Configurational eigenvalues

In the previous section we have presented the mathematical expressions that can be evaluated with the Monte Carlo algorithms to be discussed next. The first algorithm is designed to compute an approximate statistical estimate of the matrix element X0 by means of the variational estimate X(°-f0). We write1 = = mt(S) and for nonvanishing m S) define the configurational eigenvalue 2fx(S) by... 

For this to be of practical use, it has to be assumed that the configurational eigenvalue Xx(S) can be computed efficiently, which is the case if the sum over states S in = Es. can be performed explicitly. For discrete states this means that X should be represented by a sparse matrix if the states S form a continuum, Xj(S) can be computed directly if X is diagonal or near-diagonal, that is, involves no or only low-order derivatives in the representation used. The more complicated case of an operator X with arbitrarily nonvanishing off-diagonal elements will be discussed at the end of this section. 

The practical implication is that this information has to be retrieved with sufficient accuracy for small values of p, before the signal disappears in the statistical noise. The projection time p can be kept small by using optimized basis states constructed to reduce the overlap of the linear space spanned by the basis states u(> with the space spanned by the eigenstates beyond the first n of interest. We shall describe, mostly qualitatively, how this can be done by a generalization of a method used for optimization of individual basis states [3,21-23], namely, minimization of variance of the configurational eigenvalue, the local energy in quantum Hamiltonian problems. 

Besides expressions such as Eq. (5.7) one can construct expressions with reduced variance. These involve the configurational eigenvalue of G or Jtf in the same way this was done in our discussion of these single-thread algorithm. 

Since the electronic eigenvalues (the adiabatic PESs) are uniquely defined at each point in configuration space we have m(0) = m(P), and therefore Eq. (32) implies the following commutation relation ... 

Fig. 9. Two-dimensional sketch of the 3N-dimensional configuration space of a protein. Shown are two Cartesian coordinates, xi and X2, as well as two conformational coordinates (ci and C2), which have been derived by principle component analysis of an ensemble ( cloud of dots) generated by a conventional MD simulation, which approximates the configurational space density p in this region of configurational space. The width of the two Gaussians describe the size of the fluctuations along the configurational coordinates and are given by the eigenvalues Ai.

Eigenvalue problems. These are extensions of equilibrium problems in which critical values of certain parameters are to be determined in addition to the corresponding steady-state configurations. The determination of eigenvalues may also arise in propagation problems. Typical chemical engineering problems include those in heat transfer and resonance in which certain boundaiy conditions are prescribed. 

The next step might be to perform a configuration interaction calculation, in order to get a more accurate representation of the excited states. We touched on this for dihydrogen in an earlier chapter. To do this, we take linear combinations of the 10 states given above, and solve a 10 x 10 matrix eigenvalue problem to find the expansion coefficients. The diagonal elements of the Hamiltonian matrix are given above (equation 8.7), and it turns out that there is a simplification. 

To taike advantage of procedures used for configuration interaction calculations, eigenvalues of the symmetrized matrices, H -I- H, are computed. 

In Equation 1.3, the radial function Rnl (r) is defined by the quantum numbers n and l and the spherical harmonics YJ" depend on the quantum numbers l and W . When the spin of the electron is taken into account, the normalized antisymmetric function is written as a Slater determinant. The corresponding eigenvalues depend only on n and l of each single electron, which determine the electronic configuration of the system. 

When the system is made up of identical particles (e.g. electrons in a molecule) the Hamiltonian must be symmetrical with respect to any interchange of the space and spin coordinates of the particles. Thus an interchange operator P that permutes the variables qi and (denoting space and spin coordinates) of particles i and j commutes with the Hamiltonian, [.Pij, H] = 0. Since two successive interchanges of and qj return the particles to the initial configuration, it follows that P = /, and the eigenvalues of are e = 1. The wave functions corresponding to e = 1 are such that... 

In recent years density-functional methods32 have made it possible to obtain orbitals that mimic correlated natural orbitals directly from one-electron eigenvalue equations such as Eq. (1.13a), bypassing the calculation of multi-configurational MP or Cl wavefunctions. These methods are based on a modified Kohn-Sham33 form (Tks) of the one-electron effective Hamiltonian in Eq. (1.13a), differing from the HF operator (1.13b) by inclusion of a correlation potential (as well as other possible modifications of (Fee(av))-... 

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