Basis sets Gaussian function

The application of density functional theory to isolated, organic molecules is still in relative infancy compared with the use of Hartree-Fock methods. There continues to be a steady stream of publications designed to assess the performance of the various approaches to DFT. As we have discussed there is a plethora of ways in which density functional theory can be implemented with different functional forms for the basis set (Gaussians, Slater type orbitals, or numerical), different expressions for the exchange and correlation contributions within the local density approximation, different expressions for the gradient corrections and different ways to solve the Kohn-Sham equations to achieve self-consistency. This contrasts with the situation for Hartree-Fock calculations, wlrich mostly use one of a series of tried and tested Gaussian basis sets and where there is a substantial body of literature to help choose the most appropriate method for incorporating post-Hartree-Fock methods, should that be desired. 

The examples quoted in Section IV all refer to minimal basis set wave functions composed of Slater-type orbitals (best atom zetas82>) for molecules I-XIV, and of Gaussian orbitals for the others. Wave functions for compounds XV-XXI refer to a (7 s 3p 13 s) basis contracted to [2s 1 p 11 s] proposed by Clementi, Clementi and Davis33) (CD basis), while for compound XXII we have used another Gaussian basis (4 s 2p 3s) contracted to [2s 1 p 12s] proposed by M ly and Pullman (MP basis)34). 

Some applications of perturbation theory to molecular problems would benefit from the simplicity of an extended floating spherical Gaussian basis. Adamowicz and Bartlett ° have developed a procedure for projecting large conventional basis set wave functions onto a floating spherical Gaussian basis. 

Five different basis sets were used for the calculation at both the RHF and correlated levels, as shown in Tables 5.1 and 5.2, where the notation msins means that on each hydrogen atom m 5-type Gaussians are centered and are divided into n groups (contractions). The exponents and contraction coefficients of the first two sets were taken from the literature, and the exponents of the six 5-type Gaussians from another work.< > In the fourth basis set these functions were supplemented by two 5-type bond functions (2 ) centered in the middle between the hydrogen atoms, while in the fifth basis a set of three p-type atomic polarization functions p, Py, Pz) was used. The contraction coefficients and the exponents of the bond and polarization functions were optimized previously. 

Molecular orbital theories (ab initio methods (23)) were chosen and validated in this study to characterize the interaction energies between methane and water, while electron density functional theories (24) were tested, but found to be inadequate (see below). Four different ab initio methods were used in the validation MP2 (25,26), MP4(SDTQ) (27), QCISD(T) (28) and CCSD(T) (29). Three different DFT methods, BLYP (30Jl), B3LYP (32) and BPW91 (33), were used and the results compared with the ab initio methods. In addition, for each of the above methods, the effect of the size of different basis sets was investigated specifically, 6-31++G(2d,2p), cc-pVDZ, cc-pVTZ and cc-pVQZ were used. 6-31+-i-G(2d,2p) was chosen, because it was reported to yield reasonable results compared with that at near the basis set limit on this system (77). The others were chosen in order to observe the effect of systematically increasing the size of the basis set. Gaussian 94 (22) was used for all the above calculations. 

Flehre W J, Ditchfieid R and Popie J A 1972 Self-consistent molecular-orbital methods XII. Further extension of Gaussian-type basis sets for use in molecular orbital studies of organic molecules J. Chem. Phys. 56 2257-61 Flariharan P C and Popie J A 1973 The influence of polarization functions on molecular orbital hydrogenation energies Theoret. Chim. Acta. 28 213-22... 

An alternative to using a superposition of Gaussian functions is to extend the basis set by using Hermite polynomials, that is, hamonic oscillator functions [24]. This provides an orthonormal, in principle complete, basis set along the bajectoiy, and the idea has been taken up by Billing [151,152]. The basic problem with this approach is the slow convergence of the basis set. 

To deal with the problem of using a superposition of functions, Heller also tried using Gaussian wave packets with a fixed width as a time-dependent basis set for the representation of the evolving nuclear wave function [23]. Each frozen Gaussian function evolves under classical equations of motion, and the phase is provided by the classical action along the path... 

The picture here is of uncoupled Gaussian functions roaming over the PES, driven by classical mechanics. The coefficients then add the quantum mechanics, building up the nuclear wavepacket from the Gaussian basis set. This makes the treatment of non-adiabatic effects simple, as the coefficients are driven by the Hamiltonian matrices, and these elements couple basis functions on different surfaces, allowing hansfer of population between the states. As a variational principle was used to derive these equations, the coefficients describe the time dependence of the wavepacket as accurately as possible using the given... 

As usual there is the question of the initial conditions. In general, more than one frozen Gaussian function will be required in the initial set. In keeping with the frozen Gaussian approximation, these basis functions can be chosen by selecting the Gaussian momenta and positions from a Wigner, or other appropriate phase space, distribution. The initial expansion coefficients are then defined by the equation... 

In modem quantum chemistry packages, one can obtain moleculai basis set at the optimized geometry, in which the wave functions of the molecular basis are expanded in terms of a set of orthogonal Gaussian basis set. Therefore, we need to derive efficient fomiulas for calculating the above-mentioned matrix elements, between Gaussian functions of the first and second derivatives of the Coulomb potential ternis, especially the second derivative term that is not available in quantum chemistry packages. Section TV is devoted to the evaluation of these matrix elements. 

To calculate the matrix elements for H2 in the minimal basis set, we approximate the Slater Is orbital with a Gaussian function. That is, we replace the Is radial wave function... 

The solution to this problem is to use more than one basis function of each type some of them compact and others diffuse, Linear combinations of basis Functions of the same type can then produce MOs with spatial extents between the limits set by the most compact and the most diffuse basis functions. Such basis sets arc known as double is the usual symbol for the exponent of the basis function, which determines its spatial extent) if all orbitals arc split into two components, or split ualence if only the valence orbitals arc split. A typical early split valence basis set was known as 6-31G 124], This nomenclature means that the core (non-valence) orbitals are represented by six Gaussian functions and the valence AOs by two sets of three (compact) and one (more diffuse) Gaussian functions. 

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