Coefficients of wave functions calculation

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... 

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. 

Wave functions calculated for the lowest-energy levels of linear Agw are shown in Fig. 9. The coefficient of the 5s orbital for each atom is plotted versus atom number along the chain. It is apparent that the electrons in the lowest-energy level are spread almost uniformly throughout the cluster. As energy increases, the uniformity decreases since nodes appear in the wave function. The uniform spreading of electron density would be predicted by the free electron model. 

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 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. 

The END equations are integrated to yield the time evolution of the wave function parameters for reactive processes from an initial state of the system. The solution is propagated until such a time that the system has clearly reached the final products. Then, the evolved state vector may be projected against a number of different possible final product states to yield coiresponding transition probability amplitudes. Details of the END dynamics can be depicted and cross-section cross-sections and rate coefficients calculated. 

The gradient of the PES (force) can in principle be calculated by finite difference methods. This is, however, extremely inefficient, requiring many evaluations of the wave function. Gradient methods in quantum chemistiy are fortunately now very advanced, and analytic gradients are available for a wide variety of ab initio methods [123-127]. Note that if the wave function depends on a set of parameters X], for example, the expansion coefficients of the basis functions used to build the orbitals in molecular orbital (MO) theory. 

This is perhaps the easiest method to understand. It is based on the variational principle (Appendix B), analogous to the HF method. The trial wave function is written as a linear combination of determinants with the expansion coefficients determined by requiring that the energy should be a minimum (or at least stationary), a procedure known as Configuration Interaction (Cl). The MOs used for building the excited Slater determinants are taken from a Hartree-Fock calculation and held fixed. Subscripts S, D, T etc. indicate determinants which are singly, doubly, triply etc. excited relative to the... 

The expansion coefficients determine the first-order correction to the perturbed wave function (eq. (4.35)), and they can be calculated for the known unperturbed wave functions and energies. The coefficient in front of 4>o for 4 i cannot be determined from the above formula, but the assumption of intermediate normalization (eq. (4.30)) makes Co = 0. 

Note that no new operators are involved, only derivatives of the CI or HF wave function with respect to the MO coefficients. The matrix elements can thus be calculated from the same integrals as the energy itself, as discussed in Sections 3.5 and 4.2.1. 

Although a calculation of the wave function response can be avoided for the first derivative, it is necessary for second (and higher) derivatives. Eq. (10.29) gives directly an equation for determining the (first-order) response, which is structurally the same as eq. (10.36). For an HF wave function, an equation of the change in the MO coefficients may also be formulated from the Hartree-Fock equation, eq. (3.50). 

This equation is easily used for fitting. It contains calculated host wave function coefficients and, up to / = 2, it depends on just three fitting parameters rji, composed of system parameters as follows... 

A greater gain in accuracy in connection with the temperature wave depends significantly on how well we calculate the coefficients a (v). In the case where k = k u is a power function of temperature, numerical experiments showed that formula (38) is useless and formula (36) is much more flexible than (37), so there is some reason to be concerned about this. Further comparison of schemes (34) and (35) should cause some difficulties. Both schemes are absolutely stable and have the same error of approximation 0 r + h ). The scheme a) is linear with respect to the value of the function on the layer and so the value y7+i on every new layer... 

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