Expansion functions

The equations of state and expansion functions for the perturbation theories are found in paper of Fickett... 

Consider now the field scattered by an isotropic, optically active sphere of radius a, which is embedded in a nonactive medium with wave number k and illuminated by an x-polarized wave. Most of the groundwork for the solution to this problem has been laid in Chapter 4, where the expansions (4.37) and (4.38) of the incident electric and magnetic fields are given. Equation (8.11) requires that the expansion functions for Q be of the form M N therefore, the vector spherical harmonics expansions of the fields inside the sphere are... 

In Section 2.1 we derived the expression for the transition rate kfi (2.22) by expanding the time-dependent wavefunction P(t) in terms of orthogonal and complete stationary wavefunctions Fa [see Equation (2.9)]. For bound-free transitions we proceed in the same way with the exception that the expansion functions for the nuclear part of the total wavefunction are continuum rather than bound-state wavefunctions. The definition and construction of the continuum basis belongs to the field of scattering theory (Wu and Ohmura 1962 Taylor 1972). In the following we present a short summary specialized to the linear triatomic molecule. 

Equation (3.5) is a set of ordinary differential equations of second order for each radial wavefunction Xri(-R) the different expansion functions Xn (R> Ef>n) are coupled to all other functions by the real and symmetric potential matrix V. 

The vibrational expansion functions [Pg.54]

Using the orthogonality of the spherical harmonics the radial expansion functions are given by... 

Within the close-coupling approach each partial photodissociation wavefunction (R,r Ef,n) is represented by the expansion functions Xn (R >Ef,n) and the vibrational basis functions n(r) with n and n = 0,1,2,..., n. Here, n denotes the highest state considered in expansion (3.4). It is not necessarily identical with nmox, the highest state that can be populated for a given total energy. In order to simplify the subsequent notation we consider the total of the radial functions as the elements Xn n R i Ef) of a (n + 1) x (n + 1) matrix... 

The Fourier method is best suited to cartesian coordinates because the expansion functions QtkR/LR etlr Lr are just the eigenfunctions of the kinetic energy operator. For problems including the rotational degree of freedom other propagation methods have been developed (Mowrey, Sun, and Kouri 1989 Le Quere and Leforestier 1990 Dateo, Engel, Almeida, and Metiu 1991 Dateo and Metiu 1991). 

The angular expansion functions have the parity (—1 )Jp under the inversion operation the parity parameter p has the values p = 1. 

Inserting (11.6) into the time-independent Schrodinger equation and utilizing the orthonormality of the expansion functions leads to the following set of coupled equations for the radial functions Xjn(R JMp)J... 

Edmonds (1974). The radial expansion functions Xj Ci1 R) fulfill coupled... 

The coefficients Blk are related to the components of the transition dipole moment in the molecule-fixed system and the 0 are the angular expansion functions defined in (11.5). The dipole moment transforms like a tensor of rank 1 which explains why it is expanded in terms of the angular functions for an angular momentum J = 1. Since its projection on the space-fixed z-axis is independent of the azimuthal angle pn, only functions with M = 0 are allowed. Furthermore, the dipole moment has the parity —1 so that the parameter p is restricted to +1 [remember that the parity is given by (—lj p]. 

Table 1 Kinetic energy values (in hartrees) for the beryllium atom obtained in the context of LS-DFT, with the several conjoint gradient expansion functionals discussed in the text. For comparison purposes, the Hartree-Fock value [40] is included.. Tjy[p] and 7Vw[p] are the Weizsacker and non-Weizsacker components, respectively. TjJwIp] and Nwipl are the positive and negative components of the non-Weizsacker term.

The eigenvalue problem (17] of the sth power of the evolution operator U is obtained from Eq. (38) for/(U) = Us. In diagonalizations, one does not necessarily need the explicit knowledge of the operator U itself, but only its matrix elements (fm U f ) are requested. Here, f ) is a suitably selected complete set of the expansion functions fK) that form a basis, which does not need to be orthonormalized. For example, the collection of the Schrodinger state vectors represents one such basis. The two different states ) and... 

A possible computation of three and four center integrals rely upon the development of the exponential product covering by means of chosen WO-CETO basis set linear combinations. The integrals and techniques needed to optimize the expansion functions will be described here. 

A minimum basis set (MBS) includes one expansion function for each orbital occupied in the ground states of the free atoms. For example, a minimum-basis-set calculation on diatomic MgO would employ Is, Is, 3s,... 

For molecules the optimization of exponents is very important for small basis sets but it can never alone absorb the deficiency due to a lack of expansion functions. The number and secondly the kind of STO basis set functions (i.e. s, p, d,. .. type) are the most important considerations. The decreasing importance of the exponent optimization as the basis set grows is observable from Table 2.4,... 

If these can be tested directly, even inspired empirical estimates of correlated wave-functions may become a valuable subject. At a more realistic level, however, the resolution of instabilities in predictions of excited states the assessment of the claims of different systematic calculations and the assessment of rates of convergence with different systems of expansion functions will probably form the major use of the direct assessment of the accuracy of correlated wavefunctions. 

The radial expansion functions are constituted of two groups in the first group Rfi are the spherical atomic densities obtained from the radial functions R of the basis, and in the second group Rn are radial functions localized in the range r < t, < r/v+i, chosen to be piecewise parabolic. In general, only a small number of the latter functions are necessary for each (/, fi) partner in the fully symmetric representation to converge the potential. 

It is generally helpful to build into the expansion functions ( /) the symmetry properties of the system. According to the antisymmetry principle,... 

---