Coefficients of wave functions

Table 1.10. Coefficients of Wave Functions Calculated for Methyl Cation by the CNDO/2...

Coefficients of wave functions calculated for methyl cation by the CNDO/2 approximation... 

A variation of wave function coefficients is subject to constraints like maintaining orthogonality of the MOs, and normalization of the MOs and the total wave function. 

The scheme we employ uses a Cartesian laboratory system of coordinates which avoids the spurious small kinetic and Coriolis energy terms that arise when center of mass coordinates are used. However, the overall translational and rotational degrees of freedom are still present. The unconstrained coupled dynamics of all participating electrons and atomic nuclei is considered explicitly. The particles move under the influence of the instantaneous forces derived from the Coulombic potentials of the system Hamiltonian and the time-dependent system wave function. The time-dependent variational principle is used to derive the dynamical equations for a given form of time-dependent system wave function. The choice of wave function ansatz and of sets of atomic basis functions are the limiting approximations of the method. Wave function parameters, such as molecular orbital coefficients, z,(f), average nuclear positions and momenta, and Pfe(0, etc., carry the time dependence and serve as the dynamical variables of the method. Therefore, the parameterization of the system wave function is important, and we have found that wave functions expressed as generalized coherent states are particularly useful. A minimal implementation of the method [16,17] employs a wave function of the form ... 

The relationships describing the tensorial properties of wave functions, second-quantization operators and matrix elements in the space of total angular momentum J can readily be obtained by the use of the results of Chapters 14 and 15 with the more or less trivial replacement of the ranks of the tensors l and s by j and the corresponding replacement of various factors and 3nj-coefficients. Therefore, we shall only give a sketch of the uses of the quasispin method for jj coupling, following mainly the works [30, 167, 168]. For a subshell of equivalent electrons, the creation and annihilation operators a and a(jf are the components of the same tensor of rank q = 1/2... 

As in the case of LS coupling, the tensorial properties of wave functions and second-quantization operators in quasispin space enable us to separate, using the Wigner-Eckart theorem, the dependence of the submatrix elements on the number of electrons in the subshell into the Clebsch-Gordan coefficient. If then we use the relation of the submatrix element of the creation operator to the CFP... 

Let us also notice that slow variations of K with Z imply that the gauge condition K may be treated as a semi-empirical parameter in practical calculations to reproduce, with a chosen K, the accurate oscillator strength values for the whole isoelectronic sequence. Thus, dependence of transition quantities on K may serve as the criterion of the accuracy of wave functions used instead of the comparison of two forms of 1-transition operators. In particular, the relative quantities of the coefficients of the equation fEi = aK2 + bK +c (the smaller the a value, the more exact the result), the position of the minimum of the parabola Kf = 0 (the larger the K value for which / = 0, the more exact is the approximation used, in the ideal case / = 0 for K = +oo) may also help to estimate the accuracy of the method utilized. 

As the chemist has gained his sea legs in the use of wave mechanics he has attempted to define covalent and ionic character in terms of wave functions. The new point of view is presented, for example, by Coulson, who maintains, there are two distinct definitions of a covalent bond (447, p. 145). He proceeds to describe the bond wave function, first by a molecular orbital type approximation, and then by a valence bond type approximation. In either case the approximate wave functions consist of linear combinations of atomic orbitals. Partial ionic bonding is revealed by the magnitudes of coefficients which imply asymmetry of electron distribution. 

The particular iterative technique chosen by Car and Parrinello to iteratively solve the electronic structure problem in concert with nuclear motion was simulated annealing [11]. Specifically, variational parameters for the electronic wave function, in addition to nuclear positions, were treated like dynamical variables in a molecular dynamics simulation. When electronic parameters are kept near absolute zero in temperature, they describe the Bom-Oppenheimer electronic wave function. One advantage of the Car-Parrinello procedure is rather subtle. Taking the parameters as dynamical variables leads to robust prediction of values at a new time step from previous values, and cancellation in errors in the value of the nuclear forces. Another advantage is that the procedure, as is generally true of simulated annealing techniques, is equally suited to both linear and non-linear optimization. If desired, both linear coefficients of basis functions and non-linear functional parameters can be optimized, and arbitrary electronic models employed, so long as derivatives with respect to electronic wave function parameters can be calculated. 

For a macroscopic sample it is necessary to define a different set of wave functions because the spins are in a mixed state. The mixed state indicates that the wave functions of a particular nuclear spin are subject to additional molecular contributions that might differ over the whole sample. The expectation value of a mixed state now uses the averaged coefficients and is... 

For the study on the bifurcation into the ten-peak wave or the nine-peak wave, we investigate the temporal variations of the correlation coefficients of these five waves, i.e., the time-resolved correlation coefficients of these waves. For the time-resolved correlation coefficients, the periods of these five waves are plotted as a function of time, i.e., the peak positions of the waves, as shown in Fig. 18. In this figure, the abscissa is divided into eight sections by 10 ms. The time-resolved correlation coefficients are obtained by calculating the correlation coefficients of these curves in Fig, 18 in the individual sections of the abscissa and shown in Fig. 19. In Fig. 19, the time resolved correlation coefficients of waves 2-5 with wave 1 are shown, where the coefficients of the waves are normalized to be 1 at 10 ms, i.e., in the section between 10 and 20 ms. 

Thus, in the CISDt method, we construct wave functions ir) by including all singles and doubles from <>) and a small set of triples containing at least one active occupied and one active unoccupied spin-orbital indices. The Cl expansion coefficients defining wave functions are determined vari-... 

---