Terms Wave functions

The values for the dipoles, polarizabilities, and hyperpolarizabilities of the H2 series were obtained using (a) a 16-term basis with a fourfold symmetry projection for the homonuclear species and (b) a 32-term basis with a twofold symmetry projection for the heteronuclear species. These different expansion lengths were used so that when combined with the symmetry projections the resulting wave functions were of about the same quality, and the properties calculated would be comparable. A crude analysis shows that basis set size for an n particle system must scale as k", where k is a constant. In our previous work [64, 65] we used a 244-term wave function for the five-internal-particle system LiH to obtain experimental quality results. This gives a value of... 

The results for the non-BO diagonal polarizability are shown in Table XIII. Our best—and, as it seems, well-converged—value of a, 29.57 a.u., calculated with a 244-term wave function, is slightly larger than the previously obtained corrected electronic values, 28.93 and 28.26 a.u. [88,91]. It is believed that the non-BO correction to the polarizability will be positive and on the order of less than 1 a.u. [92], but it is not possible to say if the difference between the value obtained in this work and the previous values for polarizability are due to this effect or to other effects, such as the basis set incompleteness in the BO calculations. An effective way of testing this would be to perform BO calculations of the electronic and vibrational components of polarizability using an extended, well-optimized set of explicitly correlated Gaussian functions. This type of calculation is outside of our current research interests and is quite expensive. It may become a possibility in the future. As such, we would like the polarizability value of 29.57 a.u. obtained in this work to serve as a standard for non-BO polarizability of LiH. 

Expectation Values of the Hamiltonian, (//), Virial Coefficients, r, and Squared Gradient Norms, grad p. For the Ground State of (R1 = R2 = R3 = 1.65) for the 75-Term Wave Function... 

We tested a 76-term wave function for the system H3, including permutational and point group symmetry. The initial guess for the nonlinear parameters in the ECGs were generated randomly using Matlab. The Young... 

The energy obtained after optimization for the 76-term function was found to be lower than the energy obtained by Cencek and Rychlewski [49] for the 100-term wave function, illustrating the power of including correlation among aU the electrons in each basis term (see Table XV). 

Atomic orbitals are actually graphical representations for mathematical solutions to the Schrodinger wave equation. The equation provides not one, but a series of solutions termed wave functions t[ . The square of the wave function, is proportional to the electron density and thus provides us with the probability of finding an electron within a given space. Calculations have allowed us to appreciate the shape of atomic orbitals for the simplest atom, i.e. hydrogen, and we make the assumption that these shapes also apply for the heavier atoms, like carbon. 

Pj, is a projection operator ensuring the proper spatial symmetry of the function. The above method is general and can be applied to any molecule. In practical application this method requires an optimisation of a huge number of nonlinear parameters. For two-electron molecule, for example, there are 5 parameters per basis function which means as many as 5000 nonlinear parameters to be optimised for 1000 term wave function. In the case of three and four-electron molecules each basis function contains 9 and 14 nonlinear parameters respectively (all possible correlation pairs considered). The process of optimisation of nonlinear parameters is very time consuming and it is a bottle neck of the method. 

In the previous section we examined the variational result of the two-term wave function consisting of the covalent and ionic functions. This produces a 2 x 2 Hamiltonian, which may be decomposed into kinetic energy, nuclear attraction, and electron repulsion terms. Each of these operators produces a 2 x 2 matrix. Along with the overlap matrix these are... 

Figure 8.1 shows the expectation values of the electric moment for the Li+—H and covalent structures. The graph gives the negative of the moment for Li —H+ for easy comparison. In addition, the moment for the three-term wave function involving all three of the other functions is given. Although such a simple wave... 

Figure 8. 2 shows the coefficients of the three structures in the total three-term wave function. As expected, the covalent term predominates at all distances, but... 

Figure 8.2. The coefficients of the three structures in the simple three-term wave function.

HFC. The carbon orbitals are formed into the standard tetrahedral hybrids, pointing at the H atoms. There are 14 covalent basis functions and the this row gives the relative energy for the 14 term wave function. 

Mukherjee et al. [113] used a 55-term Cl wave function to compute energies for the He atom, whereas Saha et al. [114] used a 75-term wave function that contain correlation factors. They also performed calculations on ground state energies for He to Ar16+ for a variety of values of /x from zero (no screening) to an upper limit in which the system becomes unstable. 

Further evidence for the smallness of in the exact % of He comes from a better trial function. Hylleraas six-term wave function (98% of the correlation energy) gives much less fi (Fig. 3). This is apparently a real effect, since when it is taken out the energy is raised by 0.24% of Ecorr-. which, however, is negligibly small (63 cal/mole). Similar results are found > for the Hj molecule near the equilibrium distance r. (For Hj near dissociation see Section XXL )... 

A calculated from the scaled Lowdin and Redei function ( q. 91) in text. —(1)—) /j from Hylleraas six-term wave function (see Section XIX). 

Including such terms Frankowski and Pekeris [26] needed only a 246 term wave function to be more accurate that Pekeris in his original work with more than 1000 terms. The performance was even superseded by Morgan and coworkers [27] who obtained the non-relativistic He ground state energy with 13-figure accuracy. One may celebrate this as an example where a careful study of properties of the exact wave function can inspire an improved... 

Note the language subtleties At least within the field of ab initio quantum chemistry, the term wave function is almost exclusively used for a many-electron wave function T. A one-electron wave function for atoms and molecules, however, goes by the name (atomic or molecular) orbital ip, not wave function. The exception proves the rule, though. 

The numbers (15.7) when arranged in an n x n square array form a matrix (Section 7.10). If there are h symmetry operations in the molecular point group, then the h matrices of coefficients in (15.5) constitute the symmetry species of the degenerate-term wave functions in (15.5). The following letter labels are used for the symmetry species, according to the orbital degeneracy n ... 

Why then don t we just talk about high level theoretical calculations and ignore the simple theories The elementary theories are useful to us because they provide good conceptual models for the computational process. We can visualize the interactions represented by equation 1.34, as well as the physical situation suggested by equation 1.31. It is much more difficult for us to envision the interactions involved in a 50- or 100-term wave function, however. As the accuracy of the model is increased, its simplicity is decreased. 

First, there were the molecules or atomic systems of one to six electrons, for which one could effectively calculate energies as accurately as they can be measured. Second, the all too realistic prospects for faster computers allowed one to extend the range of molecules for which it would become possible to have effectively exact solutions to those with 6 to 20 electrons. Nevertheless, accurate results for these cases were achieved at the expense of visualizability even in the five-term James Coolidge function— which Coulson believed to be "the best compromise between accuracy and simplicity"— there was nothing easily visualized about the wave function, and it required a further numerical integration on an electronic computer to derive from the full 13-term wave function the electronic charge density of the electron. 

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