Kohn matrix

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. 

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... 

We may again chose a unitary transfonnation which makes tlie matrix of the Lagrange multiplier diagonal, producing a set of canonical Kohn-Sham (KS) orbitals. The resulting pseudo-eigenvalue equations are known as the Kohn-Sham equations. 

Theoretical calculations were performed with the linear muffin tin orbital (LMTO) method and the local density approximation for exchange and correlation. This method was used in combination with supercell models containing up to 16 atoms to calculate the DOS. The LMTO calculations are run self consistently and the DOS obtained are combined with the matrix elements for the transitions from initial to final states as described in detail elsewhere (Botton et al., 1996a) according to the method described by Vvedensky (1992). A comparison is also made between spectra calculated for some of the B2 compounds using the Korringa-Kohn-Rostoker (KKR) method. 

Up to this point the derivation has exactly paralleled the Hartree-Fock case, which only differs in using the corresponding Fock matrix, F rather than the Kohn-Sham counterpart, Fks. By expanding fKS into its components, the individual elements of the Kohn-Sham matrix become... 

We now need to discuss how these contributions that are required to construct the Kohn-Sham matrix are determined. The fust two terms in the parenthesis of equation (7-12) describe the electronic kinetic energy and the electron-nuclear interaction, both of which depend on the coordinate of only one electron. They are often combined into a single integral, i. e ... 

What we have not discussed so far is how the contribution of the final components of the Kohn-Sham matrix in equation (7-12), i. e., the exchange-correlation part, can be computed. What we need to solve are terms such as... 

This step is similar to what we have done in equation (7-7) where we obtained the matrix representation of the Kohn-Sham operator. If we insert expression (7-14) for the charge density in terms of the LCAO functions and make use of the density matrix P defined in equation (7-15), we arrive at... 

We can obtain a more direct comparison of the ab initio and Hiickel quantities in terms of the valence pi block of the NAO Fock matrix (or Kohn-Sham matrix) F(NA0), which provides the direct ab initio counterpart of (3.155) ... 

Diagonalization of the matrix formed by the multipliers fiy yields the Kohn-Sham orbitals and their eigen-energies ... 

The reaction field effects are easily incorporated as an additional term in the Kohn-Sham matrix, given by ... 

Beginning way back in the 20s, Thomas and Fermi had put forward a theory using just the diagonal element of the first-order density matrix, the electron density itself. This so-called statistical theory totally failed for chemistry because it could not account for the existence of molecules. Nevertheless, in 1968, after years of doing wonders with various free-electron-like descriptions of molecular electron distributions, the physicist John Platt wrote [2] We must find an equation for, or a way of computing directly, total electron density. [This was very soon after Hohenberg and Kohn, but Platt certainly was not aware of HK by that time he had left physics.]... 

One uses a simple CG model of the linear responses (n= 1) of a molecule in a uniform electric field E in order to illustrate the physical meaning of the screened electric field and of the bare and screened polarizabilities. The screened nonlocal CG polarizability is analogous to the exact screened Kohn-Sham response function x (Equation 24.74). Similarly, the bare CG polarizability can be deduced from the nonlocal polarizability kernel xi (Equation 24.4). In DFT, xi and Xs are related to each other through another potential response function (PRF) (Equation 24.36). The latter is represented by a dielectric matrix in the CG model. 

On matrix form the non-unitary transformations (27) and (30) of the previous section are easily extended to the complete Hamiltonian and have therefore allowed relativistic and non-relativistic spin-free calculations of spectroscopic constants and first-order properties at the four-component level (see, for instance. Refs. [45 7]). In this section, we consider the elimination of spin-orbit interaction in four-component calculations of second-order electric and magnetic properties. Formulas are restricted to the Hartree-Fock [48] or Kohn-Sham [49] level of theory, but are straightforwardly generalized. 

But there is a more basic difficulty in the Hohenberg-Kohn formulation [19-21], which has to do with the fact that the functional iV-representability condition on the energy is not properly incorporated. This condition arises when the many-body problem is presented in terms of the reduced second-order density matrix in that case it takes the form of the JV-representability problem for the reduced 2-matrix [19, 22-24] (a problem that has not yet been solved). When this condition is not met, an energy functional is not in one-to-one correspondence with either the Schrodinger equation or its equivalent variational principle therefore, it can lead to energy values lower than the experimental ones. 

In the Hohenberg-Kohn formulation, the problem of the functional iV-representability has not been adequately treated, as it has been assumed that the 2-matrix IV-representability condition in density matrix theory only implies an N-representability condition on the one-particle density [21]. Because the latter can be trivially imposed [26, 27], the real problem has been effectively avoided. 

The variational procedure in Eq. (100) is in the spirit of the Kohn-Sham ansatz. Since satisfies the (g, K) conditions, it is A-representable. In general, Pij ig corresponds to many different A-electron ensembles and one of them,, corresponds to the ground state of interest. However, for computational expediency in computing the energy, a Slater determinantal density matrix,... 

Z is the nuclear charge, R-r is the distance between the nucleus and the electron, P is the density matrix (equation 16) and (qv Zo) are two-electron integrals (equation 17). f is an exchange/correlation functional, which depends on the electron density and perhaps as well the gradient of the density. Minimizing E with respect to the unknown orbital coefficients yields a set of matrix equations, the Kohn-Sham equations , analogous to the Roothaan-Hall equations (equation 11). 

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