Dimension of wavefunctions

Finally, some remarks will be made concerning the dimension of wavefunctions. The bound-state orbitals are subject to the orthonormality relation 

The continuum function is normalized with the -function, see equ. (7.28f). Since the -function also has a dimension, 

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Even worse is the confusion regarding the wavefunction itself. The Born interpretation of quantum mechanics tells us that i/f (r)i/f(r) dr represents the probability of finding the particle with spatial coordinates r, described by the wavefunction V (r), in volume element dr. Probabilities are real numbers, and so the dimensions of i/f(r) must be of (length)" /. In the atomic system of units, we take the unit of wavefunction to be... 

Tel. 203-432-6288, fax 203-432-6144, e-mail dan rani.chem.yale.edu Plots of wavefunctions in three dimensions from semiempirical and popular ab initio basis sets. Silicon Graphics, Sun, VAX, Cray, and others. 

Figure 8.2 One-dimensional example of using supercells to reduce (t-point sampling. With a minimal unit cell of dimension a, the states shown on the top line require 3k-points to be sampled. Doubling the unit cell allows the first two states to be generated at (t = 0 with a unit cell dimension of 4a all three states are contained within the local cell wavefunctions.

The reliability of SOCI, coupled with its essentially a priori selection of the Cl space, makes it an attractive alternative to full Cl. Unfortunately, for reasonable active spaces the dimension of the SOCI grows very rapidly with system size and thus the method is applicable only to small molecules. Nevertheless, for quite a few molecules it is possible to use the SOCI method in conjunction with large one-particle basis sets and hence to obtain wavefunctions very close to the exact nonrelativistic Born-Oppenheimer limit. Below, we will attempt to give the reader a sense of the types of problems to which the SOCI method has been applied. Bauschlicher, Langhoff, and Taylor have already given15 an excellent review of the related CASSCF MR-CISD method, and we refer the reader to their article for a discussion of additional important studies. 

It is useful to compare these approaches when applied to a wavefunction expansion that results in a sparse density matrix. For example with a PPMC expansion, each d , with about possible unique elements, contains only about non-zero elements m(m -I- 2)/8 non-zero elements of the typ)e and mil non-zero elements of the type For m = 20 the matrix d" is only 0.29% non-zero. The inner product CSF-driven approach is clearly not suited for the sparse transition density matrix resulting from this type of wavefunction. The outer product CSF-driven approach does account for the density vector sparseness but the effective vector length is only n, the orbital basis dimension. 

As the dimension of the blocks of the Hessian matrix increases, it becomes more efficient to solve for the wavefunction corrections using iterative methods instead of direct methods. The most useful of these methods require a series of matrix-vector products. Since a square matrix-vector product may be computed in 2N arithmetic operations (where N is the matrix dimension), an iterative solution that requires only a few of these products is more efficient than a direct solution (which requires approximately floating-point operations). The most stable of these methods expand the solution vector in a subspace of trial vectors. During each iteration of this procedure, the dimension of this subspace is increased until some measure of the error indicates that sufficient accuracy has been achieved. Such iterative methods for both linear equations and matrix eigenvalue equations have been discussed in the literature . 

The conceptual simplicity of the CASSCF model lies in the fact that once the inactive and active orbitals are chosen, the wavefunction is completely specified. Such a model also leads to certain simplifications in the computational procedures used to obtain optimal orbitals and configuration-interaction (Cl) coefficients, but that is of less importance in the present context. The major technical difficulty inherent to the CASSCF model is the size of the complete Cl expansion, N(-as- ft given by the so-called Weyl formula, that is, the dimension of the irreducible representation of the unitary group U(n) associated with n active orbitals, N active electrons and a total spin... 

The electronic structure of a nanocrystal critically depends on its very size. For small particles, the electronic energy levels are not continuous as in bulk materials, but discrete, due to the confinement of the electron wavefunction because of the physical dimensions of the particles (see Figure 1.4). The average electronic energy level spacing of successive quantum levels, 6, known as the so-called Kubo gap, is given by... 

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