involves the Hamiltonian operator H together with an operator describing the scattering either the so-called T operator that is most closely connected to the scattering amplitude or the related reactance (K) or scattering (S) operators. The Kohn expression is so contrived that the portion of the expression that depends on H—a matrix element between two approximate wavefunctions—approaches zero quadratically as those wavefunctions approach... 

In chemistry, excited states and time-dependent phenomena are often of interest. Although the Hohenberg-Kohn variational principle applies only to ground states, various extensions of DFT have been developed over the years which successfully model excitation energies and time-dependent phenomena (for an overview see Refs. [47-49]). 

This theorem means that the ground state electron density, as obtained from the Hohenberg-Kohn variational principle, uniquely determines the ground state properties of the system of interest. The electron density is obtained from the variational principle... 

Zhang JZH, MUler WH (1989) Quantum reactive scattering via the s-matrix version of the kohn variational principle differential and integral cross sections for D -F H2 —> HD -F H. J Chem Phys 91(3) 1528... 

The variational approach received a major boost also when it was realised [79] that the simplest variational method - the Kohn variational principle, which is essentially the Rayleigh-Ritz variational principle for eigenvalues modified to incorporate scattering boundary conditions - is free of anomalous (i.e., spurious, unphysical) singularities if it is formulated with S-matrix type boundary conditions rather than standing wave boundary conditions as had been typically used previously. It is useful first to state the Kohn variational approach for the general inelastic scattering. Thus the variational expression for the S-matrix is... 

Derivation in one dimension. To make the derivation simpler everything will now be formulated in one dimension. In order to derive the S-matrix version of the Hulthen-Kohn variational principle (HKVP) one uses trial functions with incoming and outgoing wave boundary conditions [15, 16, 22] ... 

Kohn variational principle Miller, Zhang, Manolopolous, Wyatt and cowork-ers [69], [71] ... 

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... 

---