Gaussian wavefunctions

A few correlated Gaussian calculations have been carried out on systems with more than four particles. On example is the recent work of Komasa et al. [58] on beryllium isoelectronic ions. 

---

Some of these ideas were recently used to calculate the isotope effect for the photodissociation of N2O and O3 by Liang et al. [15]. They predict the isotope effect based on a numerical analysis in which they describe the Abs. XS divided by E (where E is the photon energy, see below) as the product of a Gaussian and a numerical Y function assumed to be the same for all isotopologues. The isotope effect is taken into account in the 3 parameters of the Gaussian wavefunction and then reported on the Abs. XS of each isotopologue. From this the related fractionation constant can be determined. 

The POLYATOM project was not the only one of its type. Indeed, in the same issue of Rev. Mod. Phys. as Barnett s paper, R. K. Nesbet of IBM Cooporation s San Jose Laboratory describes Computer Programmes for Electronic Wave Function calculations and F. E. Harris discusses Gaussian Wavefunctions for Polyatomic Molecules . 

With the force constants given by (3.21), the correlated gaussian wavefunction (3.18) is the exact solution of the harmonic Hamiltonian,... 

A trial (gaussian) wavefunction for the ground state of the hydrogen atom is... 

The square-root term is just the angular frequency obtained from a classical analysis of the vibration of two masses joined by a spring (Equation (A6.11)). So we have shown that the energy of the proposed Gaussian wavefunction is simply... 

The general Gaussian wavefunction for a N-dimensional system at time t with spatial coordinates q is... 

If all the resonance states which fomi a microcanonical ensemble have random i, and are thus intrinsically unassignable, a situation arises which is caWtA. statistical state-specific behaviour [95]. Since the wavefunction coefficients of the i / are Gaussian random variables when projected onto (]). basis fiinctions for any zero-order representation [96], the distribution of the state-specific rate constants will be as statistical as possible. If these within the energy interval E E+ AE fomi a conthuious distribution, Levine [97] has argued that the probability of a particular k is given by the Porter-Thomas [98] distribution... 

Even expression ( B3.4.31), altiiough numerically preferable, is not the end of the story as it does not fiilly account for the fact diat nearby classical trajectories (those with similar initial conditions) should be averaged over. One simple methodology for that averaging has been tln-ough the division of phase space into parts, each of which is covered by a set of Gaussians [159, 160]. This is done by recasting the initial wavefunction as... 

To remedy this diflSculty, several approaches have been developed. In some metliods, the phase of the wavefunction is specified after hopping [178]. In other approaches, one expands the nuclear wavefunction in temis of a limited number of basis-set fiinctions and works out the quantum dynamical probability for jumping. For example, the quantum dynamical basis fiinctions could be a set of Gaussian wavepackets which move forward in time [147]. This approach is very powerfLil for short and intemiediate time processes, where the number of required Gaussians is not too large. 

The total wavefunction r2,. . ., r is written as a product of single-particle functions (Hartree approximation). The various integrals are evaluated in tire saddle point approximation. A simple Gaussian fomr for tire trial one-particle wavefunction... 

We have extended the linear combination of Gaussian-type orbitals local-density functional approach to calculate the total energies and electronic structures of helical chain polymers[35]. This method was originally developed for molecular systems[36-40], and extended to two-dimensionally periodic sys-tems[41,42] and chain polymers[34j. The one-electron wavefunctions here are constructed from a linear combination of Bloch functions c>>, which are in turn constructed from a linear combination of nuclear-centered Gaussian-type orbitals Xylr) (in ihis case, products of Gaussians and the real solid spherical harmonics). The one-electron density matrix is given by... 

A computer program for the theoretical determination of electric polarizabilities and hyperpolarizabilitieshas been implemented at the ab initio level using a computational scheme based on CHF perturbation theory [7-11]. Zero-order SCF, and first-and second-order CHF equations are solved to obtain the corresponding perturbed wavefunctions and density matrices, exploiting the entire molecular symmetry to reduce the number of matrix element which are to be stored in, and processed by, computer. Then a /j, and iap-iS tensors are evaluated. This method has been applied to evaluate the second hyperpolarizability of benzene using extended basis sets of Gaussian functions, see Sec. VI. 

Wachters, A. J. H., 1970, Gaussian Basis Set for Molecular Wavefunctions Containing Third-Row Atoms , J. Chem. Phys., 52, 1033. 

Besides the Gaussian basis set for the wavefunction, an additional set of nuclear centered Gaussians, gk, can be used for expanding the electronic density [25], which can be written as... 

---