Kohn variational theory

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 

The asymptotic form of radial open-channel orbitals fps(r) is given by Eq. (8.2). Functions of this form can be represented as linear combinations of two independent continuum basis functions for each open channel. These basis functions must be regular at the coordinate origin, but have the asymptotic forms 

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A completely difierent approach to scattering involves writing down an expression that can be used to obtain S directly from the wavefunction, and which is stationary with respect to small errors in die waveftmction. In this case one can obtain the scattering matrix element by variational theory. A recent review of this topic has been given by Miller [32]. There are many different expressions that give S as a ftmctional of the wavefunction and, therefore, there are many different variational theories. This section describes the Kohn variational theory, which has proven particularly useftil in many applications in chemical reaction dynamics. To keep the derivation as simple as possible, we restrict our consideration to potentials of die type plotted in figure A3.11.1(c) where the waveftmcfton vanishes in the limit of v -oo, and where the Smatrix is a scalar property so we can drop the matrix notation. 

Kohn variational theory 8.2.3 The complex Kohn method... 

Now let us return to the Kohn variational theory that was introduced in section A3,11.2.8. Here we demonstrate how equation (A3,11,46) may be evaluated using basis set expansions and linear algebra. This discussion will be restricted to scattering in one dimension, but generalization to multidimensional problems is very similar. 

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

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... 

The Hohenberg-Kohn theory of /V-clcctron ground states is based on consideration of the spin-indexed density function. Much earlier in the development of quantum mechanics, Thomas-Fermi theory [402, 108] (TFT) was formulated as exactly such a density-dependent formalism, justified as a semiclassical statistical theory [231, 232], Since Hohenberg-Kohn theory establishes the existence of an exact universal functional Fs [p] for ground states, it apparently implies the existence of an exact ground-state Thomas-Fermi theory. The variational theory that might support such a conclusion is considered here. 

This argument shows that the locality hypothesis fails for more than two electrons because the assumed Frechet derivative must be generalized to a Gateaux derivative, equivalent in the context of OEL equations to a linear operator that acts on orbital wave functions. The conclusion is that the use by Kohn and Sham of Schrodinger s operator t is variationally correct, but no equivalent Thomas-Fermi theory exists for more than two electrons. Empirical evidence (atomic shell structure, chemical binding) supports the Kohn-Sham choice of the nonlocal kinetic energy operator, in comparison with Thomas-Fermi theory [288]. A further implication is that if an explicit approximate local density functional Exc is postulated, as in the local-density approximation (LDA) [205], the resulting Kohn-Sham theory is variation-ally correct. Typically, for Exc = f exc(p)p d3r, the density functional derivative is a Frechet derivative, the local potential function vxc = exc + p dexc/dp. 

For direct Af-electron variational methods, the computational effort increases so rapidly with increasing N that alternative simplified methods must be used for calculations of the electronic structure of large molecules and solids. Especially for calculations of the electronic energy levels of solids (energy-band structure), the methodology of choice is that of independent-electron models, usually in the framework of density functional theory [189, 321, 90], When restricted to local potentials, as in the local-density approximation (LDA), this is a valid variational theory for any A-electron system. It can readily be applied to heavy atoms by relativistic or semirelativistic modification of the kinetic energy operator in the orbital Kohn-Sham equations [229, 384],... 

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... 

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... 

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. 

Figure 1 A family tree of quantum chemistry DFT, density functional theory QMC, quantum Monte Carlo RRV, Rayleigh-Ritz variational theory X-a, X-alpha method KS, Kohn-Sham approach LDA, BP, B3LYP, density functional approximations VQMC, variational QMC DQMC, diffusion QMC FNQMC, fixed-node QMC PIQMC, path integral QMC EQMC, exact QMC HF, Hartree-Fock EC, explicitly correlated functions P, perturbational MP2, MP4, Maller-Plesset perturbational Cl, configuration interaction MRCI, multireference Cl FCI, full Cl CC, CCSD(T), coupled-cluster approaches. Other acronyms are defined in the text.

We use in Chapter 2 the Kohn variational 5—matrix formalism to probe the sensitivity of H-fH2 cross sections to small changes in the PES, to help resolve a discrepancy between experiment and theory over a possible H3 collision complex. We find the reactive scattering calculations to be very robust, and thus trust their predictions. 

Section 11.2 presents the non-variational theory. In Sect. 11.3 Kohn-Sham-like equations are obtained through adiabatic connection. Density scaling is applied to obtain a generalized Kohn-Sham scheme in Sect. 11.4. The optimized potential and the KLI methods are generalized in Sect. 11.5. The last section is devoted to illustrative examples and discussion. 

Applying the variational principle to the energy given by Eq. 1, Kohn and Sham reformulated the density functional theory by deriving a set of one-electron Hartree-like equations leading to the Kohn-Sham orbitals v().(r) involved in the calculation of p(r)15. The Kohn-Sham (KS) equations are written as follows ... 

Theoretical considerations leading to a density functional theory (DFT) formulation of the reaction field (RF) approach to solvent effects are discussed. The first model is based upon isolelectronic processes that take place at the nucleus of the host system. The energy variations are derived from the nuclear transition state (ZTS) model. The solvation energy is expressed in terms of the electrostatic potential at the nucleus of a pseudo atom having a fractional nuclear charge. This procedure avoids the introduction of arbitrary ionic radii in the calculation of insertion energy, since all integrations involved are performed over [O.ooJ The quality of the approximations made are discussed within the frame of the Kohn-Sham formulation of density functional theory. 

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